Re: Proposed definition of "possible world"
Michael,
There are a several different, but related issues involved.
The oldest one concerns the relationship between necessity
and laws. Aristotle, among others, made that observation,
and I quoted Carnap to show that a modern logician who was
an active developer of modal logic recommended the approach
of interpreting causal necessity as provability from the basic
laws of physics.
But I agree with your comment that just interpreting the
necessity operator as provability is not enough:
>.... If you interpret the modal operator Box as
>meaning "provable", then you get the modal system G. If you interpret it
>as "provable and true", you get the modal system G+S4. However, you do
>not get any of the other modal systems, like S5, K, D, so how can the use
>of provability give you a semantics for arbitrary modal theories?
That is where Dunn (1973) filled in an important piece of the
puzzle. He showed that you could replace each of the undefined
possible worlds in Kripke's semantics with a pair (L,D) of
laws L and facts D. Then he replaced Kripke's accessibility
relation between possible worlds with a condition on how the
laws and facts vary as you move from one world to another.
With that replacement, Dunn could support the same axiom
systems with constraints on the laws and facts that Kripke
could support with constraints on the accessibility relation.
For example, S4 implies that the laws of the first world
must be a subset of the laws of the second world. S5 implies
that all worlds must have the same laws (but possibly different
facts).
In general, Dunn's system lets you do everything that you can
do with Kripke's system in exactly the same way. What it adds
is the option of talking about which particular laws make
something "necessary" or "essential". That is why I recommend
it as a replacement for the notion of possible world in the SUO
ontology: instead of saying that something is essential, you
can state a law that makes it essential.
John