Dear John, (01)
It is good to see that we seem to be coming closer together again. (02)
>
> MW> The real difference is that 3D sees that what exists now is all
> > that exists, whilst 4D sees the past and the future as part of
> > what exists as well as the present. This is what it means to stand
> > outside time.
>
> I agree with that description, but you seemed to suggest that the
> notion of change does not exist in a 4d view, but I think that
> we were using different definitions of 'change'. (03)
[MW] Yes, to try to draw an analogy, a 3D view might see a point
changing its position over time, whereas in 4D you would see a line
that appears static but is extended in time.
>
> MW>> all spatiotemporal extents exist (at all times, but
> >> strictly independent of time).
>
> PH> Agreed, and a nice analysis. Putting the same point in logical
> > terms, the universe of discourse shouldn't be in a state of flux,
>
> I have no quarrel with that, but it has nothing to do with the
> definition of the concept of change. According to the most common
> definition, if time slices at t=0 and t=1 are identical, there is
> no change. (04)
[MW] That is a useful way to make 34 D neutral definition of change.
>
> Another way to say it: if the partial derivative with respect
> to the time coordinate is 0, there is no change; otherwise, there
> is change in that region of spacetime. The existence of change
> does not imply that the global 4d universe is in flux. It just
> means that there is some region in the universe where the derivative
> with respect to time is not zero. (05)
[MW] Agreed.
>
> MW>> And interestingly, I again use possible worlds as an alternative
> >> to modal logic. Not that I object to others using modal logic, but
> >> I do not see that I am obliged inevitably to do so.
>
> PH> Again, I agree that this is the best approach. I think this is
> > widely accepted, by the way: John McCarthy made the same point many
> > years ago :
> > http://wwwformal.stanford.edu/jmc/modality/modality.html
>
> I believe that what John McC, Matthew, and Pat are recommending is
> very close to Dunn's semantics for modal logic. (06)
[MW] I have always assumed this, but of course I prefer not to use
modal logic.
>
> Most AI work with "possible worlds" is actually based on metalevel
> reasoning about sets of propositions that describe those worlds,
> not with the worlds themselves. (07)
[MW] Actually, I am more interested in populating those worlds with
actual plans, than I am with reasoning, and then tracking how well
they match to reality. (08)
> Starting with any Kripke model
> K=(W,R,Phi), where W is the set of words, R is the accessibility
> relation among worlds, and Phi is the evaluation function, those
> sets can be derived:
>
> 1. For each word w in W, define the facts of w as the set of all
> propositions p that are true in w: {p  Phi(w,p) = True}.
>
> 2. Define the laws of w as the set of all propositions p that are
> necessarily true; i.e., p is true in all worlds accessible from w.
>
> 3. Define the accessibility relation R(w, w') as True iff every
> proposition p that is necessarily true in w is also true in w'.
>
> This construction replaces every world in a Kripke model with a set
> of laws and facts in a Dunnstyle model. Any theorem that can be
> proved about a Kripke model is also true of the corresponding Dunn
> model. But Dunn's version is more *usable* because it makes the
> laws and facts available for further analysis and manipulation. (09)
[MW] I have of course heard this before, and have not found anything
obviously objectionable, but I would be concerned if this involved
modal logic.
>
> PH> John's way follows Dunn's theory and is based on intensional
> > descriptions. The far more commonly used view uses Kripke's
> > possibleworlds account of modalities. Kripke's is widely accepted
> > as the standard, and certainly gives a more usable semantics...
>
> Not true. Nobody actually implements "possible worlds". (010)
[MW] What do you mean by this? For example, I expect to have an object
that represents a particular possible world (well more properly a universe)
for all the time that it exists, and then to have objects in the world as
spatiotemporal parts of it. How does this fit with what you mean? (011)
Regards (012)
Matthew West
http://www.matthewwest.org.uk/ (013)
> What they
> implement and reason with and about are sets of statements of the
> laws and facts of those worlds. Since the above construction can
> map any Kripke model into such sets, most people who implement such
> systems pay lip service to Kripke's version, but they actually use
> something that is much closer to Dunn's version.
>
> For further discussion of these and related issues, see
>
> http://www.jfsowa.com/pubs/laws.htm
> Laws, Facts, and Contexts
>
> http://www.jfsowa.com/pubs/worlds.pdf
> Worlds, Models, and Descriptions
>
> John
>
>
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