Dear Ingvar, (01)
>"Scales are possible in the first place only because there exists an
>isomorphism between the properties of the numeral series and the empirical
>operations that we can perform with the aspects of objects. This
>isomorphism is, of course, only partial. Not *all* the properties of
>number and not *all* the properties of objects can be paired off in a
>systematic correspondence. But *some* properties of objects can be related
>by semantical rules to *some* properties of the numeral series." (02)
I completely agree with this quote, and hope that nothing that I have said
conflicts with it. What I have called "magnitude of quantity" is called
"aspect of object" in this quote. (03)
We have started at a even lower level than this quote. Our "first place" is
that there is a one-to-one correspondance between the "aspects of object"
and symbols (restricted to numbers in the quote). Only afterwards do we
consider isomorphisms between "properties of the numeral series" and
"empirical operations that we can perform with the aspects of objects". (04)
My proposed definition of ordinal scale: a scale where both Q and S are
ordered, such that: (05)
f(q1) > f(q2) if and only if q1 > q2 (06)
is exactly in line with this quote. The comparison q1 > q2 is an "empirical
operation that we can perform with aspects of objects". The comparison f(q1)
> f(q2) is a "property of the numeral series". The function f is an
isomorphism with respect to order. (07)
Best regards,
David (08)
At 14:41 12/08/2009 +0200, you wrote:
>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. 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
>
>Whatever kind of definition of 'scale' the information sciences in the end
>will find good and useful, everyone ought to be aware of the fact that the
>definition above is not what one finds in traditional philosophy of
>science literature on measurement (whose terminology, BTW, I have been
>using). Here is a quotation from the man (S. S. Stevens) who first made
>the distinctions between nominal, ordinal, interval, and ratio scales
>explicit:
>
>"Scales are possible in the first place only because there exists an
>isomorphism between the properties of the numeral series and the empirical
>operations that we can perform with the aspects of objects. This
>isomorphism is, of course, only partial. Not *all* the properties of
>number and not *all* the properties of objects can be paired off in a
>systematic correspondence. But *some* properties of objects can be related
>by semantical rules to *some* properties of the numeral series."
>
>I have taken the quotation from the latest overview book of measurement
>that I know of: D. J. Hand, "Measurement Theory and Practice. The World
>Through Quantification" (Arnold 2004; quotation p. 41).
>
>If one accepts such a definition of 'scale' (which I do), then David
>Leal's term 'set of magnitudes of quantity' is already implicitly
>presupposing a scale. Without a scale (in the traditional sense) there can
>be no magnitudes.
>
>Ingvar J
>
>
>
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> (09)
============================================================
David Leal
CAESAR Systems Limited
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