stuart@gargoyle.UChicago.UUCP (Stuart Kurtz) (09/07/84)
Wayne Christopher (faustus@ucbvax) suggests that the notion of an omnipotent being is logically inconsistant, pointing out the apparent paradox "Can God create a rock too heavy for Him to lift?" A few months ago, a suggested a simple resolution to this paradox -- Yes, because an omnipotent being should, being all powerful, be able to limit his own power, thereby ceasing to be omnipotent. The notion of omnipotence is certainly fraught with logical difficulties, and some care must go into deciding what omnipotent means. For example, should an omnipotent entity be able to enforce a logical contradiction, e.g. "0 = 1"? What would it mean to have such a contradiction enforced? Stu
hawk@oliven.UUCP (09/12/84)
~For example, should an omnipotent entity be able to enforce a ~logical contradiction, e.g. "0 = 1"? What would it mean to have ~such a contradiction enforced? It simply wouldn't be contradictory. -- rick (Rick Hawkins @ Olivetti ATC) [hplabs|zehntel|fortune|ios|tolerant|allegra|tymix]!oliveb!oliven!hawk
steiny@scc.UUCP (Don Steiny) (09/16/84)
*** The sentence: This sentence is false. is considered logically meaningless because of a confusion of logical levels. The sentence is on the "object level". Whether or not the sentence is true is a meta-level, because is is talking about the sentence. Saying that the universe was created is a statement about the universe. If the creator is one of the things in the universe it is the same level confusion. Likewise with "omnipotance". If we say X is omnipotant and if X makes the statement I am omnipotant then we have the Epimenidies paradox. Emimenidies was a Createn who said "All Createns are liars". The "paradox" is not really a paradox because when a lower logical level makes statements about a higher logical level the statments are considered logically meaningless. They are neither true nor false, but ill-formed and undefined. -- "Damn, more bats ... Someone get me a ruler!" Don Steiny - Personetics @ (408) 425-0382 109 Torrey Pine Terr. Santa Cruz, Calif. 95060 ihnp4!pesnta -\ fortune!idsvax -> scc!steiny ucbvax!twg -/
stuart@gargoyle.UChicago.UUCP (Stuart Kurtz) (09/19/84)
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
mark@uf-csg.UUCP (mark fishman [fac]) (10/11/84)
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Seems curious that no one on the SIG's ever been exposed to the most
primitive rudiments of mathematical
logical -- which is to day, specifically, type theory. It's an altogether
trivial, primitive and not very esoteric observation that the question, "Can an
all-powerful being create a rock HE/SHE/IT can't lift" is semantically
ill-formed in the worst sense, and demonstrates only the cognitive deficiencies
of the poser. Look: The modal form "able to" can apply only to sentences that
don't include negations of itself. Russell recognized this umpteen years ago
in the Principia. You'd think that somebody else might have heard of this only
seminal result in the history of twentieth-sentury logic. But then again,
maybe not in a user population capable of taking seriously questions of
magicness and fanciful superstition, in the first place.
I really believe that anybody is entitled to nurture any exotic delusions
he or she likes, provided they don't lead to his or her acting to crush and
maim other people (religion too often conduces to this), but why pretend
rationality is upward compatible with the silly, atmospheric fuzziness?
....innocuously yours..
----------
The opinions herein expressed are irrelevant, inflammatory and possibly
fattening. They do not represent those of the University of Florida or of any
known biological organism or mythical construct.mark@uf-csg.UUCP (mark fishman [fac]) (10/11/84)
<>
<>
Seems curious that no one on the SIG's ever been exposed to the most
primitive rudiments of mathematical
logic -- which is to say, type theory. It's an altogether trivial, primitive
and not very esoteric observation that the question, "Can an
all-powerful being create a rock HE/SHE/IT can't lift?" is semantically
ill-formed in the worst sense, and demonstrates only the cognitive
deficiencies of the poser. Look: The modal form "able to" can apply only to
sentences that don't end up including, thereby, explicit negations of the
resultant proposition. Russell recognized this umpteen years ago in the
Principia. You'd think that somebody else might have heard of this *merely
seminal* result in the history of twentieth-sentury logic. But then again,
maybe not in a user population capable of taking seriously questions of
magicness and fanciful superstition, in the first place.
I really believe that anybody is entitled to nurture any exotic
delusions he or she likes, provided they don't lead to his or her acting to
crush and
maim other people (religion too often conduces to this), but why pretend
rationality is upward compatible with the silly, atmospheric fuzziness?
....innocuously yours..
----------
The opinions herein expressed are irrelevant, inflammatory and possibly
fattening. They do not represent those of the University of Florida or of
any
known biological organism or mythical construct.