Message-Id: <199602090250.EAA27255@xiron.pc.helsinki.fi>
Date:         Thu, 8 Feb 1996 20:52:38 -0600
Reply-To: "Steven M. Belknap" <sbelknap@UIC.EDU>
Sender: Lojban list <LOJBAN@CUVMB.BITNET>
From: "Steven M. Belknap" <sbelknap@UIC.EDU>
To: Veijo Vilva <veion@XIRON.PC.HELSINKI.FI>
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la djan ko.uin cusku dihe
>I am now proposing the addition of two conventions for handling fuzzy logic:
>
>1) for fuzzy truth statements, a convention that "je'a xi <number>" reflects
>the degree of truth of the statement in which it is embedded, thus:
>
>1)      mi je'a xi pinomu blanu
>        I am (5%-true) blue.

I'm still spending a lot of time trying to puzzle out exactly what the
tersely defined cmavos mean, so maybe I'm really misunderstanding things.
As I understand it, <jeha> is a cmavo of the NAhE selmaho class, thus it
would apply to the bridi, in this case, <mi blanu>. <xi> is the subscript
operator which attaches a number or letteral string onto a grammar
structure; presumably <xi> is attached to <jeha>, rather than <blanu>.
Thus, the naive reading would be:

<mi jeha xi pinomu blanu> <-> <jeha xi pinomu mi blanu>

which means something like:

"In the affirmative to the extent 0.05, I am blue."

or

"The statement 'I am blue' is five percent true."

Presumably, this would be logically equivalent to:

<nahe xi pisomu mi blanu>

"The statement 'I am blue' is ninety-five percent false."

Fuzziness applies here to the truth value of the bridi, <mi blanu>, rather
than to the selbri, <blanu>. This might be o.k., as far as it goes. I am
seeking a general formalism that will describe the fuzziness of a grammer
structure, such as a selbri, rather than a fuzzy formalism whose scope is
the entire bridi. Supose I use And's experimental fuzzy operator, but with
the <xoi> *before* the grammer structure it is to modify, that is with
structure:

x1 is <fuzzy number> on scale/in quality x2

<mi xoi pinomu blanu>

"I am fuzzily 0.05 blue."

Clearly, the two expressions:

"The statement 'I am blue' is five percent true."

and

"I am fuzzily 0.05 blue."

are related, but distinct, ideas; the latter has two layers of reference,
while the former has just one. There are several potential ways of avoiding
a metareference, obviously. One could make a lujvo using something like
<murse> + <blanu> = mursyblanu and apply the number to that.

<mi peha pinomu mursyblanu>

This would be idiomatic and we would have to adopt the convention that
murse used in this way would be a fuzzifier. We could come up with a new
gismu for fuzzy and do the same thing. (I suggested <fudji>, but there is
resistance to new gismu. I used <kerfa>, but some people thought this was a
malglico. It doesn't seem much worse to me than <murse>, but some people
thought <murse> was better.) I could use a fuhivla, I suppose, like
<lojyfuzis>

I am still not sure why one couldn't say:

<mi pinomu blanu>

as I'm not sure what *else* this could be referring to but either nonsense
or a fuzzy sort of blue, but for some reason this didn't fly either. (Why
not?)

>This is distinct from saying I have a 5% probability of being blue; probability
>does not enter into it.  A die has a 1/6 probability of coming up deuce, but
>it does so with a degree of truth that is 1 (with 1/6 probability) or 0
>(with 5/6 probability): not fuzzy at all.

That is exactly right.


>I had previously proposed this convention.

I think you have slightly changed your example, because I thought your
example previously had the <jeha> at the front, as I've transposed above.
But, as I understand it, the meaning is the same.

>
>Claims using this convention can be rewritten to use "jei" thus:
>
>        da je'a xi <number> broda = le jei da broda du li <number>
>

<le jei mi blanu du pinomu>


>2) for scalar claims generally, a new cmavo of selma'o MOI (for discussion
>purposes, "fiu'i"), with tentative place structure:
>
>        x1 is at <number> location on scale x2 (of type x3?)
>
>I'm not sure if x3 is useful; it is meant to be filled with things like
>"cardinal", "interval", etc.
>
>Comments?

If you are building on <fihu>, the fraction cmavo, I think you mean
<fihuhi> instead of <fiuhi>, . Since three of Guttman scales already have
cmavo, it seems to me to be more elegant to express "nominal" (la),
"ordinal" (moi), "interval" (?) and "ratio" (sihe) with existing cmavo
rather than use the x3 place. But you probably thought of that, so I'm
probably using these cmavo incorrectly.

But assuming I am using the selmaho [MOI] cmavo correctly, these could be
combined with a fuzzy operator. I'll use <xoi> for discussion purposes,
since its still experimental unless a decision is made to include it in the
baseline :-). This <xoi> is, I think, a little different than &'s.


The place structure for this <xoi> is

x1 is at <number> on scale x2

<mi xoi re fihu mu blanu>

"On a fuzzy interval scale of granularity six, I am a two in fuzzy blueness."

or

"I am 2/5 fuzzily blue."

(I am assuming an interval scale as the default, since that is most common
in my usage of fuzziness, but see below.)

If the denominator, n, is the number of intervals then the granularity is
n+1 because there are n+1 positions, in this case, 0/5,1/5,2/5,3/5,4/5,5/5.
The fraction should not be reduced so as to preserve the granularity:

<mi xoi mu fihu mu blanu> and <mi xoi ci fihu ci blanu> are not equivalent,
because the granularity of the former is higher. Thus the first statement
is "completely fuzzy blue" to a greater granularity or tolerance than the
latter.

The scale could then be included by using compound cmavo:

<mi xoi sihe re fihu mu remna tsali>

"On a fuzzy ratio scale of granularity six, I am 2/5 in fuzzy human strength."

(That is, on a ratio scale where 0/5 is *no* strength (quadriplegic?) and
5/5 is maximal human strength, my strength is fuzzily 2/5 of the maximum.
This differs from an interval fuzzy scale in that the zero is
non-arbitrary.)

<levi rozgu xoi moi re fihu mu rozgymelbi>

"This-here thing I am calling a rose is a 2 out of 5 on the ordinal fuzzy
rose-beauty scale."

or for Peter :-)

levi rozgu xoi la re fihu mu rozgymelbi

"This-here rose is fuzzily in the second of our five rose-beauty
categories, no ordering of these categories being implied."

(Of course, using fihu for nominal and ordinal scales, as I've done here,
is flat-out wrong; we need something like "2 of 5", to apply to things that
are ordered, but not necessarily evenly distributed. I plead ignorance of
what to use instead of fihu for nominal and ordinal scale scalar
expressions)

These expressions so far refer to only a single selbri. It would also be
useful to refer to fuzzy semantic space between two selbri.

<lo cinta xoi re fihu mu blanu be crino>

"On a fuzzy interval scale of granularity six where 0/5 is blue and 5/5 is
green, the paint is fuzzily 2/5 blue-green."

Fuzzy logic is to true/false logic as rational numbers are to integers.
>From that perspective, it shouldn't be very surprising that a separate
cmavo is necessary to express fuzziness elegantly! I won't repeat my
longwinded exhortations regarding the importance of fuzzy logic. But it
seems to me that any language which claims to be a "logical language" ought
to be rich in logical operators. Certainly potential speakers who are
familiar with fuzzy sets will be singularly unimpressed by a lack of an
elegant mechanism for fuzziness.

As I've tried to indicate above, I am still struggling with comprehension
of how the cmavo interact, so this probably has some mistakes, and is not a
formal proposal. If someone(s) can diminish my ignorance, perhaps we could
get closer to having a formal proposal for expressing fuzziness.

cohomihe 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