Re: SUO: Model theory for modal logics
Stefano,
I think we are converging on agreement. We're now getting
into minor qualifications and degrees of emphasis.
JS>> In natural
>> languages, people freely intermix multiple modal operators
>> in the same sentence, but Kripke semantics makes it very
>> difficult to support more than one kind of modality.
>
SB> Now we know this is not true. Plenty of multi-modal logics (for Kripke
> frames) have been developed during the 90s and in several cases they
> have nice formal properties.
>
> A good source is the forthcoming book by Dov Gabbay, Agi Kurucz, Frank
> Wolter, and Michael Zakharyaschev "Many-dimensional modal logics: theory
> and applications", Elsevier Science, 2003.
> The results are quite surprising when you compare them to what was known
> in the 70s or 80s.
Now we are arguing about how difficult "difficult" may be.
I agree that there are various extensions of the Kripke approach.
As an example, I discuss the version by Cohen & Levesque at the end
of Section 7 of my paper. (I also sent a copy to Hector Levesque,
who agreed that my summary of their approach was fair.)
The point I made there is that anything you can do with Kripke-style
semantics can be done in a formally equivalent way with Dunn-style
semantics. But as with the single modality semantics, the accessibility
relation must be assumed as an unexplained primitive, but it can be
related to something more basic -- namely the laws and facts --
with a Dunn-style approach.
In addition, however, Dunn's approach allows you to do various kinds
of metalevel reasoning. You can even introduce the notion of
"entrenchment" -- as in belief revision systems. You can, for example,
say that logical necessity is the most strongly entrenched, physical
necessity is next, and even have a partial order of legal necessity:
constitution > federal laws > state laws > city laws > regulations.
You can reason about such things at the metalevel in a way that is
very common in a court of law. But I would claim that doing that
kind of reasoning with any Kripke extension is "very difficult".
> Btw, the problem is not with the term law only. It won't take long
> before somebody asks what a "fact" is in this approach and why we can
> have different sets of facts and why a set of facts has to be maximally
> consistent...
Yes, I said that you could consider the term "law" as a metaphor.
If you prefer a term that is more noncommittal, you could just
call them propositions at different levels of entrenchment.
And I agree that there is no reason why the facts must be maximally
consistent. That was Hintikka's assumption for his original model
sets, but you could relax that assumption, as I pointed out in
Section 6 of the paper. See Figure 12 (attached as worlds.gif).
At the upper left is a Kripke-style world w, at the lower left
is a model set M, at the upper right is an excerpt called a
situation s, and at the lower left is a nonmaximal set of
propositions called a context c. I assume that the mappings
make Figure 12 a commutative diagram.
John
PS: For anyone who has forgotten the previous thread, the corrected
version of the paper is at http://www.jfsowa.com/pubs/laws.htm
