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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.

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"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
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