fuzzy matters

["Steven M. Belknap" <sbelknap@UIC.EDU>]

>> >You tell me that
>> >"John is to fuzzy extent 0.5 taller than normal". I then tell you that
>> >"Mike is to fuzzy extent 0.6 taller than normal". Are we allowed to
>> >conclude that Mike is taller than John?
>>
>> Well, it depends on the granularity intended by the fuzzy statement. If I
>> am claiming that there are more than 10 categories of fuzzy height in the
>> interval 160 to 200 cm, then yes, such differences are distinguishable.
>
>I wasn't objecting to the fine granularity. Say you say "John is to fuzzy
>extent 1/2 taller than normal", and I say "Mike is to fuzzy extent 3/4 taller
>than normal". Can we now compare their heights? Of course not, since we don't
>know what each other means by "1/2" and "3/4". In that case, without having
>a pre-agreed upon scale, such numbers cannot be used for comparison.

I really like math. Math has been very, very good to me. In this way, I
belong to a different culture than those who do not like math. So using
numbers in this way seems very natural to me. Could the relative
comfort/discomfort people feel with using numbers this way be a sort of
Sapir-Whorf effect? Suppose that we agreed to use three categories:

slightly taller than normal
moderately taller than normal
extremely taller than normal

Suppose then, that you and I independently assigned 1000 tall persons to
each of these categories. I assert that there would be a correlation
between your assignments & my assignments. Would the correlation
coefficient be 0.99? (that is, very good?) Probably not. Would it be 0.1?
(very bad?) I would be surprised to fuzzy extent 0.9 if it were. I assert
that the correlation would be sufficiently high enough to justify using a
numerical means of ordering this data, and that *even if we had no prior
agreement* that we would be able to meaningfully talk about height in terms
of:

ti pafi'ucisi'e xoi bardi le clani
ta refi'ucisi'e xoi bardi le clani
tu cifi'ucisi'e xoi bardi le clani

Would there be misclassifications? Yes. Would borderline cases be more
likely to be misclassified than middle-of-category cases? Yes. Yet I assert
that such conversations would be natural and intuitive if the linguistic
apparatus was present in lojban to have such conversations. I also believe
that in moving from an ordinal scale (slightly, moderately, extremely) to
an interval scale, that the universe of potential discourse is enriched in
a very important way. For math-nerds like me, it would be great.
>
>> Perhaps if you expressed the 175 cm tall John as 3/8
>> fuzzily-tall + or - 1/8 you would feel more confident about not using a
>> calculator.
>
>I just don't see the point of using the word "tall" in that case. If
>I'm talking about objective heights, even if given unprecisely with large
>error bars, why bring in a subjective word that doesn't add anything?
>
Agreed. But often we don't have exact measurements at hand, yet it might
still be possible to convey meaningful information in the form of an
interval or ratio scale. I think the subjective:objective distinction is a
false dichotomy.

>Say that instead of "tall", which has the objective {mitre} alongside,
>we were to talk about {melbi}. We agree on a scale: Mike is 0-melbi
>and John is 1-melbi. Now I tell you "Susan is 1/2 melbi" and you tell
>me "Phil is 2/3 melbi". Can we meaningfully say that we agree that Susan
>is prettier than Phil? No, because {melbi} is a subjective notion, and
>so my {melbi} is not directly comparable to yours.

Guess what. Someone did this experiment. And guess what they found. It
turns out that there *is* considerable consensus about some aspects of
beauty among observers! Using computer morphic blendings of many different
faces, the investigators concluded that humans are quite sensitive to
asymmetry in human faces. Apparently assymmetry is considered ugly a priori
by the brain. So, perhaps one could use an ordinal scale for beauty. We are
already too dominated by physical appearance as a society, so I would
eschew doing this. I find beauty pageants and the like mildly repugnant.
Yet many humans seem quite fond of these beauty contests. Aren't the judges
just placing the contestants on an ordinal scale?
>
>> The "lets get real" version of the above
>> conversation might go something like this:
>> Person 1: Mary is short.
>> Person 2: She's somewhat short, but not as short as Phil.
>> Person 1: Yeah, Phil's definitely short.
>> Person 2: And that Robert, he's quite tall.
>> Person 1: Do you think so? Elizabeth is taller than Robert.
>> Person 2: Well, sure, Elizabeth is definitely tall.
>> Person 1: Don't you think John is fairly tall?
>> Person 2: I'd say so.
>
>That's fine, no numbers are involved.

Sure they are. Just because person 1 and person 2 are not using a measuring
tape and recording the heights on their laptop computers doesn't mean that
their brains are not measuring the heights. And somewhere in the brain,
this information is represented in some (analogue?) numerical fashion or
other.

>Your assumption is that there is some math to be done when using terms
>like "somewhat", "rather", "fairly", "quite", "reasonably", "definitely"
>and a myriad others. I don't think people are doing math, even
>subconsciously, to come up with those terms. Perhaps you can model the
>use of those terms using math, but that is not the same thing.

A difference which makes no difference is no difference.

>Why not give examples using {melbi} rather than {clani}, which has no
>objective counterpart to rely on.

<clani> is simpler to explain, and people are less emotional about height
than beauty. Except for Peter. (Joking!!, Joking!!) Clarity is served
through the use of a simple example as a demonstration of proof of concept.

>Dividing a continuous
>spectrum into a discrete set can always be done, but that is not what
>fuzzy truth values are about.

We're dividing the continuous spectrum into fuzzy sets, not discrete sets.

>> la djan vofi'uzesi'e xoi bardi le clani < ratio scale>,<linear>,<fuzz
>>160-200>
>>
>> For this tall example, the required <fancu> is zero below {160,0}, slopes
>> up to {200,1}, and then stays at one above 200.
>
>You would need to use the word {mitre} or some other unit, otherwise the
>numbers 160 and 200 are meaningless.

Hmm. You're right. A valid point. As long as people understand that these
are fuzzy meters.

la djan vofi'uzesi'e xoi bardi le fudjimitreranji [1.6,2.0]
(*fudji is a made up gismu. its not in the dictionary. yet.  mitre & ranji are*)

>>
>> if you used a word meaning "flat-topped fuzzy function" you could clearly
>> express this with only 4 numbers and this is about the most complicated it
>> would get.
>
>Says who? Why can't I use non-linear scales, or even piecewise linear,
>e.g. in your notation {{0,150},{1/2,160},{1,180},{1/2,200},{0,210}}?
>There are an infinity of possible scale transformations, but that's
>something for mathematics or for specific applications, not for everyday
>language.

Sure. You could. There might even be situations where these sorts of things
would be necessary. But there could be a simple default. Linear functions
are used most of the time anyway. They're good enough for government work.

>> <flat-topped fuzzy function with corners 150,165,175,190>
>>
>> What could be simpler?
>
>Well, for a start you would have to say whether your starting value
>is 0 or 1 (or something else?). Why are you assuming that the first
>lap (0-150) is flat at 0 and not at 1 or increasing from 0 to 1 or
>decreasing from 1 to 0. Just there you would have four possibilities.

Sure. If someone wants to use a non default, he/she will have to do more
linguistic heavy lifting. As is always the case.

>
>> When you start to see things fuzzily, it is really amazing how intuitive it
>> gets.
>
>It is easy to oversimplify, too.

Agreed. Garbage in, garbage out.

>What seems natural for the heights
>of astronauuts may not work for other things. Isn't it much easier
>to say "you have to be between 150 cm and 190 cm to be an austronaut,
>and optimally between 165 cm and 175 cm", rather than some involved
>convention with flat-topped functions that may not work for other
>things. For example, how would you use functions to say how beautiful
>you have to be in order to be a model?
>
As I recall from college, the male chauvinist method for evaluation of
womanly pulchritude employs a 10 position ordinal scale. Typically the
scores are presented by drunken fraternity members holding up cards with
numbers on them. I thought this was funny at one time, until, to my
surprise, a woman friend of mine actually cried about being scored! A group
of my friends had a long discussion about this issue at the time, and the
women in the group convinced me that this sort of thing isn't humorous, and
is a actually a part of the daily existence of many women. I've indicated
above that I think this is demeaning and insulting to women, as such
scoring implies that the chief value of women lies in their physical
attractiveness. But that's a different soapbox.

>> English gets in the way of fuzzy thinking, partially because we don't
>> have good predicates and formalisms in English for expressing things
>> fuzzily.
>
>I believe English is well equipped to deal with the notion. (Isn't most
>of the stuff about fuzzy logic written in English, anyway?)

Speaking as someone who prefers thinking about the world in fuzzy numerical
terms, I find English to be chafing for expressing these types of ideas. I
prefer either diagrams or mathematical notation. Really, Jorge, English
gets in the way. Most of the articles and books about fuzzy sets get
numerical and use diagrams fairly early on for I believe the same reason.
>
>> If we put such constructs in lojban, I believe we will allow
>> lojban speakers to describe such matters in a manner which is intuitive for
>> human brains.
>
>Most of what you talk about is already there in Lojban. It's more a matter
>of finding it.

We agree. So far only two additions have been proposed: the selma'o <xoi>
and the gismu <fudji>. lojbab has half-convinced me we don't need <fudji>
as <klani> could be used, but he didn't give an example, and I'm still
baffled by <klani>. Even if klani works (fuzzy meter zone =
<klanimitreranji>?) it might be useful to have a separate word for fuzzy.
We still need a formalism for handling numbers, like in this example:

la xorxes pafi'ucisi'e xoi <fudjimitreranji [1.6,2.0]>

Any ideas?


co'o mi'e. la 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