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Re: [ontolog-forum] Thing and Class

To: "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Matthew West <dr.matthew.west@xxxxxxxxx>
Date: Sat, 13 Sep 2008 13:43:15 +0100
Message-id: <48cbb5cf.0c92100a.3e0c.5b9c@xxxxxxxxxxxxx>
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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