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)
Best,
Leonid - http://ototsky.mgn.ru/it (02)
>
>
> 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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>>
>>
>> --
>>
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>
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