Amount Abstractor: ni

["Robert J. Chassell" <bob@gnu.ai.mit.edu>]

You mentioned awhile back some concerns with {ni}.  It is worth
remembering that {ni} must work with all four types of metric that
humans use for measurement.  Here is the brief:

On Guttman scales

A metric does not necessarily require rational numbers.  As Guttman
pointed out in 1944, all forms of measurement belong to one of four
types of scale:  categorical, ordinal, interval, and ratio.
(Actually, forms of measurement can belong to more than four, but
people conflate them into these four.)

The four scales 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.  

Thus you can say this stone weights twice as much as that stone (ratio
scale), but you cannot meaningfully say this Fahrenheit temperature
(interval scale) is twice that temperature since the Fahrenheit scale
has an arbitrary zero.  But you can add ten F. degrees to a
F. temperature.  Similarly, 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).
Likewise, you can say that topaz is harder than quartz (Moh's ordinal
scale of hardness for minerals) but not how many degrees harder.
Finally, you can say that one animal is a cat and another one is a
dog.

Much progress in science comes from changing the type of scale used in
a measurement: from `it is cold outside' to `it is colder today than
yesterday' to `it is 10 F. degrees colder today than yesterday' to
`the thermal energy content of this piece of iron is 0.6% less than it
was yesterday'.

As for truth: if you are using a categorical scale, you may say that a
proposition belongs to the category of truthful propositions or the
category of false propositions.  If you use such a scale, you are not
saying how much truth there is in a proposition, only that it is true,
not false.  Much logic is based on there being only two categories,
true and false; it makes the mathematics simpler.  The various fuzzy
logics are a formal attempt to add interval or ratio scales to logic.

Or you can say that this first proposition is more credible than that
second proposition, and that second proposition is more credible than
a third.  This is an ordinal scaling.  In a court case, a jury may
have to judge whether one person's testimony is more credible than
another's (ordinal scale) so as eventually to place the defendant in
one of the categories `guilty' or `not guilty'.

In artificial intelligence programs, numbers may be used to indicate
the quality of the evidence for a proposition.  Even though the
numbers appear to suggest a familiar ratio scale, as used in measuring
weight or density, the computer program often limits operations on the
the numbers to a more restrictive set of axioms than that used by
rational numbers.


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                   e.g. Model T Ford
                                 Contingency          Specify species of 
              Equivalence          correlations         animal, e.g. cat  
                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


  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