On 9/10/08 12:21 PM, leo@xxxxxx wrote:
> Colleagues ,
>> Nice thought, but the problem is that these notions are all defined
>> purely in extensional terms, i.e. in terms of membership. Every
>> intensional type has a corresponding extension, which is the set of all
>> things which are of the type; and Michael's taxonomy, above, only refers
>> to the extensions.
> I wrote already ( http://www.ototsky.mgn.ru/it/21abreast.htm - 3.
> Mid-range objectives.) that from the Classification Theory(CT) point of
> view there mast be not only the Taxonomy but the dual Meronomy part !
> And this duality must be taken into account.
> Only two items from the CT :
> " - Any Classification System has two Dual parts - "Taxonomy" and
> "Meronomy". The first one is "external" and connected with ordinary set
> theory relations (unions, intersections, hierarchy (a subclass of)) etc..
> - The second one is "internal" and connected with Properties/Parts
> structure (archetype)." (01)
That is a different distinction. None of the current discussion
is about part/whole relationships. The intension/extenstion
question is rather whether subclass (for example) really is a
purely set-theoretic relationship. In RDFS for example it is
treated intensionally, so that subCLASS entails subSET of the
extensions but not the reverse. In an intensional framework, two
different classes might have the same extension. (02)
Pat (03)
>
> Best,
> Leonid - http://ototsky.mgn.ru/it
>
>
>>
>> tim.glover@xxxxxx wrote:
>>> Pat, noting John Sowas comments about the difference between
>>> extensional sets, and intensional types, would the situation be
>>> improved by substituting "type" for "set" throughout, eg
>>>
>>> X: the universal type
>>> Y: the type of things that have members
>>> Z: the type of things that do not have members.
>>> U: the type of things whose members are all of type Y
>>> V: the type of things whose members are all of type Z
>>>
>> Nice thought, but the problem is that these notions are all defined
>> purely in extensional terms, i.e. in terms of membership. Every
>> intensional type has a corresponding extension, which is the set of all
>> things which are of the type; and Michael's taxonomy, above, only refers
>> to the extensions.
>>> U and V might be useful types, even if they are not useful sets?
>> Maybe I spoke too harshly if I said they were useless. They are
>> obviously widely used notions. But it is dangerous to assume that they
>> are the only possibilities.
>>
>> Pat
>>>
>>> Tim.
>>>
>>>
>>>
>>> ------------------------------------------------------------------------
>>> *From:* semantic-web-request@xxxxxx
>>> [mailto:semantic-web-request@xxxxxx] *On Behalf Of *Pat Hayes
>>> *Sent:* 09 September 2008 18:16
>>> *To:* [ontolog-forum]
>>> *Cc:* [ontolog-forum]; James Leigh; SW-forum; Michael F Uschold
>>> *Subject:* Re: [ontolog-forum] Thing and Class
>>>
>>> At 2:35 PM -0700 9/8/08, Michael F Uschold wrote:
>>>> Azamat said:
>>>>
>>>> Sum up: If Thing goes as the universal class, of which everything is a
>>>> member, it will equivalent to Class, as the class of all classes. Other
>>>> interpretations will be inconsistent, asking for many questions.
>>>>
>>>> Whatever you choose to call these things, I find the following
>>>> distinctions helpful:
>>> I don't find them helpful. They embody and give credence to several
>>> common errors; they do not conform to established usages that go back
>>> half a century or more; and they aren't properly defined.
>>>
>>>> 1. X: the set of all things in the universe of discourse
>>>>
>>>> 2. Y: the set of all things that have member individuals
>>>>
>>>> 3. Z: the set of all things that do NOT have member
>>>> individuals
>>>>
>>>> 4. U: the set of all things whose members do not themselves
>>>> have members
>>>> (i.e. the set of all things whose members
>>>> are members of the class Z)
>>>>
>>>> 5. V: the set of all things whose members also have member
>>>> individuals
>>>> (i.e the set of all things whose members are all
>>>> members of the class, Y)
>>>>
>>>> The names I find most useful for these things are (substituting into
>>>> the text above)
>>>>
>>>> 1. THING: the set of all things in the universe of discourse
>>>>
>>> What do you mean here by 'universe of discourse'? That sounds like a
>>> semantic term. A semantic universe of discourse can be very small,
>>> e.g. {A} is a possible universe of discourse.
>>>
>>>> 2. CLASS: the set of all things that have member individuals
>>>>
>>> Does that exclude the empty set/class? (I hope not.) And do the
>>> members have to be individuals, or can they also include classes? As
>>> stated, this seems to exclude METACLASS, whose elements don't have
>>> individuals as members. I presume you did not intend this, though.
>>>
>>>> 3. INDIVIDUAL: the set of all things that do NOT have member
>>>> individuals
>>>>
>>> But not including the empty set/class, I presume.
>>>
>>>> 4. ORDINARYCLASS: the set of all things whose members do not
>>>> themselves have members
>>>> (i.e. the set of all things whose members
>>>> are members of the class INDIVIDUAL)
>>>>
>>>> 5. METACLASS: the set of all things whose members also have
>>>> member individuals
>>>> (i.e the set of all things whose members are all
>>>> members of the class, CLASS)
>>>>
>>> It might be worth remarking that your 'ORDINARYCLASS' and 'METACLASS'
>>> classifications are not recognized by set theory, and seem to have no
>>> purpose that I can determine; that the distinction between them isn't
>>> exhaustive, since it omits the (common) case of a set containing both
>>> sets and non-sets; and that it has been shown quite rigorously that if
>>> set theory is consistent at all, then it is also consistent to assume
>>> that there are no INDIVIDUALs in the above sense, i.e. that everything
>>> can be treated as a class.
>>>
>>> More importantly, the term "individual" as used in logic does not have
>>> anything like this meaning: it simply means anything in the universe
>>> of discourse, i.e. anything that one is quantifying over. This may
>>> include classes, of course, just as it may include anything else.
>>> Logical individuals can be any kind of thing: the term does not act
>>> there as a classifier. It is therefore very misleading and confusing
>>> to treat it as one, as you do here. Looked at from a logical point of
>>> view, it seems crazy to single out lack of one particular relation as
>>> being a defining characteristic for individual-hood. Why choose lack
>>> of class-membership? Why not say that anything that isn't married is
>>> an INDIVIDUAL, or anything that doesn't have more than three parts, or
>>> ...?
>>>
>>> The recognized term in set theory for INDIVIDUALs in your sense is
>>> 'ur-elements', from the German "urelemente". Maybe we could call them
>>> "irreducible individuals" or some such formulation, to avoid this
>>> (recurrent and exasperating) confusion? Or simply NON-CLASS ?
>>>
>>>> Here is the class hierarchy:
>>> Um... it's not a hierarchy.
>>>
>>>> THING (the most general anything)
>>>> CLASS (the most general class)
>>>> ORDINARYCLASS
>>>> METACLASS
>>>> INDIVIDUAL (the top of the ordinary class hierarchy)
>>> This is a horribly misleading mis-use of a widely used, almost
>>> universal, nomenclature. Why screw up a terminology that has worked
>>> well for 50 years? Its not even the terminology that is used in set
>>> theory itself, which (unlike logic) actually deals with these
>>> distinctions.
>>>
>>>> INDIVIDUAL and CLASS form a partition of THING
>>>> ORDINARYCLASS and METACLASS form a partition of CLASS
>>> No, they most definitely don't. If A is an individual then {A, {A}}
>>> isn't either ordinary or meta.
>>>
>>> Pat
>>>
>>>> THING and CLASS have all of the five things below as members:
>>>>
>>>> * THING
>>>> * CLASS
>>>> * ORDINARYCLASS
>>>> * METACLASS
>>>> * INDIVIDUAL
>>>>
>>>> ORDINARYCLASS has members:
>>>>
>>>> * INDIVIDUAL (any any of its subclasses)
>>>> *
>>>>
>>>> METACLASS has members:
>>>>
>>>> * CLASS
>>>> * METACLASS
>>>> * ORDINARYCLASS
>>>>
>>>> INDIVIDUAL has members that are inherited from any of its subclasses
>>>> (e.g. individual persons, or companies, or drugs, depending on the
>>>> domain).
>>>> Michael
>>>> On Thu, Aug 28, 2008 at 12:58 AM, Azamat <abdoul@xxxxxxxxxxxxxx
>>>> <mailto:abdoul@xxxxxxxxxxxxxx>> wrote:
>>>>
>>>> As a devil's advocate, seemingly unsanctioned with the 3WC,
>>>> Richard is doing
>>>> a useful work raising sometimes justified objections for SW
>>>> candidates
>>>> looking for canonization (standardization).
>>>> As for James's reading of Thing and Class, it looks more as
>>>> punning,
>>>> possibly intentionally.
>>>>
>>>> The interrelations of classes as well as classes and things are
>>>> actually
>>>> more subtle and deep, than generally presented in various
>>>> specifications.
>>>> A member of a class may itself be a class. For example, the class
>>>> of humans
>>>> is a member of the class of species of animals. An individual
>>>> human, even
>>>> being a member of its class, is not a member of the latter one,
>>>> the class
>>>> of species of animals. For a human is not a species of animal.
>>>> Whatever the number of human beings, it will not affect the number
>>>> of
>>>> species of animals. This goes as a kind of ontological rule of all
>>>> taxonomies: whatever the number of instances, objects,
>>>> particulars, it will
>>>> not change the number of classes of things. Again, this means that
>>>> relationships of class inclusion (subsumption) and class
>>>> membership have
>>>> some principal differences. Namely, the class inclusion is a
>>>> transitive
>>>> relation, while the CLASS MEMBERSHIP IS NOT TRANSITIVE. This
>>>> fundamental
>>>> fact is missing in some large scale, common sense ontologies,
>>>> making the
>>>> whole hierarchy just as invalid for computing applications.
>>>> Sum up: If Thing goes as the universal class, of which everything
>>>> is a
>>>> member, it will equivalent to Class, as the class of all classes.
>>>> Other
>>>> interpretations will be inconsistent, asking for many questions.
>>>>
>>>> Hope this will be of use,
>>>> Azamat Abdoullaev
>>>>
>>>>
>>>> ----- Original Message -----
>>>> From: "James Leigh" <james-nospam@xxxxxxxxxxx
>>>> <mailto:james-nospam@xxxxxxxxxxx>>
>>>> To: "Richard H. McCullough" <rhm@xxxxxxxxxxxxx
>>>> <mailto:rhm@xxxxxxxxxxxxx>>
>>>> Cc: "Semantic Web at W3C" <semantic-web@xxxxxx
>>>> <mailto:semantic-web@xxxxxx>>
>>>> Sent: Thursday, August 28, 2008 4:30 AM
>>>> Subject: Re: Thing and Class
>>>>
>>>>
>>>> >
>>>> > Hi Richard et al.
>>>> >
>>>> > Here is an informal interpretation of some of the spec written
>>>> in plain
>>>> > English.
>>>> >
>>>> > Class stands for classification.
>>>> > We use Class to classify things.
>>>> > Class is a set of Things.
>>>> > "I am a Human" - I just classified myself as Human (I hope I'm
>>>> right).
>>>> > "I am a Thing" - that is true for everything.
>>>> > Human is a classification of all people.
>>>> > Thing is a classification of all things.
>>>> > Every Human is a Thing. Therefore Thing is a super set of Human.
>>>> > Is Human a Thing? No! its a Class!
>>>> > Everything Thing is an individual.
>>>> > Human is not an individual, it is a classification of
>>>> individuals.
>>>> > Thing is not an individual, it is a classification of
>>>> individuals.
>>>> > Can we classify Classes? Yes we can! Human is a classification
>>>> - I just
>>>> > classified Human as a classification.
>>>> > Human is a Class.
>>>> > Thing is a Class.
>>>> > Are all Things Classes? No! I am a Thing, but I am not a
>>>> classification.
>>>> > Is Thing the same as Class? No! Human is not a Thing, but Human
>>>> is a
>>>> > Class.
>>>> >
>>>> > Hope this helps,
>>>>
>>>> > James
>>>> >
>>>> >
>>>> >
>>>> >
>>>>
>>>>
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