Dear Colleagues, (01)
John and Pat seem determined to leave me in some uncomfortable
ontological place. Since this is not the case I will continue my
defence.
John wrote:
>
> They define the type, and every type determines a set. But it is
> possible to have two types that happen to have exactly the same
> set of instances. Plato used the two terms 'featherless biped'
> and 'animal with speech', which happen to have the same extension.
>
> But that happens to be a coincidence. To emphasize the point,
> Diogenes the Cynic plucked a chicken and threw it into the Academy
> while shouting "Here is Plato's man."
>
> Therefore, the criteria for type identity are stricter (or finer)
> than the criteria for set identity. That is the point that Church
> made in the quotation of my previous note:
>
> Alonzo Church> It is possible, however, to allow two functions to
> > be different on the ground that the rule of correspondence is
> > different in meaning in the two cases although always yielding
> > the same result when applied to any particular argument. When
> > this is done we shall say that we are dealing with functions
> > in intension.
>
> The usual distinction between types and sets is that the identity
> criteria for types are based on the axioms and definitions for
> those
> types. But the identity criteria for sets are based solely on
> their
> members. Since we can't observe the future or any possible world,
> we have to rely on the definitions and axioms because the entities
> that constitute the sets are unobservable. (02)
And Pat said: (03)
> The point is what ontological discipline you use for deciding when
> properties or relations are identical. Speaking strictly
> mathematically, a relation is (usually assumed to be, or to be
> mathematically modeled as) a set of pairs <relata, relatee>, and a
> predicate or property is modelled as a set, the set of things that
> have the property or that satisfy the predicate. For those of the
> extensionalist persuasion, these set-theoretic definitions are
> taken
> to be the absolute identity criterion, so that relations are
> treated
> as a species of set, and have similar identity criteria. So if two
> different ways of specifying a relation or property turn out to
> define
> the same set or set of pairs - that is, if the very same things
> stand
> in the relationship to one another, under the two descriptions -
> then
> on this view, these both define the same relation or property.
> Classical examples include being human = being a hairless biped.
> The
> other, intensional, view wants to give relations and properties a
> more
> robust notion of identity, and treats the mathematical set-
> theoretic
> account as being only a mathematical model. On this view, the
> property
> of being human is one thing, and the property of walking on two
> legs
> and having no body hair is another, and the accidental fact that
> they
> happen to coincide on this planet right now is not sufficient
> grounds
> to declare them to be identical *as properties*. After all, they
> *could* be different: one can imagine a non-human hairless bipedal
> creature. Maybe Neanderthals were examples. OK, lets not get into
> that
> debate (which hasnt been settled in the last two millennia), just
> observe that the difference of opinion exists. However, it does
> have a
> (small but not invisible) practical consequence for ontology
> reasoners, because you get different logics in the two cases. RDF,
> RDFS and OWL-Full (and ISO Common Logic) all have a semantics based
> on
> the second, intensional perspective: OWL-DL and classical FOL both
> are
> based on the first, extensionalist view. The practical difference
> is
> that the intensional logics are slightly weaker than the
> extensionalist ones, and hence somewhat easier to implement. (One
> gets
> the extensionalist logics by adding a lot of conditional equations
> to
> an intensional logic.)
<snip>
> For the extensionalist, however,
> this
> is a real problem, as the general task of determining extensional
> identity is NP-hard. And this issue does arise in practice, when
> reasoners are required to check subsumption relationships between
> ontological specifications. Part of the motivation for using
> description logics (as in OWL-DL) is to restrict expressivity in
> order
> to keep this problem manageable. (04)
MW: The good news for me is that having taken on board possible worlds,
I get a very good return for my ontological commitment. (05)
To return to the tail of the 2 eyed and 4 legged sheep (or human and
featherless biped) if I want to talk about 2 eyed sheep and 4 legged
sheep and determine if they are necessarily the same, then I need only
use sets that that go across all possible worlds. Thus I find that there
are some worlds in which there are one eyed sheep and 3 legged sheep and
I can see that these sets are not the same, and that whilst these
definitions might pick out the same set in my field, this is not the
most general case. (06)
However, although I am able to deal with the situations that Pat and
John say will cause me trouble without any trouble at all, I am still
only using set theory with extension as the basis for identity. (07)
Regards (08)
Matthew West
Information Junction
Tel: +44 560 302 3685
Mobile: +44 750 3385279
matthew.west@xxxxxxxxxxxxxxxxxxxxxxxxx
http://www.informationjunction.co.uk/
http://www.matthew-west.org.uk/ (09)
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