David Leal wrote: (01)
>>> 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. (02)
The distinction I see is between 'particular quantities' in the first
group, and categories of particular quantity in the second. 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. (03)
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)))
)) (04)
"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'. (05)
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)))
)) (06)
>> 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. (07)
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. (08)
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))
))
)) (09)
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'. (010)
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. (011)
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. (012)
This is the idea that underlies the International System of Units (SI). (013)
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. (014)
Pat noted this, but he devised an alternative model: (015)
> 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. (016)
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. (017)
I hope this is clearer. (018)
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.) (019)
-Ed (020)
--
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 (021)
"The opinions expressed above do not reflect consensus of NIST,
and have not been reviewed by any Government authority." (022)
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