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Re: [ontolog-forum] Context, at last!

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Chris Menzel <cmenzel@xxxxxxxx>
Date: Fri, 17 Jun 2005 16:59:14 -0500
Message-id: <20050617215914.GL64829@xxxxxxxx>
On Fri, Jun 17, 2005 at 12:26:04PM -0700, Duane Nickull wrote:
> Chris Menzel wrote:
> >ID: if x = y, then anything true of x is true of y. 
> > 
> Disagree.      (01)

Well, we haven't really fixed the context rigorously enough for there to
be anything to disagree about.  In the context of standard first-order
logic with its usual extensional semantics, the above principle is
rendered schematically as follows:    (02)

  For any well-formed formula A not containing the variable y, the
  following is an axiom:    (03)

  ID_FOL:  x = y -> (A -> A'),    (04)

  where A' is the result of replacing every free occurrence of x in A 
  with an occurrence of y.      (05)

And the fact is that, on the standard, extensional semantics for
first-order logic, every instance of ID_FOL is valid.  That's not
something about which one can rationally disagree; it's just a
mathematical fact about first-order languages and their models.    (06)

What one might disagree with is that the principle ID (properly
formalized) is valid in every logical context.  And that is exactly what
the counterexamples I mentioned were designed to show -- ID fails, or at
least appears to fail, in contexts involving belief and necessity (among
others).    (07)

> Most things may be the same but it is still instance y as opposed to
> instance x, therefore his axiom has a logic error.      (08)

You lost me there, I'm afraid.    (09)

> They are still two different things.  It may be better to state:
> if x = y, then x is y and both should be called x.    (010)

That rather seriously confuses the *value* of the variable "x" with "x"
itself.  It also seems implausible -- Mark Twain is Sam Clemens, but I
don't see any particular reason why he *should* be called by either
handle.  (And aside from these points, the development of theories that
include semantical notions like "called" are fraught with difficulty.)    (011)

> Also - an important consideration of context is perceiver.      (012)

Yes, that can be important.  That is one reason there is a lot of active
research on formal theories of context at the moment.    (013)

> To you and I, a coffee table is a solid item, to a neutrino, it is a
> lot of open space interspersed with a few bits of solid matter...    (014)

True enough, so a neutrino's ontology of your front room will probably
look a lot different than your front room ontology!  Note, though, that
the notion of context is often just assumed in the background of an
ontology -- we often create ontologies *from the perspective of a given
perceiver or set of perceivers*, e.g., domain experts in a mfg shop
floor.  For these cases, the notion of context needn't play an explicit
role.  More and more, though, we have a need to represent multiple
ontologies arising out of different contexts within a *single*
framework, and this requires explicit mechanisms that enable us to group
pieces of information according to context -- and that's exactly what
formal theories of context are attempting to provide.    (015)

Cheers!    (016)

Chris Menzel    (017)

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