Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!mnetor!seismo!mcnc!duke!mps From: mps@duke.cs.duke.edu (Michael P. Smith) Newsgroups: sci.philosophy.tech Subject: Re: The nature of knowledge Message-ID: <9882@duke.cs.duke.edu> Date: Wed, 8-Jul-87 00:21:35 EDT Article-I.D.: duke.9882 Posted: Wed Jul 8 00:21:35 1987 Date-Received: Sat, 11-Jul-87 00:56:21 EDT References: <3587e521.44e6@apollo.uucp> <680@gargoyle.UChicago.EDU> <1022@water.UUCP> <51@thirdi.UUCP> <121@cavell.UUCP> Reply-To: mps@duke.UUCP (Michael P. Smith) Distribution: world Organization: Duke University, Durham NC Lines: 54 Keywords: belief, truth, omega-inconsistency Summary: Belief systems 'omega-inconsistent' In article <121@cavell.UUCP> jiml@cavell.UUCP (Jim Laycock) writes: > ... It seems to me that >not all of my beliefs reflect true propositions (unless I'm incredibly >skilled in choosing what to believe). Nonetheless, it is not true of >any particular belief p that I consider it to be false, otherwise I would >reject it and believe ~p. Let us also assume that I have a finite number >of beliefs. > Consider a much smaller scenario--one in which I have but three beliefs: > > 1. Bel(p) > 2. Bel(q) > 3. Bel(~(p^q)) > >Surely if such a situation were to come about, you'd have no trouble >considering me to be inconsistent. Yet my proposal is that we all >entertain a much greater version of precisely the same notion. Are >we inconsistent, or just unreflective (are certain beliefs not questioned)? >Is this to deny > > 4. Bel(Bel(p)) > >for some p? >-- > Jim Laycock Philosophy grad, University of Alberta > alberta!Jim_Laycock@UQV-MTS > OR > decvax!bellcore!ulysses!mhuxr!mhuxn!ihnp4!alberta!cavell!jiml A closer analgoue might be *omega-inconsistency*. A formal system is omega-inconsistent when for some open formula Px, its existential closure (Ex)Px as well as the denial of each instance, ~Pa, ~Pb, etc., are provable. No restriction to finitude required. Godel's first incompleteness theorem as originally proven applied to omega- consistent systems. Take a universe of propositions, let P stand for 'is false', and substitute 'believed (by me)' for 'proveable'. Since we have substituted a fuzzy notion for a precise one (in proof theory), however, we only have an analogy here. Doxastically, we are *like* omega-inconsistent systems. I don't see the relation to B --> BB, though. OK, so I believe that some of my beliefs are false. Suppose, *per impossibile*, that I have particular beliefs I believe to be false. Still, by hypothesis I have these beliefs, and I see nothing to prevent my believing that I have them. ---------------------------------------------------------------------------- "The truth seems to be like the proverbial [side of a barn] which no one can fail to hit, ... but the fact that we can have a whole truth and not the particular part we aim at shows the difficulty of it." Aristotle Michael P. Smith ARPA mps@duke.cs.duke.edu ----------------------------------------------------------------------------