Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site gargoyle.UChicago.UUCP Path: utzoo!watmath!clyde!burl!ulysses!mhuxj!ihnp4!gargoyle!stuart 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 References: <213@laidbak.UUCP> <1804@ucbvax.ARPA> <192@gargoyle.UChicago.UUCP> <149@scc.UUCP> Organization: U. Chicago - Computer Science Lines: 39 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