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Path: lukasiewicz.cc.nd.edu!scharle
Newsgroups: sci.logic
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References: <1992Apr5.213320.27618@morrow.stanford.edu> <1992Apr7.124948.5325@vax.oxford.ac.uk> <1992Apr8.100146.689@gacvx2.gac.edu>
From: scharle@lukasiewicz.cc.nd.edu (Scharle)
Reply-To: scharle@lukasiewicz.cc.nd.edu (Scharle)
Organization: Univ. of Notre Dame
Subject: Re: Set theory and modal logic
Keywords: 

In article <1992Apr8.100146.689@gacvx2.gac.edu>, logic@gacvx2.gac.edu writes:
|> In article <1992Apr7.124948.5325@vax.oxford.ac.uk>, loader@vax.oxford.ac.uk (R Loader) writes:
|> 
|> > S4 and S5 (and most other common modal logics), are subsystems of classical
|> > (non-modal) logic extended with the axiom: A iff \box A.
|> 
|> Other stuff deleted....
|> 
|> > Ralph Loader,
|> > Mathematical Institute, Oxford.
|> 
|> S4 and S5 are supersystems, not subsystems, of classical logic..  Neither
|> contain the axiom "A iff \box A", both contain the axiom "if \box A then A".
|> 
|> Al Hart
    There is a problem with the terms "supersystem" and "subsystem".  
S4 and S5 are both subsystems and supersystems of classical logic, 
depending on the map one uses.
    Every statement true in classical logic is true in S4 or S5.
Hence classical logic is a subsystem.
    Every statement true in S4 or S5 is true in classical logic,
provided we interpret the box as the identity operator.  That, I
believe, is what R. Loader was saying.  Hence classical logic is a
supersystem.

-- 
Tom Scharle                |cm65n6@irishmvs(Bitnet)
Room G003 Computing Center |scharle@lukasiewicz.cc.nd.edu(Internet)
University of Notre Dame  Notre Dame, IN 46556-0539 USA



-- 
Tom Scharle                |cm65n6@irishmvs(Bitnet)
Room G003 Computing Center |scharle@lukasiewicz.cc.nd.edu(Internet)
University of Notre Dame  Notre Dame, IN 46556-0539 USA