Sunday, September 14, 2008 7:48 PM, John Sowa wrote: (01)
Intensions can be derived from extensions, but the extensions may lose or
ignore some important distinctions in the definitions. I
> use the Greek letter delta for a *denotation operator* that maps a type t
> and a world w to the set of instances of t in the world w: (02)
> delta: Types x Worlds -> Sets (03)
> However, the mapping by delta is not one-to-one, since for a given world
> w, the set delta(HumanBeing,w) may be the same as the set
> delta(FeatherlessBiped,w). Also, delta(t,w) for any type t that has no
> instances in the world w would be the empty set. (04)
John, (05)
I hold with you when you talk of semantic priority of Intension (sense,
content, comprehension, context) over Extension (reference, denotation,
denomination, volume), even though, these two basic parts of meaning are
inversely correlated. And just would like to focus on one principal semantic
distinction. For the totality of all individuals, or the total reference
class, we have a semantic relation (operator, map, or function) of
denotation or designation (for conceptual individuals). Here comes your
delta operator.
But for the totality of all properties, we have another fundamental semantic
relation, representation, say, ro operator, holding between all symbolic
structures (predicate letters, sentences, codes, languages, natural or
formal ) or (constructs, concepts, statements, theories) and real
properties (states, qualities, quantities, laws, rules, or relationships).
This semantic distinction accounts for the reason why in our deap ontologies
one needs Intensional Classification, generally formalized as a lattice of
intensions (properties) arranged with partial order relationship (which
seems anti-isomorphic to the corresponding lattice of predicates?). This
difference makes all the difference. Take ''number''. Many good minds still
thinking that it is a class, a multitude of units, although in Universal
Arithmetic, following Euclid's Geometry, it was indicated: (06)
''By Number we understand not so much a Multitude of Unities, as the
abstracted Ration of any Quantity, to another Quantity of the same kind,
which we take for Unity "
So, a number (natural, rational, or real) as a ratio is a sort of relation.
It has richer classification prospects, while viewing Number as a property
(possessed by an indefinite quantity of units, items or individuals and
zero) exemplified as individual numbers, all sorts of numerals, or
mathematical sets, abstract collection of numbers (or symbols). What, again,
can be ordered as a lattice of number-related properties. (07)
Azamat (08)
---- Original Message -----
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
Sent: Sunday, September 14, 2008 7:48 PM
Subject: Re: [ontolog-forum] Thing and Class (09)
> Azamat and Leonid,
>
> Intensions can be derived from extensions, but the extensions may
> lose or ignore some important distinctions in the definitions. I
> use the Greek letter delta for a *denotation operator* that maps
> a type t and a world w to the set of instances of t in the world w:
>
> delta: Types x Worlds -> Sets
>
> However, the mapping by delta is not one-to-one, since for a given
> world w, the set delta(HumanBeing,w) may be the same as the set
> delta(FeatherlessBiped,w). Also, delta(t,w) for any type t that
> has no instances in the world w would be the empty set.
>
> Therefore, a taxonomy defined by intensions is more fine grained
> than a taxonomy defined by extensions, because the sets may blur
> important distinctions in the definitions.
>
> Different applications that might use the same taxonomy may depend
> on those distinctions. For example, a taxonomy for the history of
> proposed airplane designs might have the type FlappingWingAircraft,
> but a taxonomy of airplanes that were actually built and survived
> their first test flight might have no instances. But a taxonomy
> defined by descriptions (intensions) could be used for both purposes.
>
> > Accordingly, there is extensional classification, called taxonomy,
> > and intensional classification, called meronomy, or mereology.
>
> The confusion between those two classifications is usually caused
> by terminologies that make a vague distinction of broader/narrower.
> That distinction should be clarified by using different dyadic
> relations, subtypeOf and partOf, which define distinct partial
> orderings. Both of them should accommodate hypothetical things
> like unicorns, which have a horn as part, and airplane designs
> that depend on the details of how the parts are assembled.
>
> People might dismiss unicorns as mythical or fictional animals,
> but biologists commonly talk about, describe, and search for
> instances of fossils of hypothetical organisms. An example
> would be the first tetrapod that crawled out of the water and
> became the ancestor of the amphibians.
>
> John
>
>
> _________________________________________________________________
> Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
> Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
> Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
> Shared Files: http://ontolog.cim3.net/file/
> Community Wiki: http://ontolog.cim3.net/wiki/
> To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx
> (010)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx (011)
|