guttman scales

[Logical Language Group <lojbab@ACCESS.DIGEX.NET>]

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.  Maybe if someone can explain
colors to me in terms of Guttman scales, I will feel I understand
Guttman scales.  The predicate "tall" is probably easier, but I would
also like to see it explained.

The essential paradigm for colors I am envisioning is essentially what
Chassell describes as a categorical scale.  There are a large number of
colors, and one presumably assigns a colored object as being in one of
those categories.  But this is NOT what I think happens.  First of all,
the semantic space of the different color categories overlap - certainly
for different people, and I think perhaps even for the same people with
certain colors.  Example:  pink and red.  For Russians everything pink
is red because Russian does not have a major color word for "pink".  But
do two English speakers necessarily agree on the boundary between pink
and red - I suspect that the boundary is quite fuzzy, and that as you
move from a pure pink to a pure red, you will start getting some number
of English speakers who will start calling the color of the object "red"
rather than pink.  I am fairly sure that this gradient is not an
interval scale per Guttman, but I am not sure whether it is an ordinal
scale.

Complicating this, there are English speakers with more categories than
others - who might start throwing in other color words like "rose" or
"cerise" or "carmine".  As an occasional stamp collector, I am familiar
with the fact that such collectors use perhaps several dozen color
words, most of which I have only a vague definition of unless I refer to
a color reference chart.

Now someone who adds in "carmine" into their color vocabulary,
presumably modifies the boundaries of what they call "pink" and "red",
in order to fit that color into the scale.  And if forced to call
something carmine either "red" or "pink", I suspect that they will
choose one or the other of the two, and call all things within their
"carmine" domain either "red" or "pink".  This has thus changed the
boundary between red and pink, since those boundaries were adjusted in
making room for carmine in the first place.

The complicating factor is that I might be more likely to use "carmine"
if asked to describe a stamp color, than, say, if asked to describe the
color of a bird, an autumn leaf, or the sky at sunset, and the
boundaries might be different for me as a sole observer even if I would
use the color name "carmine" in all of those circumstances.

What I see coming out of all this is a kind of n-dimensional categorical
"scale" with a variable number of categories, and each category serving
as a semantic gravity well of some strength depending on the importance
of the color word - "Red" is more all encompassing a color than "myrtle
green".  The result of this dimensionality is something neither
categorical or ordinal.  When we want to assign a color label (or
predicate) to something, we implicitly measure it on SEVERAL ordinal
scales, one for every color word we consider as possibly applying to the
domain of the object.  And we then categorize the object based on which
categories score the highest ordinal "score", where that score may be
weighted by the "gravity" or power of the color word in description.

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.  I also suspect that the pattern of color selections is
closer in mathematical properties to an ordinal or interval scale.

For the simpler case of "tall and "short" (or "not-tall"), I think we
only have two categories, but again, the boundary between the two is
context and indvidual dependent.  I am "tall" according to my 5'2" wife,
but I don't think of myself, at 6'1 as being "tall".  My daughter at
4'8" is tall when the context is 9 year old girls, but not-tall in terms
of general humanity.  Clearly an ordinal situation occurs here in which
it IS possible to rate any two objects such that one is taller than the
other by some amount (assuming we aren't measuring explicitly), but
there is an underlying pair of categories "tall" and "not-tall" that
correlates with this ordinal scale, but with a fuzzy context dependent
boundary.  Once you have set that boundary for a given context, you KNOW
that all object "taller" than any object that is in the category "tall",
is also "tall" - hence the ordinality of the scale still applies, and
perhaps, if a ruler is used, interval or even ratio can be applied.

The color situation seems to be to also have this property - that 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 (or even interval or ratio scale if you are using the
standard hue/saturation style of color measurement) are also "red".  And
this is NOT a property of a simple categorical scale, as I understand it
from Bob's description.

Is there a 5th type of scale animal here?  Has mathematics attempted to
describe this kind of space/scale?  And what are the implications for
"fuzzy logic" and "fuzzy Lojban"?

lojbab