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Re: [ontolog-forum] Is modal logic first-order?

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Tue, 27 Feb 2007 13:24:23 -0600
Message-id: <C58BFD51-8D47-46EB-96A5-6C1A153F3155@xxxxxxxx>
> MG> Furthermore, this modal logic [G] is not first-order
>> definable (since the additional axiom requires that the
>> accessibility is well-founded).
> That's fine, if you're concerned about non-well-founded
> sets.    (01)

Michael's point has nothing whatever (inherently, anyway) to do with  
non-well-founded sets, John.   A Kripke structure, as you know,  
consists of a pair <W,R> such that W is nonempty and R is a binary  
relation over W; all Michael is saying is that Loeb's axiom is valid  
in a Kripke structure <W,R> only if R is well-founded, i.e., there  
are no infinitely descending R-chains, otherwise put, for every  
"proposition" over W (i.e., subset of W) there is an R-least world in  
which it is true.  This, again, has nothing at all inherently to do  
with well-founded sets -- though in fact it turns out to be quite  
fruitful to think of Kripke structures as certain sets of non-wf sets  
(where R is the membership relation on those sets), thanks notably to  
lovely work by Larry Moss and the late great Jon Barwise.    (02)

> CM> I am quite certain that your claim about the number
>> of papers published about S5 is not even close to true,
>> given vast literature on provability logics, temporal
>> logics, logics of belief, and many other applications
>> of modal logic where S5 is inappropriate.
> Yes, I was reacting to the significant number of papers
> that try to apply S5 to databases and ontologies.   The
> database field, in particular, has practitioners who
> have inflicted many short-sighted disasters on us and
> theoreticians who publish papers that are irrelevant
> to anything that could ever be used in practice.
> CM> Yes, of course, because S5 is meant to reflect a
>> context where "law" means something like "logical
>> necessity".  It's perfectly appropriate for laws in
>> that sense to be constant across all possible worlds.
> Unfortunately, S5 has been applied in fields where the
> "laws" are domain-dependent constraints -- and that
> makes the applications impossible to generalize.
> CM> Which, if true, simply goes to show that S5 is not
>> the right modal logic for modeling DBs.
> Or knowledge bases, or ontologies, or....    (03)

I strongly disagree with you there.  The relevant necessity in many  
ontological contexts is logical necessity, in which case S5 seems  
perfectly appropriate.  It even seems appropriate for physical  
necessity, frankly, i.e., where "[]p" is true at a world w just in  
case p is true in all worlds that share w's natural laws.    (04)

> What applications can S5 support other than publishing
> papers in the Journal of Tenure and Promotion?    (05)

Please, John, that's ridiculous.  For one thing, if you were to look,  
you'd find that, even in philosophy, a huge amount of work has been  
done on logics other than S5, notably, deontic and temporal logics.   
And the emphasis on S5 itself has been quite reasonable.  The reason  
S5 has been popular, once again, is that it is arguably the best  
logic for logical necessity.  Logical necessity in turn has been  
dominant in contemporary metaphysics for the past forty years largely  
because of its relevance to issues in the philosophies of logic,  
language, mind, and science that were dramatically revitalized not  
only by the rise of possible world semantics but also by the so- 
called "new" theory of reference developed by Kripke, Putnam, and  
others.  To allude to some sort of Great Academic Conspiracy that is  
selectively limiting and controlling the direction of academic  
research in modal logic is, frankly, beneath you.  There are  
excellent reasons for studying S5 in both pure and applied contexts  
and excellent reasons for studying other modal logics as well --  
research on *both* sides here has been very robust and well-motivated.    (06)

-chris    (07)

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