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Guttman Scales

There are several newgroups and www pages devoted to discussing fuzzy
logic, including relative merits, etc.
for example is pretty decent. The NASA shell fuzzyCLIPS is interesting.
(ai.iit.nrc.ca/fuzzy/fuzzy.html) You can download some software which does
fuzzy stuff if you want. In my experience, Mathematica is the best tool for
doing things fuzzy, although there are several other pretty good packages
for MatLab and (I hear) also for Maple. I would refer those interested in
learning more about fuzzy logic (or in trashing fuzzy logic) to those
sites. I am more interested in how fuzzy logic can be elegantly described
in lojban.

There are broader issues raised by the fuzzy logic question which may also
be of some importance to lojban's implementation of numerical description.
I have been reading the most recent preliminary version of the document
describing mathematical description that John Cowan just uploaded - _lojbau
mekso: Mathematical Expressions in Lojban Revision: 3.31_ contained in
mex.tex on the lojban ftp site:
(If you try to menu your way into this paper you will get stuck as there is
some kind of glitch, so just list the whole entry down to access/lojbab
from the get-go and you will get to the ftp index where mex.tex appears.)

This document is an interesting description of the mekso system in lojban.
Its quite well written. Its author acknowledges the absence of much
experience (by anyone) with usage of the mekso. I think it would be
interesting to explore mekso usage, so perhaps some of you could comment on
some things I've been thinking about, and how they could be well-expressed
in lojban.

Robert Chassell recently told me about the Guttman Scales, which is
apparently the original source of an interesting meme I remember from high
school physics.  According to Guttman, [Louis Guttman, 1944, _A Basis for
Scaling Qualitative Data_, American Sociological Review 9:139-150]
measurement scales come in four flavors: nominal, ordinal, interval, and

Some Examples:

 Scale      Example
Nominal    Animal species
Ordinal    Mineral Hardness
Interval   Celsium Temperature
Ratio      Length, Mass, Density

Corresponding mathematical structures and valid operation:

 Scale       Mathematical Structure       Operations
Nominal     Equivalence Relation        a=b?
Ordinal     Linear Ordering             a > or = or < b?
Interval    Abelian Ordered Group       a1-a2 > or = or < b1-b2?
Ratio       Archimedean Ordered Field   a1/a2 > or = or < b1/b2


 Scale     Descriptive      Tendency       Distribution
Nominal   Frequency        Mode           Membership (True/False)
Ordinal   Cumulative Freq  Median         Standard Deviation
Interval                   Mean           Coefficient of Variation
Ratio                      Geometric Mean

Robert Chassell on Guttman Scales:
>The four Guttman 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
>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.

The controversy regarding the utility of fuzzy logic is centered on which
scale is appropriate to use for logical analysis. Aristotle proposed a
principle now called the law of the excluded middle which stipulates that
an element must either be a member or a non-member of the set, that is, it
must be one or the other, and it can't be both.

In the Aristotlean formulation there are actually four states which can
obtain regarding set membership: True, False, Indeterminate, & Undecidable.
Indeterminate applies to situations when there is not enough information;
Undecidable applies to situations when there is paradoxical or conflicting
information. Thus, Aristotlean logic is an example of a four-valued nominal
scale. (Some might object to my inclusion of Indeterminate & Undecidable as
scale states because they are metareferential.)

The USA criminal justice system is often also cited as being an example of
a nominal scale: Guilty, Innocent, & Mistrail being the possible outcome
states. But perhaps this is not actually true: many criminal cases are
resolved through plea-bargaining, in which the quality of the evidence, the
heinousness of the offense, the political sensitivity of the crime, the
closeness to election time, and other factors are used during negotiations
by prosecutors and defendants (or their proxies) to calculate the
punishment. Also, sentencing is separate from judgement, and sentencing
outcome varies within limits set by law.  The scale of criminal justice
seems more like an (imperfect) interval scale than a nominal scale.

There is an evolving proposal for an ordinal fuzzy scale, which I believe
xorxes introduced several months ago in response to fuzzy ship of Theseus
tread, and which <lojbab>, <and>, <jcowan> and others have been shaping. It
goes something like this:


This is an ordinal scale, but is not an interval scale, since the elements
do not form an Abelian ordered group: no requirement is being made of being
evenly distributed along the range of false to true. These constructs are
useful, but there is some disagreement as to whether implementation in
<jeu> is the proper course.

Several months ago, when I was thinking about the Fuzzy Ship of Theseus, I
tried to find a gismu which would allow me to express an interval scale,
but came up empty. So did jorges, which considering his mastery of lojban
is more relevant than my own failed efforts.

mi cusku di'e
>> It is possible that there is some easy way to do this already in lojban,
>> although I haven't seen any posts that quite get at what I want to do.
la xorxes cusku di'e
>What predicate would you like in order to say that? Something like:
>        x1 is x2 units (default=1) on scale/property x3
>Unfortunately, I can't find a gismu with that structure. I remember that
>someone once proposed that {gradu} should have that structure, but it
>seems that it wasn't changed. That the place structure of all the
>specific units, it is really weird that the generic unit doesn't have
>that structure.

Agreed. Is there a good reason why gradu has its assigned place structure?
It seems less useful than it could be, and seems inconsistent with related
gismu. The nipponized english for fuzzy is something like <fudje>.I believe
that the Chinese and Russian words are also derived from the English (but I
don't know what they are) Would <fudje> be a possible lojban word for fuzzy
logic? What score would it get?

mi cusku di'e
>> Now, perhaps I saw some shadows which on a 0 to 7 scale I am 4ish certain
>> corresponded to the defendant and the errant politician, and suppose I saw
>> these shadows perform the alledged defenestration to certainty 3ish on a 0
>> to 4 scale.
la xorxes cusku di'e
>If I understand correctly, here you want an observative to mark how certain
>you are of what you are saying. How about this:
>        sei vofi'uzesi'e selbirti
>        With 4/7th certainty
>        sei cifi'uvosi'e selbirti
>        With 3/4th certainty

The use of rational numbers in this way seems like a good way to express
ratio scale things, particularly if this were combined with and's

la and cusku di'e
>I am persuaded by these arguments, and conclude
>that we therefore need a new one-member selmao that takes
>a number expression (like MOI and ROI do) and yields a NA.
>I will call this cmavo {xoi} in selmao XOI.

> mi pi mu xoi clani

"I am 0.5 tallish" or "I am fuzzily tallish to extent 0.5"

>pi mu xoi ku mi clani

Perhaps xoi could be one of a multi-member selmao which could distinguish
which of the 4 Guttman scales was being used.

mi <nominal selmao> clani

"I am categorically tall"

mi cifi'uvosi'e <ordinal selmao> clani

"On a (not necessarily evenly distributed) scale of 7 parts,I am 4 tall."

mi muzefi'ubiresi'e <interval selmao> clani

"On an (evenly distributed, but not necessarily ratioed) scale of 82 parts
I am 57 tall."

 mi cifi'uzesi'e <ratio selmao> clani

"On a ratioed scale of 7 parts I am 3 tall"


"I am 0.5 tallish."

(The above two translations are both accurate. A ratioed seven point scale
would have values 0,1,2,3,4,5,6. Because it is a ratio scale, we can
legitimately divide numerator by denominator.)

Thus, the meaning of <scale unspecified> utterances would be unchanged;

do clani

"You are (without reference to a particular scale) tall."

I probably have messed up something about selmao, but I hope the benefits
of this or some similar scheme are clear.

co'o mi'e. stivn.

Steven M. Belknap, M.D.
Assistant Professor of Clinical Pharmacology and Medicine
University of Illinois College of Medicine at Peoria

email: sbelknap@uic.edu
Voice: 309/671-3403
Fax:   309/671-8413