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 4-d 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 spatio-temporal 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 3-4 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 space-time. The existence of change
> does not imply that the global 4-d 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://www-formal.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 Dunn-style 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
> > possible-worlds 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
spatio-temporal parts of it. How does this fit with what you mean? (011)
Regards (012)
Matthew West
http://www.matthew-west.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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