various fuzzy matters

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

>la stivn cusku di'e
>> Actually, I'm not sure what
>> la djan clani
>> means.
la xorxes cusku di'e
>It means: "John is long (in longest dimension, by some standard)."
>
Yes, but it would also seem to mean:
"John is short (in longest dimension, by some standard)."

I believe there is a dichotomy here between lojban & English. In English,
saying "John is long" means something like: "In my experience of seeing
people, John is a long person." This implication is *not* present in
lojban, as far as I can tell from the definitions and what I've read so far
about the grammer. I think both you and _and_ are mistranslating between
tall & <clani>.

>> Would this translate as "John has height."?
>
>Not really. A better way to say that would be:
>
>        la djan ckaji le ka ke'a sraji mitre
>        John has the property of being something in vertical meters.
>

This does not seem a better way. I think
la djan clani
means " John has the property/attribute of height." just as
la djan cerda
means "John has the property/attribute of being an heir."

>John "having height" does not imply that he is tall. Even the shortest
>person has height.
>
Agreed. <clani> is a property of tall people and short people alike.

>> la djan cu barda le clani
>>
>> seems closer to "John is tall." in the sense of an unspecified external
>> standard.
>
>{le clani} means "the long one(s)",

Are you sure about this? I think <le clani> means "the one(s) having height".

>so {la djan cu barda le clani} means
>"John is big in the long dimension" which seems fine.
>
Agreed.

>> How about
>>
>> la djan barda le clani mi
>>
>> "John is tall compared to me."
>
>That seems ok. In the case of {barda} the wording of the definition
>suggests that the x3 is what you compare with the x1.
>
>> la djan pi mu xoi barda le clani le cnano
>>
>> "John is to fuzzy extent 0.5 taller than normal."

>> Note how numbers are being used here. There is not a 1:1 correspondence
>> between height and the fuzzy tallness <xoi bardi le clani> of the person!
>> (There seems to be a pervasive misunderstanding of this point.)
>
>I'm afraid that the pervasive misunderstanding persists. 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 categories of fuzzy height in the
interval 160 to 200 cm, then yes. If less, then no. This is what I was
getting at with my note a few days ago about granularity and fuzzy
confidence intervals. I'm not sure of the best method to deal with this.
One way would be to say that you'd have to be within one fuzzy class of the
intended class. Then if we had ten evenly distributed intervals, Mike and
John would be the fuzzily the same height in the example you gave.

>> If we were
>> explicitly specifying a fuzzy tallness function, it might be something
>> like:
>>
>> 0 for all persons shorter than 160 cm
>> 1 for all persons taller than 200 cm
>>
>> linearly increasing from 0 to 1 for all persons between 160-200 cm in height.
>
>Ok, let me get out my calculator, Mike is 1.75m, so in fuzzy he comes out
>to be (175-160)/40 = 0.37. I was wrong, I should have said "Mike is to fuzzy
>extent 0.37 taller than normal", and now we can conclude that he is shorter
>than John, who is 0.5 fuzzy tall (or 1.80m).

Almost. This is a little different than my last lojban example. I would say
Mike is fuzzily-tall to extent 3/8. If he were to grow 5 cm to 180 cm, you
might say Mike is fuzzily-tall to extent 1/2. Note the difference between
"taller than normal" and "fuzzily-tall to extent" (The difference in
calculations is trivial, except that the fractional expression of the
extent of fuzzily tall seems more doable by a person without a calculator.)
>
>Obviously, to use this specific scale we need a calculator handy, so we
>won't be using it in general.

I don't think that a calculator would be necessary, unless you were using a
very fine granularity. Suppose that only three fuzzy categories were used
on the interval from 160 to 200 cm. Don't you think you could do a pretty
good job of classifying people into their fuzzy category without a
calculator? 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.
>
>> Thus, this is an interval scale, not a ratio scale, as we are using
>> arbitrary cutoffs for fuzzy tallness.

I was being too conservative. My scale is at least an interval scale. It
could also be a ratio scale as well.

>>But where did the specific choices of
>> 160 and 200 cm come from? From a prior implicit or explicit understanding
>> between speaker and listener, of course! I might imagine the following
>> conversation:
>>
>> Person 1: I want to talk about human tallness.
>>
>> Person 2: O.K. What fuzzy norm should we use?
>>
>> Person 1: I think that anyone shorter than 160 cm is definitely not tall,
>> and that anyone taller than 190 cm is definitely tall.
>>
>> Person 2: Actually, I would choose 200 as the top cutoff.
>>
>> Person 1: O.K. And I want to use a linear function for instances of tall in
>> the interval 160-200 cm.
>>
>> Person 2: O.K., I accept that. So we agree on a fuzzy norm.
>>
>> Person 1: la djan papino xoi barda le clani le cnano
>>
>> Person 2: go'i
>
>Can you really imagine that conversation taking place in real life?

I made the conversation long, detailed & explicit to demonstrate the idea.
I think we do have these types of conversations fairly often. I think we
*should* have them far more often, as such conversations would eliminate
much misunderstanding. 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.

Remember that both speakers know all these people, so they have referents
for the fuzzy terms they are using. If you asked them to give numerical
heights they would give up, thinking that was too hard. (There are many
possible reasons for this, including, possibly, poor educational methods
regarding quantitative measurements, but that's another story.)

I believe that in such apparently pointless discussions, we are actually
negotiating a mutual fuzzy scale for tallness. Young children seem
particularly fond of such discussions. (I have a young niece and nephew who
were discussing how hot the pizza was with their mother the other day. The
three of them seemed to distinguish 5 fuzzy degrees of hotness inedibly
cold, a little cold, O.K., a little hot, and inedibly hot.) Often we can
successfully omit this negotiation with someone else, relying on common
experience and many prior conversations as a guide. We might have some sort
of internal dialogue:

#Is this other guy a raving lunatic when it comes to matters of height? No,
doesn't seem like it. I guess that my usual fuzzy scale for tallness will
suffice.#

>what about people who can't do all the necessary math in their head?
>Isn't it much easier for us just to talk about the actual heights?
>
This would be another method of discussing the same point. But for some
reason, people seem to prefer fuzziness, and they do the math in their head
automatically. Its amazing how good people are at making measurements, and
how bad they think they are. I think that apparent precision of
calculations such as you made is misleading.). Perhaps a granularity of 8
is stretching it a bit (3/8), but fourths or fifths are doable by nearly
everybody.

>> Of course, speaker and listener might choose to leave the exact fuzzy norm
>> unspecified, or might use a different function;
>
>If they leave it unspecified, how can they understand each other?
>"John is 0.43 fuzzy tall", "No he's not, he's obviously a 0.56",
>"What do you mean 0.56, can't you tell he's a 0.43?" etc.
>
Same method. "John is 2/4 fuzzily-tall" is distinguishable from
"John is 1 fuzzily-tall" Given the appropriate choice of granularity, a
fuzzy lojban speaker ought not be off by more than one fuzzy class. I think
you have convinced me that fractions are better than decimals because they
avoid implying unwarranted precision.

>> in this case, any
>> monotonically increasing function mapping [0,1]:>[0,1] will do.
>
>I guess you mean [0,large enough):>[0,1]

I used a normed domain. Unnormed, it would be [160,200]:>[0,1].
>
>> I am unsure how to elegantly describe the fuzzy function explicitly in
>>>>lojban.
>You could probably describe it as elegantly as in English, but when would
>you want to use it?

I think I want to use it right now! Perhaps we should plug the fuzzy
function into X3 slot of <bardi> where the standard is expressed. How do
you think we could do this? Would we use <linjifancu>? Maybe we don't need
to specify the scale and function type explicitly every time. So all we
would need is a container for 160 and 200.

la djan vofi'uzesi'e xoi bardi le clani < ratio scale>,<linear>,<fuzz 160-200>

John is 4/7 fuzzily tall <using a ratio scale and a linear function where
things get fuzzy between 160 and 200 cm>

In most discourse, the X3 would be omitted, as it would be implicitly
determined by cultural consensus. If there were a problem, then it could
then be made explicit.

For this tall example, the required <fancu> is zero below {160,0}, slopes
up to {200,1}, and then stays at one above 200. There are other
possibilities. Astronauts, for example need to be of "optimal" height. (I'm
going to make these numbers up, but I think I'm close.) If you are shorter
than 150 cm or taller than 190 cm then you can't be an astronaut at all. If
you are between 165 cm and 175 cm you are of optimal astronaut height. So
we have a flat-topped function with sloping sides. If we use a linear
function, it would be mapped out by

{{0,150},{1,165},{1,175},{0,190}}

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.

<flat-topped fuzzy function with corners 150,165,175,190>

Many instances would be simpler. Say that astronauts were optimally 170 cm:

{{0,150},{1,170},{0,190}}

This sort of model is both richer and more intuitive than other ways of
expressing the same idea. Consider, for example using IF THEN ELSE
constructs:

IF height<150 or height>190
 THEN reject
 ELSE
   IF (150<height<165) or (175<height<190
   THEN possible
   ELSE accept (* because 165<height<175 must be true*)

Th IF-THEN-ELSE formulation is unwieldy, the break-points are out of order
and it loses the information that 160 is better than 150.

Imagine something like this:

break-points[160,200]
break-points[150,170,190]
break-points[150,165,175,190]

What could be simpler?

When you start to see things fuzzily, it is really amazing how intuitive it
gets. English gets in the way of fuzzy thinking, partially because we don't
have good predicates and formalisms in English for expressing things
fuzzily. 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.

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