On Sep 10, 2009, at 2:55 AM, David Leal wrote: (01)
> Dear Hajo,
>
> This is on fringe of set theory - see http://en.wikipedia.org/wiki/Power_set
> .
>
> The powerset (or powerclass) of the set S is the set of all subsets
> of S.
> Sometimes the powerset/powerclass of S is denoted S*. (Neither the
> term
> "powerclass" nor the notation S* is used in the Wikipedia page.) (02)
BTW, in case y'all ever come across it, this is also sometimes written
as 2|S, ie 2 to the power S. This is because a subclass of S can be
thought of as a function from S to the set {0,1} - which is the
standard way in set-theoretic foundations of mathematics to encode the
number two - the elements of the subset being those elements of S that
are mapped to the ! rather than the 0. This is why it is called the
'power set'. (03)
BTW, the usage of "meta" in your diagram is highly nonstandard in set
theory and logic, though I know it is widely used in current ontology
engineering. I mention this only to avoid possible future
misunderstandings. (04)
Pat Hayes (05)
>
> Hence if S is {a, b, c}, then S* is {{}, {a}, {b}, {c}, {a, b}, {a,
> c}, {b,
> c}, {a, b, c}}.
>
> L* is the set of all subsets of "length as a class of particular
> quantity".
> Therefore any set of particular lengths, chosen on any basis, is a
> member of L*.
>
> A useful set of particular lengths can be defined that consists of ALL
> particular lengths that are equal in magnitude to a chosen reference
> particular length. The set of particular lengths called "1.3 metres"
> is an
> example of this. This set is a member of L*.
>
> The set of all sets of this type with different magnitudes is a
> subset of
> L*. This subset is called "length as a class of magnitude" in the
> picture.
>
> The object "the metre" is not a particular length but a set of
> particular
> lengths that are equal in magnitude. Therefore it is a member of
> "length as
> a class of magnitude".
>
> The set "length as a class of magnitude" seems to have some useful
> properties which enable us to define one member with respect to
> another.
> Hence we can define the member "1.3 metres" with respect to "the
> metre".
>
> Best regards,
> David
>
>
> At 11:11 10/09/2009 +0200, you wrote:
>> Dear David,
>>
>> Thanks for the picture showing the objects and their meta-levels.
>> Please
>> allow me to ask a few questions, for my understanding:
>> 1. What is the difference between Q1* and Q1?
>> 2. What is L*? A symbol, perhaps?
>> 3. What is the difference between Q3 and Q3*?
>>
>> Thanks in advance.
>>
>> Best regards, Hajo
>>
>> ________________________________
>>
>> Van: uom-ontology-std-bounces@xxxxxxxxxxxxxxxx namens David Leal
>> Verzonden: zo 6-9-2009 11:36
>> Aan: uom-ontology-std
>> Onderwerp: Re: [uom-ontology-std] FW: Quantity kinds
>>
>>
>>
>> Dear Mike,
>>
>> Alas, everything is abstract except the members of
>> particular_quantity.
>> The
>> different abstractions are at different meta-levels, which it is
>> why it
>> is
>> so confusing.
>>
>> I have taken Hajo's example, and shown the objects and their meta-
>> levels
>> in
>> http://www.caesarsystems.co.uk/uom_ontology_progress/uom_ontology_progre
>> ss-f
>> iles/quantity_meta-levels.gif .
>>
>> The bottom left of the diagram has exactly what Hajo wishes. The top
>> right
>> of the diagram is entirely abstract and may seem unnecessary, but I
>> do
>> not
>> think that this is so. Its utility is justified as follows:
>>
>> 1) "1.3 metres" and "the metre" are both members of "magnitude of
>> quantity"
>> and of "length as a class of magnitude" (neither are members of
>> "length
>> as a
>> class of particular quantity" because both are abstractions and not
>> members
>> of "particular quantity");
>> 2) that "1.3 metres" can be expressed in terms of its relationship
>> with
>> a
>> unit, such as "the metre", is a property of the class "length as a
>> class
>> of
>> magnitude";
>> 3) a subclass of "Q4, kind of quantity as class of magnitude" can be
>> defined
>> that contains all members for which a simple relationship with a unit
>> exists.
>>
>> Best regards,
>> David
>>
>> At 00:39 05/09/2009 +0100, you wrote:
>>> Would it be less confusing if one of those had a parent class of
>>> "Abstract Thing" and the other had a parent class of " Concrete
>>> thing"
>>> or "Particular thing" or some such? Because I think you've hit the
>>> nail
>>> on the head about what the actual distinction is, and therefore it
>>> should be possible to represent that in the upper ontology.
>>>
>>> More broadly, all these distinctions would surely lend themselves to
>>> modeling such that the model, and not the reader's understanding of
>> some
>>> paragraph, is what captures or fixes the meaning?
>>>
>>> Mike
>>>
>>> Ed Barkmeyer wrote:
>>>> Rijgersberg, Hajo wrote:
>>>>
>>>>> 1. Maybe the most important question is (and probably only for my
>> information and understanding): why do we base the vocabulary on VIM?
>>>>>
>>>> VIM is the International Vocabulary for Measurement, a
>>>> publication of
>>>> BIPM -- the people who maintain the international standards for
>>>> units
>> of
>>>> measure. http://www.bipm.org/en/publications/guides/vim.html
>>>> It seemed like adopting their vocabulary would be good, and we have
>>>> tried to stay close to that.
>>>>
>>>>
>>>>> (I'm very interested in studying VIM, but couldn't find the
>> document.
>> Could someone please send me a link?) There are also other documents
>> that we
>> could base our vocabulary on.
>>>>>
>>>>>
>>>> Yes. For example, we could use the model in DOLCE or Cyc or SUMO
>>>> or
>>>> some other published upper ontology that provides formal axioms for
>> the
>>>> quantity and measurement concepts. We have a work item that says
>>>> we
>>>> should look at these.
>>>>
>>>>
>>>>> 2. But then I ignore VIM, and repeat myself: couldn't we have:
>>>>>
>>>>> "length of my table" ---"member of" ---> "length quantity"
>>>>> "length quantity" ---"subclass of" ---> "quantity"
>>>>>
>>>>> This is in accordance with some standard documents I know.
>>>>>
>>>>>
>>>> And it is in accordance with the model that David has produced.
>>>> But
>> he
>>>> uses "instance of", instead of "member of" ("instance" is an
>>>> "intensional" characterization; "member" is an "extensional"
>>>> characterization; although probably only an ontology purist would
>> care.)
>>>>
>>>>
>>>>>
>>>>> 3. Also, according to these documents, e.g. "1.3 metre" is not a
>> quantity, but a measure.
>>>>>
>>>> Agreed. VIM calls it a "quantity value", which is an "expression
>>>> of
>> the
>>>> magnitude of a quantity". VIM uses "measure" to talk about a
>>>> process
>>>> for determining a quantity value and for the results of that
>>>> process.
>>>>
>>>>
>>>>> (A quantity is like a variable, it is a reference to a value, like
>> "1.3
>> metre".)
>>>>>
>>>>>
>>>> We are careful to distinguish the class "quantity", whose instances
>> are
>>>> particular quantities, like "the length of your table", from the
>> class
>>>> "quantity magnitude", whose instances are the abstractions --
>>>> amounts
>> of
>>>> length stuff. So if you have two chairs next to your table and they
>> are
>>>> identical in size and shape, "the height of the first chair" is a
>>>> (particular) quantity, and "the height of the second chair" is a
>>>> (particular) quantity, and they are _different_ instances of
>> "quantity",
>>>> but they have the same "magnitude". That is, there is one amount
>>>> of
>>>> 'length' that they both have.
>>>>
>>>> "1.3 metre" is a quantity value, which is an expression/name for a
>>>> "quantity magnitude". If your chairs are 1.3 metres high, then
>>>> "1.3
>>>> metres" is a name for the magnitude of the height of each chair.
>>>> And
>> of
>>>> course, the same magnitude can have other names, such as "51
>>>> inches".
>>
>>>> "51 inches" and "1.3 metres" are two different quantity values, but
>> they
>>>> express the same "magnitude". That is the model we are proposing.
>>>>
>>>> But then there are two (or more) different classifiers that are
>> called
>>>> "length". length-1 is a subclass of "(particular) quantity". The
>>>> length of your table is a length-1, the height of the second
>>>> chair is
>> a
>>>> length-1. But the magnitude of the height of the second chair is
>>>> not
>> a
>>>> length-1. length-2 is a subclass of 'quantity magnitude'. The
>>>> magnitude of the height of the second chair, which is the same as
>>>> the
>>>> magnitude of the height of the first chair, is a length-2. So we
>> must
>>>> choose one of these to be what we mean by "length". VIM is very
>> clear
>>>> that it means length-1. But then we must not say that "1.3 metres
>>>> expresses a length"; we must say (as VIM does) that "1.3 metres
>>>> expresses the magnitude of a length".
>>>>
>>>> I personally think this usage is confusing for everyone. I would
>> prefer
>>>> that we use "particular quantity" to be the class of things like
>>>> the
>>>> height of the second chair, and use "quantity" to be what VIM calls
>> the
>>>> "magnitude". But, on the other hand, you can't ever measure a
>>>> "magnitude"; you can only measure a particular quantity -- the
>>>> height
>> of
>>>> the second chair. The "magnitude" itself is not a physical
>> phenomenon;
>>>> it is an abstraction. So it is very important that we agree on
>>>> what
>>>> definition we choose for each term we use. Otherwise, everyone is
>> confused.
>>>>
>>>>
>>>>> Again, hopefully this discussion is appreciated; it is meant to be
>> constructive, with the aim of working together to achieve a high-
>> quality
>> vocabulary.
>>>>>
>>>> Me, too. But we keep going around these same issues, because
>>>> people
>>>> don't understand the definitions.
>>>>
>>>> -Ed
>>>>
>>>>
>>>
>>>
>>> --
>>> Mike Bennett
>>> Director
>>> Hypercube Ltd.
>>> 89 Worship Street
>>> London EC2A 2BF
>>> Tel: +44 (0) 20 7917 9522
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>>> www.hypercube.co.uk
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>>>
>>>
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>>>
>>
>> ============================================================
>> David Leal
>> CAESAR Systems Limited
>> registered office: 29 Somertrees Avenue, Lee, London SE12 0BS
>> registered in England no. 2422371
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>> mob: +44 (0)77 0702 6926
>> e-mail: david.leal@xxxxxxxxxxxxxxxxxxx
>> web site: http://www.caesarsystems.co.uk
>> <http://www.caesarsystems.co.uk/>
>> ============================================================
>>
>>
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>
> ============================================================
> David Leal
> CAESAR Systems Limited
> registered office: 29 Somertrees Avenue, Lee, London SE12 0BS
> registered in England no. 2422371
> tel: +44 (0)20 8857 1095
> mob: +44 (0)77 0702 6926
> e-mail: david.leal@xxxxxxxxxxxxxxxxxxx
> web site: http://www.caesarsystems.co.uk
> ============================================================
>
>
>
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> (06)
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