Re: guttman scales

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

   Having now read Chassell's essay on Guttman scales, I find it
   comprehensible, but cannot quite place the kind of fuzzy category
   we use for colors as being one of the 4 types.

   ... The essential paradigm for colors I am envisioning is
   essentially what Chassell describes as a categorical scale.

Right.  Colors such as `green', `blue', are different categories.  The
Sapir-Whorf effects have to do with how people are influenced by the
language they speak in fixing exactly where a boundary lies.  Painters
and other experts see more categories, and may reach common agreement
on fixing boundaries that are unknown to you or me.

As an everyday matter, I personally tend to think of colors as part of
a spectrum with a specific sequence: xunre, narju, pelxu, crino,
blanu, zirpu.  In this everyday thinking of mine, the scale is
ordinal.  Blue is `more' than green, which is `more' than yellow, but
I don't know how much, except in a vague ratio-scale way, that violet
has not quite twice the frequency of red.

When I have used a spectroscope, I have actually been able to relate
colors to wavelengths, and treat them as a ratio scale.

In _The Adapted Mind_, by Barkow, Cosmides, and Tooby, R. N. Shepard
maps color categories onto a color sphere in terms of lightness, hue,
and saturation, in which each axis is a ratio scale.  The boundaries
of each color are vague, but people tend to have `best example' focal
colors.

Shepard argues that three dimensions for color enable a human viewer
to perceive a surface as looking the same under widely differing kinds
of illumination -- direct sun (yellowish), indirect sun (bluish), dawn
and dusk (reddish).  Two dimensions do not work as well three,
although two are not too bad (hence the large numbers of color blind
people).


   What I see coming out of all this is a kind of n-dimensional categorical
   "scale" with a variable number of categories, ...

Yes; different numbers of categories, and different scales, in
different circumstances.

   To simply call this a categorical scale oversimplifies, because the
   number of categories is not fixed and the "distance" between them is
   also not fixed.

Correct.  As soon as you start comparing categories, you are working
with some other scale, often a multi-dimensional ordinal scale.


   ... for any object, once you have decided it is "red", then you
   also have implicitly decided that a whole bunch of objects that are
   "more red" on some ordinal scale ... are also "red".  And this is
   NOT a property of a simple categorical scale, as I understand it
   from Bob's description.

Correct.  When you sort apples into a box for red apples, you compare.
But in the end, all the red apples are identical in the only way that
matters in this case, which is that they are in the `box for red
apples' rather than being in the `other box'.

As for fuzziness; don't confuse `fuzziness' with type of scale.
Fuzziness is something else.  Fuzzy logic uses non-categorical scales
in conjuction with categorization as a tool for circumstances where it
is inconvenient to be tied to logics involving pure categorization.


    Robert J. Chassell               bob@gnu.ai.mit.edu
    25 Rattlesnake Mountain Road     bob@rattlesnake.com
    Stockbridge, MA 01262-0693 USA   (413) 298-4725