Axioms for Guttman scales [short; hard] 4 of 4

["Robert J. Chassell" <bob@GNU.AI.MIT.EDU>]

This is the 4th of 4 related messages.

This describes the axioms for Guttman's categorical, ordinal,
interval, and ratio scales; it is written in a mostly non-mathematical
way.

See my three other messages regarding Guttman scales.


A categorical scale requires following axioms;

  * reflexive, such that an instance is identical with itself;

  * symmetric, such that when one instance is identical with a second,
    the second is identical with the first; and

  * transitive, such that when one instance is identical with a
    second, and the second with a third, the first is identical with
    the third.

An ordinal scale requires additional axioms: three that are variations
on the first three, plus:

  * connected, such that every instance fits into line somewhere; none
    are undefined

An interval scale requires yet more axioms:

  * additive identity, such that the addition of nothing to an
    instance leaves the instance as it was

  * associative, such that variations in the clumping of instances
    does not change the total (if I give you two mice and then three
    mice, you receive the same number of mice as if I had given you
    four mice and then one mouse.)

  * inverse, such that you can take away an instance, so it is not
    there any more.

  * communitive, such that it does not matter to the total if I give
    you first three then two mice, or first one then four mice.

(These axioms produce what is called an Abelian group.)

  * order preserving, such that if I give you three mice and then one
    mice, you end up with fewer mice than if i give you four mice and
    then one mouse.

Finally, a ratio scale requires all that goes before, plus:

  * a second kind of Abelian group, witha different kind of operation,
    such as multiplication/division rather than addition/subtraction

  * distributivity, which is a way to clump the two operations
    together so that different clumpings are identical,
    most commonly: ab + ac = a (b + c)

  * order preservation, as before, but more sophisticated

  * the Archimedean property, such that there are no infinitisimally
    small instances that you cannot perceive

The operations need not be the everyday addition, subtraction,
multiplication, and division with which we are familiar, but must fit
the criteria for operations as defined by the axioms.