Re: various fuzzy matters

[Jorge Llambias <jorge@PHYAST.PITT.EDU>]

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

No, that's {tordu}.

"x1 has length" is given by {mitre} or {gutci} or {minli}.

{clani} and {tordu} have the added subjective information that the
length is large or small, respectively.

> >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 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.

> 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?

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.

> 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. John is fairly tall and Robert
is quite tall. I don't know which one is taller, but I know what you
mean. If you had given them numbers, I would know who was taller, but
I couldn't use the scale myself, if you wanted me to be able to make
comparisons with people I know but that you don't

> >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.

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.

> Its amazing how good people are at making measurements, and
> how bad they think they are.

But subjectivity doesn't have to do with making measurements. How can
you measure beauty?

Why not give examples using {melbi} rather than {clani}, which has no
objective counterpart to rely on. You seem to be using the subjective
word {clani} with the objective notion {mitre}. Dividing a continuous
spectrum into a discrete set can always be done, but that is not what
fuzzy truth values are about.

> 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. Given that, isn't it much easier
to stick with it rather than convert to other units?

> {{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.

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.

> <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.

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

It is easy to oversimplify, too. 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?

> 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?)

> 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.

Jorge