Pat Hayes wrote: (01)
> On Aug 11, 2009, at 1:04 PM, ingvar_johansson wrote:
>
>> Pat Hayes wrote:
>>
>>> On Aug 11, 2009, at 5:51 AM, ingvar_johansson wrote:
>>
>>>> 1. Ratio scales, interval scales, and ordinal scales require
>>>> different
>>>> formalisms.
>>>
>>> Surely not. We plan to produce an ontology all written in one
>>> formalism.
>>>
>>> Let me test my understanding of these terms.
>>
>> I can tell you how I have been using the terms, which I think is in
>> conformity with much philosophy of science writing on these issues.
>> But I
>> can't promise that it is in conformity with everything that is written
>> today.
>>
>>> A scale is a set of items used to represent measurements.
>>
>> I would say: 'a set of NUMERALS+UNIT used to represent properties,
>> amounts, measurements or observations'. Even realists (not only
>> operationalists and pragmatists) accept scales.
>
> Well, Im not so concerned with philosophical scruples, but I wonder at
> the notion that every scale has a unit and must use numerals. Is it
> really necessary to assume this? (02)
No, ordinal scales can equally well use, for instance, the alphabet. But
interval scales and ratio scales must have a unit and use numerals since
they allow arithmetical operations. (03)
>
>>
>>> An ordinal scale is a scale with a total order on its elements.
>>
>> I would say: 'a scale with ONLY a LINEAR order on its elements'.
>
> Total = linear; and allowing there to be unused extra structure is
> just good mathematical style. It would be very hard to establish that
> something were an ordinal scale if it were required to have no other
> structure than the ordering. (04)
The point is that an ordinal scale in itself by definition contains no
more structure. If it is possible to add structure, then one may obtain
other and more informative scales such as interval scales and ratio
scales. (05)
>
>>
>>> An interval scale is a scale with a difference function from pairs of
>>> scale items to ... (what? Real numbers?
>>
>> I would say: an interval scale is from a formal point of view an
>> ordinal
>> scale plus a function that tells what is to be regarded as EQUALLY
>> LARGE
>> INTERVALS in the order.
>
> Interesting. So let me press this point a little. Take this example:
> The scale (1 2 3 4 5 6 7 8) where we know that (12) = (4 5) = (7 8)
> and also that (2 3) = (5 6), but we know no other identities
> concerning equally large. Is that an interval scale? (06)
I wouldn't call it an interval scale, since in such it is assumed that all
the intervals you mention are equally large. If a scientist should meet
the kind of situation you present, I think she or he would apply other
numerals to what you call (2 3) and (5 6). (07)
>>
>>> Some other scale? Could there
>>> be an interval scale in which differences were restricted to natural
>>> numbers, for example?).
>>
>> In my opinion: yes.
>
> OK, good.
>>
>>> A ratio scale is a scale which has a zero element and a
>>> multiplication
>>> operation by rational numbers.
>>
>> If it is a scale in physics, then the zero element has from a physical
>> point of view to represent an ABSOLUTE zero of the quantified
>> dimension in
>> question. The interval scales of Celsius and Fahrenheit have a zero
>> element, but these are completely conventional.
>
> Oh sure, I meant a zero in the multiplication operation, not an
> arbitrary scale point. (08)
Interval scales contain by definition no multiplication operation at all. (09)
Ingvar (010)
>
>>
>> With your formal interests, I think you may benefit a lot from
>> looking at
>> Henry Kyburg's
>
> Henry does seem to keep cropping up. He was my departmental chair at
> Rochester, and later a colleague at IHMC. I have looked at this many
> years ago but I should re-read it, indeed, thanks for the reminder.
>
>> "Theory and Measurement" (published 1984, but reprinted now
>> 2009 by Cambridge UP)
>
> Pat
>
>
> (011)
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