Path: utzoo!attcan!uunet!mcsun!ukc!mucs!mccuts!zlsiial
From: zlsiial@uts.mcc.ac.uk (A.V. Le Blanc)
Newsgroups: comp.ai.philosophy
Subject: Re: Consistency theorems
Message-ID: <2398@mccuts.uts.mcc.ac.uk>
Date: 1 Mar 91 14:57:56 GMT
References: <16462.9102272325@s4.sys.uea.ac.uk> <1991Feb28.172555.12897@sics.se>
Reply-To: zlsiial@cms.mcc.ac.uk (A.V. Le Blanc)
Organization: Computing Centre and Philosophy Dept, University of Manchester, UK
Lines: 19

It is not necessarily the case that all proofs of consistency of the
type you mention are trivial.  There is an interesting case of
three theories A, B, and C, in which B and C contain all of the
rules and axioms of A, C contains all of the rules and axioms of B, 
B contains rules and axioms which are not valid in A, and C contains
axioms which are not valid in B.  With the addition of a further axiom
-- essentially a highly restrictive assumption about the domain of
discourse, which is nevertheless consistent with the axioms and rules
of A, B, and C, it is possible to construct a model of C, and hence of
B, in the original A.  Hence the consistency of B and of C is
demonstrated relative to the much weaker system A.

An article discussing this proof appeared in the Journal of
Symbolic Logic, perhaps about 1967? entitled
`The Consistency of Mereology' by Prof. Czeslaw Lejewski.
The proof dates from about 1930.

				A. V. Le Blanc
				ZLSIIAL@uk.ac.mcc.cms