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Re: [uom-ontology-std] retitled: magnitude of a quantity

To: edbark@xxxxxxxx
Cc: uom-ontology-std <uom-ontology-std@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Thu, 16 Jul 2009 06:23:15 -0500
Message-id: <3A69F1F5-71F9-4BCC-A2D6-A61FB726F4F9@xxxxxxx>

On Jul 15, 2009, at 6:42 PM, Ed Barkmeyer wrote:    (01)

> David Leal wrote:
>
>>>> Probably we need to look at two "kinds of quantity":
>>>> - categories such as that which includes Ed Barkmeyer's height,  
>>>> width of the
>>>> Thames at London Bridge, the diameter of the earth's orbit;
>>>> - categories such as that which includes ultimate tensile  
>>>> strength, yield
>>>> strength in tension, yield strength in compression (all are  
>>>> stresses).
>> Can anyone elaborate on the distinction here? I don't follow it.
>
> The distinction I see is between 'particular quantities' in the  
> first group, and categories of particular quantity in the second.    (02)

I guess I was assuming there was more to it than this. I can speak of  
the ultimate tensile strength of a given material (or object?) just as  
I can speak of the width of my third finger, and that is just as  
particular a value. David, correct me if I have this wrong, but were  
you meaning to draw a distinction between for example the category of  
lengths, and the category of strengths (stresses?)? If so, can you  
expand on what you see as the difference between them?    (03)

Pat H    (04)

>  Each of the examples in the first group is an individual physical  
> phenomenon; no example in the second group is a particular quantity  
> -- each of the examples in the second group is a class of  
> phenomena.  So if I understand David correctly, one 'kind of  
> quantity' is a class whose instances are individual phenomena; the  
> other 'kind of quantity' is a class whose instances are classes of  
> phenomena.
>
> VIM has only one category of the second kind: 'kind of quantity'.
> The instances of 'kind of quantity' are categories of the first  
> kind: each instance of 'kind of quantity' is a subclass of the  
> general class 'particular quantity'.
> (forall (k)(if (kind-of-quantity k)
>                (forall (x) (if (k x) (particular-quantity x)))
>            ))
>
> "length", as the class of phenomena that are one-dimensional spatial  
> displacements, is a 'kind of quantity'.  That is, "length" is a  
> subtype of 'particular quantity' and an instance of the class 'kind  
> of quantity'.
>
> Further, a 'kind-of-quantity' has the property that all particular  
> quantities of the same kind are comparable.
> (forall (k)(if (kind-of-quantity k)
>                (forall (x y) (if (and (k x) (k y))
>                                  (comparable x y)))
>            ))
>
>>> There is a set of categories that are
>>> 'kinds of quantity', such that all instances of any 'kind of  
>>> quantity'
>>> category are comparable and no pair of instances from two different
>>> kinds of quantity are comparable.
>> Why the second requirement? If length and width are different  
>> quantities (why not?) they are nevertheless comparable.
>
> As I said above, the VIM only introduces one class whose instances  
> are subtypes of 'particular quantity', but that class does not  
> include all subtypes of 'particular quantity'!  To make this clearer  
> we might define 'kind of quantity' with only the axioms above, and  
> then 'length' and 'width' and 'circumference' and 'diameter', etc.,  
> would all be instances of it.
>
> But then we need the concept VIM has:  'special kind of quantity',  
> which is a subtype of 'kind of quantity' with the additional  
> property that no pair of instances from two different special kinds  
> of quantity are comparable.
> (forall (s1 s2)
>         (if (and (special-kind-of-quantity s1)
>                  (special-kind-of-quantity s2))
>             (forall (x y) (if (and (s1 x) (s2 y))
>                               (not (comparable x y))
>                           ))
>         ))
>
> The reason for this is the concept 'system of measurements'.  A  
> system of measurements selects a set of special kinds of quantity to  
> be base quantities (reference dimensions), and it assigns to each  
> 'base quantity' a particular measurement unit to be the reference  
> unit for all instances of that 'base quantity'.
>
> So 'length' is a 'base quantity' or 'reference dimension' of SI and  
> the MKS system assigns 'metre' as the reference unit for 'length'.   
> MKS does not assign units to 'width' and 'diameter' and  
> 'circumference', etc., because they are all subtypes of 'length' and  
> can all be measured in metres.
>
> Now, velocity is not a base quantity; it is a 'derived quantity'.  
> 'Derived quantities' are all the special kinds of quantity that are  
> not base quantities.  Each derived quantity can be defined/expressed  
> as a relationship among base quantities, i.e., in terms of the  
> reference dimensions.  So velocity can be described as length/ 
> duration.
>
> This is the idea that underlies the International System of Units  
> (SI).
>
> We need these "maximal subtypes" of 'particular quantity' to create  
> a uniform scheme for magnitudes of things that can be compared.
> But the VIM has no need to deal with all the other subtypes -- those  
> are just the particular relations, like 'circumference' and  
> 'ultimate tensile strength', which are irrelevant to the system of  
> measurements and the system of units.
>
> Pat noted this, but he devised an alternative model:
>
>> It would make sense to allow each KOQ to have an associated  
>> dimension, but the association be many-one, so that length, width,  
>> distance, etc. all are different KOQs but all associated with the  
>> one physical dimension, which is spatial distance. But that  
>> requires relaxing the 'incomparable with other kinds' restriction  
>> you mention above. NOt sure about this.
>
> The VIM model is 1-to-1 by choosing a maximal KOQ that represents  
> the dimension.  That makes the KOQ and the dimension the same  
> concept.   The alternative model, which Pat sketched, separates the  
> two concepts: KOQ and dimension, and makes the relationship many- 
> to-1.  In that model, there is no concept for the maximal SKOQ, but  
> the comparability property is then based on the dimension, not the  
> KOQ; so the reason for the SKOQ may also disappear.
>
> I hope this is clearer.
>
> One other observation:  SI treats mass and energy as 'special kinds  
> of quantity', with the property that no mass is comparable to an  
> energy. We have known for almost 100 years that this is not correct  
> at the atomic particle level.  But there have been as yet no  
> international commerce measurement issues that have had to be  
> concerned about this. So, an ontology that supports SI implicitly  
> supports only Newtonian physics!  (But mass is a base quantity,  
> while energy is a derived quantity, so perhaps that can somehow be  
> fudged.)
>
> -Ed
>
> -- 
> Edward J. Barkmeyer                        Email: edbark@xxxxxxxx
> National Institute of Standards & Technology
> Manufacturing Systems Integration Division
> 100 Bureau Drive, Stop 8263                Tel: +1 301-975-3528
> Gaithersburg, MD 20899-8263                FAX: +1 301-975-4694
>
> "The opinions expressed above do not reflect consensus of NIST,
> and have not been reviewed by any Government authority."
>
>    (05)

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