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Re: [ontolog-forum] Thing and Class

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Fri, 12 Sep 2008 02:11:58 -0400
Message-id: <48CA082E.2070404@xxxxxxxxxxx>
Dear Matthew and Pat,    (01)

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.    (02)

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>> all spatio-temporal extents exist (at all times, but
 >> strictly independent of time).    (04)

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,    (05)

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.    (06)

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.    (07)

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.    (08)

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    (09)

I believe that what John McC, Matthew, and Pat are recommending is
very close to Dunn's semantics for modal logic.    (010)

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.  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:    (011)

  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}.    (012)

  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.    (013)

  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'.    (014)

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.    (015)

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...    (016)

Not true.  Nobody actually implements "possible worlds".  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.    (017)

For further discussion of these and related issues, see    (018)

    Laws, Facts, and Contexts    (019)

    Worlds, Models, and Descriptions    (020)

John    (021)

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