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. (01)
Pat
>
> (02)
> 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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