David Leal wrote:
>> I agree except for one thing - a scale is not a set of items/symbols
>> in itself, but a mapping from a set of "magnitudes of quantity" to a
>> set of items/symbols.
>>
Pat Hayes wrote:
> Well then let us use a different name for that set of items/symbols,
> which is the thing I am wanting to describe for the moment, although I
> would rather not specify its metaphysical nature. What name would you
> suggest? (Scale value?) I will go on using 'scale' until this group
> comes to a different consensus.
>
>
The distinction I would make is that a "scale" associates the "set of
symbols" with a set of interpretations. (01)
The VIM speaks of a "scale" as a "sequence of marks", each "mark" being
associated with a quantity value. That is, the "mark" that is "5" or
"5cm" is associated with "5 centimetres" and therefore with the
magnitude of length that is expressed by "5 centimetres". In a simlar
way, the second of four little lines on my thermometer between the line
labeled "20" and the line labeled "25" is a mark that is associated with
the quantity value "22 degress C" and refers to the magnitude of
temperature that is expressed by "22 degress C". (02)
The fundamental idea of VIM scales is that they are _sequences_ of marks
-- the marks are ordered -- and the mark ordering corresponds to an
ordering of quantity magnitudes. (03)
David's idea that a scale is a _set_ of symbols (marks) is too weak.
> [DL]
>> Hence re-expressing the consensus in these terms we have:
>>
>> scale: a mapping f from Q (set of magnitudes of quantity) to S (set of
>> symbols - commonly numbers), such that:
>>
>> f(q1) = f(q2) if and only if q1 = q2
>>
> [PH]
> Iff is surely too strong. And if true, this makes the two sets S and Q
> exactly equivalent, so why are we talking about two of them?
>
>
Whether this is too strong or not depends on what S is and what f is. (04)
In particular, a scale may be continuous or discrete. It only has the
marks it has. A "continuous" scale has one mark for each real number in
some interval, which makes the quantity values real numbers and creates
a nominal 1-to-1 and onto mapping from symbols/marks to quantity
magnitudes. A discrete scale has marks that are associated with
quantity values whose number parts are integers, or rationals with a
fixed denominator. (05)
For example, my temperature scale distinguishes 21 degrees from 22
degrees, but it has no mark that corresponds to 21.5 degrees -- that
magnitude will be interpreted as either 21 degrees or 22 degrees,
according to some convention. (If I talk about interpolating between
the scale marks, I really mean I have a conceptually continuous scale
that happens to have a physical implementation that only looks discrete.) (06)
So there are two mappings involved here. The principal mapping is:
m maps the set of marks S into the set of quantity magnitudes Q (07)
m is a function (OWL:functionalProperty), and it is 1-to-1
(OWL:inverseFunctionalProperty), but it is not onto unless m is
continuous (and S is 1-to-1 with Real quantity values). (08)
David refers to a function f such that
f maps the set of quantity magnitudes Q to the set of marks S (09)
What David says is only true if m is continuous. (010)
If f is understood to be the inverse of m and S is discrete, then f is
only defined on a countable subset of Q. When q1, q2 are in the image
of m, then David's axiom above holds:
f(q1) = f(q2) iff q1 = q2.
But for arbitrary q1 in Q, f(q1) is almost always undefined. (A
mathematician would say: f is undefined "almost everywhere".) (011)
Now in the discrete case, f may be an "extended inverse" of m, which
maps every q in Q to some s. For example, f may be defined to map q to
the mark s in S such that:
m(s) <= q and m(successor(s)) > q
(On my thermometer, this f maps every temperature in the half-open
interval [21 C, 22 C) to the 21 mark.)
Such an f is a function from Q to S that is well-defined everywhere,
that is:
for all q1, q2 in Q, if q1 = q2, then f(q1) = f(q2)
but David's axiom does not hold: many different q's can be mapped to the
same mark. (012)
David wrote:
> [DL]
>> ordinal scale: a scale where both Q and S are ordered, such that:
>>
>> f(q1) > f(q2) if and only if q1 > q2
>>
According to the VIM, S (the sequence of marks) is always ordered, and
quantity magnitudes of the same 'kind of quantity' are always ordered
(that is what is meant by "comparable"). (013)
The VIM requires:
for all s1, s2 in S, s1 < s2 iff m(s1) < m(s2) (014)
And it is safe to assume that for an extended inverse f
for all q1, q2 in Q, q1 <= q2 iff f(q1) <= f(q2)
But unless m is continuous, and thus f is 1-to-1, we cannot conclude the
inequality:
for all q1, q2 in Q, if q1 < q2 then f(q1) < f(q2) (015)
[DL]:
>> ratio scale: a scale where ratios can be defined for both Q and S,
>> such that:
>>
>> r.f(q1) = f(q2) if and only if r.q1 = q2
>>
> [PH]
> What is r here? Any real number? What does it mean to multiply a
> number and a symbol? And do we really want to say that multiplication
> can be done directly to a magnitude?
>
>
Pat is right. The above is badly misstated. What a ratio scale
requires is:
(1) S is a quantity scale, a sequence of marks denoting quantity values
So each mark is associated with a (number, unit) pair.
(2) The magnitude q expressed by each such quantity value (n, unit) has
the property: m/unit = n
Which means that the notion "ratio of magnitudes" must be well-defined
for the kind of quantity measured by the scale. That is, there is a
function "ratio" that maps any pair of quantities (q1, q2) in Q into
some number field, usually the real numbers. And instead of
ratio(q1,q2), we use the notation q1/q2. This ability to construct a
ratio of magnitudes is a stronger requirement than the ability to
compare them. (Note that m and unit are both magnitudes of the same
quantity kind, which makes m/unit meaningful.) (016)
All the consequences for the scale fall out of these two requirements. (017)
It supports the idea of the scale as a kind of vector space over the
real numbers R in which, for r in R and q in Q, r*q is interpreted to
mean that quantity q1 in Q such that q1/q = r. But that model assumes
that the scale is continuous, i.e., that the number part of the quantity
values is Real, not Rational or Integer, and that m is a continuous
mapping from the number part of each s in S into Q. There are weaker
forms of this that work for rational scales. (018)
[PH]
> In this formulation, the (sets of) magnitudes and symbols are so
> similar that it hardly seems worth distinguishing them. They would be
> isomorphic, considered as mathematical categories. One could for
> example swap them without changing anything. Surely this cannot be
> right.
>
It isn't. (019)
Also, we have to be careful about the meaning of "ratio" for scales that
are logarthmic instead of affine. This is why we need an ontology that
covers the problem space, even if most of our interest is in continuous
scales based on a linear real axis. Such an ontology would allow us to
say that "hardness" satisfies axioms 1 thru 6, but not 7, and therefore
most of the rest of the ontology doesn't apply. (020)
-Ed (021)
--
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 (022)
"The opinions expressed above do not reflect consensus of NIST,
and have not been reviewed by any Government authority." (023)
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