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Guttman scales [long; not too easy] 1 of 4
There has been a debate here recently about truth, fuzzy logic,
numbering systems, and the like.
This debate has to do with how `number' is defined, and how people
perceive scales of measurement. Furthermore, it has to do with how
people make judgements regarding the truth of a predication.
In this message, I am going to discuss the kinds of measurement and
number that people perceive and use. These kinds reflect basic
mathematical entities and influence how people make judgements.
Also, I discuss Lojban briefly (basically, I argue that Lojban already
can express what we want it to), and I list some references .
I sent an earlier version of this message to several people; Steven
Belknap quoted a part of it. I hope you find this version easier to
follow, especially if you are not a mathematician.
I am also sending three separate messages:
* more examples of the scales discussed here [easy].
* a nifty metric for dealing with certainty and uncertainty
[not so easy, but worthwhile].
* the axioms for categorical, ordinal, interval, and ratio scales,
written in a mostly non-mathematical way [but hard nontheless,
unless you studied number theory].
But first, the context.
Context
-------
Peter L. Schuerman wrote:
.. the problem is that the numbers aren't being used as numbers
(in other words, a 3 restaurant isn't really 2 times worse than a
6 restaurant), they're just being used as symbols for words like
"awful", "good", "bad", "wonderful" etc.
Peter is requiring that you should always be able to find a ratio
between two numbers, else the entity is not properly defined as a
number.
He says this even more vividly with regard to degrees of pain:
... I bet that any of your patients could give a description, in
words, of what 4-pain is, and what 8-pain is, but to conclude that
8-pain is exactly twice as "bad" as 4-pain is absurd... pain is a
complex phenomenon, and not numerical in a simple, scalar fashion.
Finally, he says that "You can't do statistics with" this kind of entity:
On a scale of 1 to 10, 1 means "not much" and 10 means "a lot" and
so forth. However, you don't treat these numbers as numbers. You
can't do math with them. You can't do statistics with them.
Peter is right in saying that a `scale of 1 to 10' is not a ratio
scale. And people get into trouble when they think of it as one.
However, mathematicians include in their definitions of number those
entities for which you cannot find a ratio; and you can do statistics
with entitites for which you cannot find a ratio.
For example, a `scale of 1 to 10' is most commonly an ordinal scale, a
ranking. You cannot calculate a mean, standard deviation, or
coefficient of variation on such a scale; those statistics cannot be
applied. But you can find the median of an ordinal scale, which is a
useful statistic.
Likewise, a distinction between "true' and "false" is categorical or
nominal scale, a `go' or `no-go' metric. You cannot calculate a
median on such a scale. But you can calculate the number of cases,
which is also a useful statistic.
Cloudy truths and fuzzy logics have to do with the kinds of metric you
use in your logic. For practical purposes among humans, there are
only four. Louis Guttman pointed them out.
Guttman Scales
==============
In 1944, Louis Guttman observed that all forms of measurement are seen
to belong to one of four types of scale:
* categorical belongs or does not belong to class
* ordinal ranked against other instances
* interval constant distance of unknown size between instances
* ratio comparable in all ways
(Actually, forms of measurement can belong to more than four, but
people conflate them into these four.)
These scales reflect how people perceive the world.
Also, interestingly, and I think, truthfully, Alan Page Fiske
hypothesises that these four mathematical structures underlie the
myriad different ways humans organize their social structures.
What do I mean by `scales'?
---------------------------
You can say this stone weights twice as much as that stone (ratio
scale). This what people most often think of as a `scale'.
But you cannot meaningfully say this Fahrenheit temperature (interval
scale) is twice that temperature since the Fahrenheit scale has an
arbitrary zero. However, you can add ten Fahrenheit degrees to a
Fahrenheit temperature. And there is no doubt that a Fahrenheit or
Celsius scale is useful.
You can say that a captain in the Army is superior (ordinal scale) to
a lieutenant but you cannot say by how much he is superior (and
indeed, the `how-muchness' is irrelevant).
People often do not think of a rank ordering as a scale, but it is a
kind of scale in a mathematical sense. The only purpose of arguing
whether a ranking should be defined as a scale is to clarify what is
meant by the word. Here I am using the word in a manner intended to
show the connections between different mathematical structures.
Finally, you can say that one animal is a cat and another one is a dog
(categorical scale).
These four types of scale are different primary mathematical
structures: equivalence relation, linear ordering, ordered Abelian
group, and Archimedean ordered field. They are different
axiomatically, but all serve as means of measurement.
Here is a table:
Scales of Measurement
====================
Scale Basic Empirical Permissible Statistics Examples
Operations (invariantive)
Name of mathematical
structure
--------------------------------------------------------------------------
Categorical Determination of Number of cases Assign model numbers
(or Nominal) equality Mode Specify species of
Contingency animal
Equivalence correlations
relation
Ordinal Determination of Median Hardness of minerals
greater or less Percentiles Quality of leather,
Order correlation lumber, wool
Linear ordering (type O) Pleasantness of odor
Interval Determination of Mean Temperature
equality of Standard deviation (Fahrenheit and
intervals or Order correlation Celsius)
differences (type I) Calendar dates
Product-moment
Ordered Abelian correlation
group
Ratio Determination of Geometric mean Length, weight, density,
equality of ratios Coefficient of resistance
variation Loudness scale (sones)
Archimedean Decibel
ordered field transformations
The four types of scale or metric themselves form an ordered scale in
which each additional type is distinguished mathematically by being
defined with additional axioms.
Of course, people seldom think of a categorical scale as a scale at
all! After all, the instance is either in or out. But from a
mathematical point of view, an `in/out' category is a type of metric,
merely different from, and with fewer available operations, than a
metric based on rational numbers.
Why are these metrics important? Because they have to do with how you
make judgements yourself and how you communicate to others.
Much progress comes from changing the type of scale used in
measurement:
>From `it is cold outside' (categorical scale)
to `it is colder today than yesterday' (ordinal scale)
to `it is 10 Celsius degrees colder today than yesterday' (interval scale)
to `the thermal energy content of this mass of air
is 2.7% less than it was yesterday'. (ratio scale)
[In another message, I provide more examples of each type of scale.]
Lojban discursives
==================
Lojban has the following discursives:
je'u truly
la'a probably
ju'o certainty
As far as I can see these should work fine with different scales.
{je'u} works with categorical and fuzzy truth.
{la'a} works with probabilities.
{ju'o} works with certainty factors, such as McAllister's.
I don't see any trouble in adapting the existing ordinal scale for attitudes:
__|__________|__________|__________|__________|___________|__________|__
cai sai ru'e cu'i nairu'e naisai naicai
Thus, you can say:
je'u meaning `truly'
je'u sai meaning `fairly true'
la'a meaning `high probability, but less than 1.0'
la'a sai meaning `fairly high probability, but less than {la'a}'
ju'o meaning `certainly'
ju'o sai meaning `fairly strong suggestion of certainty'
Also, you can use subscripts for interval and ratio scales:
ju'o xi pi bi meaning `certainty factor of 0.8'
Of course, this scheme does not define the semantics of each expression.
It does not say that you must interpret {juo} as a McAllister
certainty factor. Nor does it tell you whether the number in the
subscript is a part of an interval scale or a ratio scale.
You could use {klani} for that:
klani
x1 measured thing (quantity of something)
x2 measurement (the number that describes that quantity)
x3 scale (dictates the interpretation of x2)
Or we could go further, as Steven Belknap has suggested. I actually
favor going further, because scales are so fundamental; but lack the
time to offer a strong argument.
Robert J. Chassell bob@gnu.ai.mit.edu
25 Rattlesnake Mountain Road bob@rattlesnake.com
Stockbridge, MA 01262-0693 USA (413) 298-4725
Bibliography;
From:
S. S. Stevens, 1951, _Mathematics, Measurement, and Psychophysics_,
in Handbook of Experimental Psychology_, S. S. Stevens, Ed., NY: Wiley
See also:
Louis Guttman, 1944, _A Basis for Scaling Qualitative Data_,
American Sociological Review 9:139-150
Patrick Suppes, 1957, _Introduction to Logic_, NY: Van Nostrand
S. S. Stevens, 1958, _Measurement and Man_, Scienc 127:383-389
Louis Narens and R. Duncan Luce, 1986,
_Measurement: the Theory of Numerical Assignments_,
Psychological Bulletin, Vol. 99 No. 2, p. 166-180
Alan Page Fiske, 1991, _Structures of Social Life_, NY: Macmillan