On Oct 7, 2009, at 2:51 AM, Matthew West wrote: (01)
> Dear Colleagues,
>
> 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.
>
> And Pat said:
>
>> 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.
>
> MW: The good news for me is that having taken on board possible
> worlds,
> I get a very good return for my ontological commitment. (02)
FWIW, I agree that taking on board possible worlds is a good strategy.
As John McCarthy said many years ago: modalities, si; modal logic, no.
By which he meant, formalize the modalities using Kripke-style
semantics (possible worlds) in FOL. Which is exactly what Matthew is
doing, and what we did in IKRIS project, where we were obliged to
interface with explicitly modal logics. (03)
> 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.
>
> 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 (04)
> only using set theory with extension as the basis for identity. (05)
Indeed, and I apologize for not expanding on this point in my summary,
above. Adding modal content, expressed in terms of possible worlds, is
one way to get a version of intensional identity without sacrificing
semantic extensionality. It is popular for exactly that reason.
However, intensional identity is not *obliged* to take this route. The
CL/RDF version of intensionality is much more radical, and simpler: it
says that two relations or properties are the same just when they have
the very same definition, or when explicit assertions of identity are
made which entail that they are identical. That is, the logic simply
says nothing about relational identity, and it is up to users to write
axioms if they want any nontrivial identities to hold. This is (a)
simplest of all to use and reason with, and (b) absolutely requires
people to state their assumptions explicitly in axioms. Also in
practice, it seems to be quite workable. Most of the time people
simply are not interested in identity between relations, only with the
accuracy with which relations are applied to things. (06)
I don't say your place is uncomfortable, Matthew, but I think there
are softer beds we can lie on. (07)
Pat (08)
>
> Regards
>
> Matthew West
> Information Junction
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