SUO: Re: Monoclonal Antebodies
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Robert,
Let me try this 1 bit at a time.
To me a "sort" is something done
to a list, and "Ents" are sentient
arboroids that live in Middle Earth.
My teachers some time ago deprecated
this other use of the word "sort"
for some reason that I've now
forgotten, if I ever knew,
so my mind is open, if {}.
Let's say that I have a working, if naive, knowledge of
sets, functions, relations, and so on. I understand in
concrete terms about cartesian products X_1 x ... x X_k,
and relations L c X_1 x ... x X_k, and I'm vaguely if
somewhat uncertainly aware that some people call that
list of relational domains a "signature", I think
that's the gist of it, not sure, let me know.
I think of that list of domains as giving
the "type" of the relation.
One more bit, while I have it in mind.
I think of functions, things of type
f : X -> Y, as species of relations,
and it is a mixed "convenience" to
have a whole separate sublingo for
functions, but I have learned how
to get around all that, and to
keep uppermost in my mind the
information that is invoked
by any notations that refer
to more or less the same
set-theoretic objects.
Indeed, one of the main reasons that I find so utterly useful
every bit of category theory that I can manage to pick up is
that it helps me to see the isomorphisms between things that
are in actuality only superficially distinct, while FO logic,
by comparison, is constantly bogging me down in ossified and
reified and syntactic superficialities.
So my question is, what am I missing about
these things of sorts of which you speak,
over and above what I already have in
objects in categories and types?
Jon
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Robert E. Kent wrote:
>
> ----- Original Message -----
> From: "Jon Awbrey" <jawbrey@oakland.edu>
> To: "Robert E. Kent" <rekent@ontologos.org>
> Cc: "Patrick Cassidy" <pcassidy@bellatlantic.net>; "SUO"
> <standard-upper-ontology@ieee.org>
> Sent: Thursday, June 12, 2003 11:44 AM
> Subject: Re: Monoclonal Antebodies
>
> > o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> >
> > Robert,
> >
> > I enjoy a good parallel worlds story as much as any of my altered egoes,
> > but I think that a more central and applicable way of thinking about this
> > is to regard it as a question of diverse descriptions of the same
> universe,
>
> Yes and no. The IFF notion of a model(-theoretic structure) refines and
> extends the traditional notion (Chang & Keisler) of a sorted model (Chris
> Menzel http://grouper.ieee.org/groups/suo/email/msg01792.html
> -- I seem to reference this a lot -- thanks Chris), since it classifies a
> set of entity instances by a set of entity types (sorts) and it classifies a
> set of relation instances (abstract tuples) by a set of relation types
> (predicates). The traditional case, where tuples are concrete and the tuple
> set is the given set of all concrete tuples, would seem to correspond to a
> universe. The more refined notion would seem to include both the notion of
> multiple universes (a multiverse or polycosmos) and the notion of diverse
> descriptions of the same universe.
>
> > and this will bring us back to the theory of manifolds, which is what made
> > it possible for math and physics folks to think about the appropriate maps
> > between different coordinate charts for different reference frames, and to
> > start looking out for the proper sorts of invariants that tell of
> objective
> > realities.
>
> The IFF defines not only the notion of a model(-theoretic structure), but
> also the notion of a morphism of model(-theoretic structure)s. Again this
> refines and extends the traditional notion. These comprise a category called
> Model. Perhaps the isomorphisms in this category correspond to (or at least
> are a first approximation to) invariants -- certainly isomorphic models
> induced isomorphic model theories; that is, an expression is true in one
> isomorph iff its isomorphic copy is true in the other isomorph.
>
> Robert E. Kent
> rekent@ontologos.org
>
> > See my besotted comment:
> >
> > http://suo.ieee.org/ontology/msg04782.html
> >
> > Jon
> >
> > o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> >
> > Robert E. Kent wrote:
> > >
> > > ----- Original Message -----
> > > From: "Patrick Cassidy" <pcassidy@bellatlantic.net>
> > > Cc: <standard-upper-ontology@ieee.org>
> > > Sent: Tuesday, June 10, 2003 11:45 AM
> > > Subject: SUO: RE: SUO Ballot with 2 Questions -- "monolithic"?
> > >
> > > [snip]
> > >
> > > > The term "monolithic" may be interpreted by some to mean
> > > > "having no contradictory assertions", but inasmuch as
> > > > (almost) every commentator seems to agree that any
> > > > Standard Upper Ontology will have to have a provision
> > > > for representing alternative possible worlds (e.g.
> > > > fictional contexts, alternative theories of physics,
> > > > alternative theories of history, etc.), (are there
> > > > dissenters? Perhaps Pierre doesn't agree?) the continued
> > > > use of "monolithic" suggests that the term includes
> > > > ontologies with provisions for possible worlds.
> > > > Rather than get hung up on debates over term definitions,
> > > > I would use the term "polycosmic" to refer to an ontology
> > > > that has a provision for alternative possible worlds,
> > > > and includes some alternative logically contradictory
> > > > theories as applying to alternative possible worlds.
> > > > Whether a polycosmic ontology is *one* ontology or many is
> > > > perhaps merely an issue of terminology, but there are,
> > > > I think, useful distinctions to be made depending on how
> > > > high up in the class hierarchy the contradictory alternatives
> > > > are, and whether they are truly contradictory or merely
> > > > alternative but not inconsistent representations of the same
> > > > world.
> > >
> > > [snip]
> > >
> > > > Of course, a specialized domain ontology will usually
> > > > be not only "monolithic" but "monocosmic" -- it will have no
> > > > alternative possible worlds.
> > >
> > > [snip]
> > >
> > > > I think that motion #2 contemplates the kind of polycosmic
> > > > merged ontology, with mappings to other ontologies, that I think
> > > > is needed at the present time. So I hope the motion passes.
> > >
> > > ______________________________
> > >
> > > >From Merriam-Webster Online:
> > >
> > > cosmos
> > > Function: noun
> > > Etymology: Greek kosmos
> > > Date: 1650
> > > - an orderly harmonious systematic universe;
> > > - a complex orderly self-inclusive system
> > >
> > > universe
> > > Function: noun
> > > Etymology: Latin universum, from neuter of universus entire, whole
> > > Date: 1589
> > > - the whole body of things and phenomena observed or postulated
> > > - a distinct field or province of thought or reality that forms a closed
> > > system or self-inclusive and independent organization
> > > - a set that contains all elements relevant to a particular discussion
> or
> > > problem
> > > ______________________________
> > >
> > > The library of modules (theories) envisioned by motion #2 is situated
> within
> > > the context of a lattice of theories (generalization/specialization
> > > hierarchy) and its correlated structure known as the truth concept
> lattice.
> > > A library of modules in the IFF is represented by an indexed collection
> or
> > > diagram of theories (and theory morphisms).
> > >
> > > In the IFF the notions of a lattice of theories and a truth concept
> lattice
> > > are very polycosmic in nature. A model theory is defined as the theory
> > > associated with a model(-theoretic structure); it consists of all
> > > expressions satisfied by that model. Clearly any diagram of theories
> more
> > > generic than a particular model theory is monocosmic.
> > >
> > > However, in the IFF representation for a lattice of theories, there are
> > > possible diagrams of theories where any two theories are either
> equivalent
> > > or mutually inconsistent. Each of these theories lies at the lowest
> level
> > > in the lattice of theories strictly above the bottom inconsistent theory
> > > containing all expressions. Such diagrams of theories are very
> polycosmic.
> > >
> > > So it seems that from the metatheoretic standpoint the notion
> > > of a polycosmos is very natural.
> > >
> > > Robert E. Kent
> > > rekent@ontologos.org
> >
> > o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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