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

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Fri, 23 Feb 2007 12:17:46 -0600
Message-id: <FCB5C6FD-62C1-4177-83D3-71D024CB5A15@xxxxxxxx>
Well, John, I almost wrote a ps to my post saying "Expect a reply  
from John Sowa on Dunn's semantics for modal logic shortly!".  :-)   
I'll not bother repeating past debates (see e.g., http://suo.ieee.org/ 
email/msg10546.html).  But just a couple of points:    (01)

> By using Dunn's semantics, modal logic can be formalized in a first- 
> order metalanguage about a first-order object language. After you  
> do that, you can then collapse the two levels and define a single- 
> level Tarski-style model for the combined system.  That collapsed  
> form is less readable than the original, but it demonstrates that  
> you can map the modalities into a first-order framework.  See
>     http://www.jfsowa.com/pubs/laws.htm
>     Laws, Facts, and Contexts
> With Dunn's semantics instead of Kripke's semantics, each world w  
> is replaced by a pair (M,L), where M is the set of propositions  
> true in w (called facts) and L is the subset of facts (called laws)  
> that are necessary in w.
> Both versions are formally equivalent,    (02)

If as you say Dunn's semantics is entirely first-order, then the two  
cannot be formally equivalent.    (03)

> And by the way, I'd like to make a comment about "truly first-order  
> modal logic like S5".  The fact that there are more papers  
> published about S5 than any other axiomatization is a very strong  
> argument *against* Kripke semantics.    (04)

I haven't a clue what argument you have in mind but, be that as it  
may, on the whole 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.  What is much more likely true is that S5 is typically  
the modal logic of choice (for solid, though hardly uncontroversial)  
reasons in the rather narrow, and more philosophical, context of  
modality in the "broadly logical", or metaphysical, sense -- the  
context, notably, of DOLCE's use of S5.    (05)

> Even C. I. Lewis said that S5 was too strong to be realistic,    (06)

For some contexts, yes; for all, highly debatable.    (07)

> and in terms of Dunn's semantics, S5 implies that all possible  
> worlds have exactly the same laws.    (08)

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.    (09)

> There are many papers that apply S5 to database systems, where the  
> laws are the DB constraints.  But in such systems, it would be  
> impossible to update the constraints -- the most unrealistic  
> assumption imaginable.    (010)

Which, if true, simply goes to show that S5 is not the right modal  
logic for modeling DBs.    (011)

> People who publish such stuff give aid and comfort to those who  
> think logic is useless.    (012)

Anyone finding comfort in such a spectacular non sequitur is a lost  
cause anyway. :-)    (013)

-chris    (014)

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