I just happened to re-read the following post and noticed it's full of
hasty errors and poor exposition. A cleaned up version follows. (01)
On Sep 14, 2008, at 11:37 PM, Christopher Menzel wrote:
> On Sep 14, 2008, at 5:48 PM, Pat Hayes wrote:
>> ...
>> I'm still waiting to hear how to map form Kripke to Dunn.
>
> I can't see how this can be done uniquely. Consider a very simple
> Kripke model with two worlds w1 and w2 where all atomic sentences
> true in w0 are true in w1 (but not vice versa -- assume also that at
> least one atomic sentence is true in w0) and both worlds are
> accessible to themselves and w1 is accessible to w0. The problem for
> turning this into a Dunn model is: when do we have a "mere"
> necessary truth and when do we have a law? Nothing seems to
> determine an answer to this question; it is simply stipulated in a
> given model. In particular, presumably, starting with a Kripke model
> M, we map each world w0 of M to a world u0 in a Dunn model whose
> facts are the truths of w0. Given this, it seems we can have one
> Dunn model where every fact of u0 is also a law of u0 and another
> model where u0 is a "lawless" world, i.e., where no fact of u0 is a
> law of u0. By Dunn's definition, this will make both worlds
> accessible to themselves and u1 accessible to u0. (In both cases,
> suppose also that every fact of u1 is a law of u1 -- this will
> prevent u0 from being assessible to u1, since some of u1's laws are
> not facts of u0.) (02)
Corrected version: (03)
I can't see how this can be done uniquely. Consider a very simple
Kripke model M with two worlds w0 and w1 where all atomic sentences
true in w0 are true in w1 (but not vice versa -- assume also that at
least one atomic sentence is true in w0) and both worlds are
accessible to themselves and w1 is accessible to w0. The problem for
turning this into a Dunn model is: when do we have a "mere" necessary
truth and when do we have a law? Nothing seems to determine an answer
to this question; it is simply stipulated in a given model. In
particular, presumably, starting with a Kripke model M', we map each
world w of M' to a world u in a Dunn model whose facts are the truths
of w. But nothing seems to determine which necessary truths at each
world we declare as facts. Thus, returning the little model M above,
it seems we can have one corresponding Dunn model D where every fact
of u0 -- the world of D corresponding to w0 -- is also a law of u0 (by
construction, the facts of u0 are all necessary in u0) and another
model D' where v0 -- the world of D' corresponding to w0 -- is a
"lawless" world, i.e., where no fact of v0 is a law of v0. Suppose
now that every fact of u1 (v1) is a law of u1 (v1). By Dunn's
definition, in both models D and D', the worlds of the models are
accessible to themselves, and in D, u1 is accessible to u0 and in D',
v1 is accessible to v0 (but not vice versa, since some of u1's (v1's)
laws are not facts of u0 (v0)). There is nothing that I can see in
the original Kripke model M that enables us to say that either of the
two Dunn models is somehow "the" model corresponding to M. So it
looks there is no unique mapping from Kripke models to Dunn models.
Not that there is anything wrong with that. :-) (04)
That said, there is actually a pretty intuitive choice among the
possible mappings -- let the laws of a world in a Dunn model be the
"non-tautological" necessary truths, the ones that aren't simply
analytical true, true in virtue of the meanings of the logical
constants; but that is still a *choice* -- it isn't determined by the
definition of a Dunn model alone. (05)
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