fuzzy threads

["John E. Clifford" <pcliffje@CRL.COM>]

I am still not back, but I had a quiet weekend to do a little catching
up, so I read up on fuzzitude, a set of threads where I remember
my name coming up from time to time.  Herewith some scattered
comments.

Although the general theory of fuzzy sets allows characteristic
functions which never take the values 0 or 1 (as well as other very
strange sets), for most sets we meet in practical situations, most
things would receive one or the other of these values.  That is,
even fuzzy sets are rarely fuzzy to the core (never getting to 1) or
fuzzy to the horizon (never getting to 0); for most, the class of
objects which take values in the internal open interval is rather
small. One of the reasons that bivalent logic has endured so well is
simply that it works most of the time. Indeed, being fair to
Zoroastrianism would make the fuzzy part of natural languages
belong more nearly to it: most of the world belongs absolutely
either to The True or The False with a small segment (human souls
in the religious case) that fall between the two.

The size of the fractionally valued part varies a lot from set to set.
There are very few things, I think, that are not either definitely
cows or definitely non-cows -- a few mutants, accidents, and
hybrids (though these tend to get absorbed).  On the other hand, as
a matter of statistical theory, a fairly large percentage of a
population P will be in the fractional area for "tall for P." Even in
this latter case, though, most members of P fall definitely in or
definitely out, get a 1 or a 0 from the characteristic function.  To
think otherwise is to confuse the categorical notion of "tall" with
the the relative notion "taller than."  The fact that one member of P
is taller than another does not necessarily mean that the first is
more into (gets a higher value from the characteristic of) the set
than the other, they may both be in the 0 or 1 category.  Of course,
if one does get a higher characteristic value than the other, he must
be taller (on this notion -- I recognize that there is another notion
which takes relative width into account).  And the fact that the two
are of different height does allow that they may get different scores
for some related sets: "very tall," "sorta tall," "short (polar opposite
of tall)" and so on.

Much of the talk about Guttmann scales, though carried out in the
fuzzy threads, seems to me to be more relevant to the comparative
notions (which might, I suppose, be fuzzy, too, though the internal
interval gets very little workout in typical cases of these sorts) or,
rather, to the continua that underlie them.  "Ugly" is not identical
with 0 or even with very low scores on the characteristic function
of "beautiful," although presumably every ugly thing gets a 0 of the
characteristic of "beautiful." But so do many intermediate things,
too beautiful to be ugly but still too ugly to be beautiful.  (The
doctor will know the common complaint that I may be better but I
sure ain't well.).  It is usually this whole spectrum that gets
chopped up in some way; though it may be named for one of its
poles, it is not generally chopping up the charateristic of just that
pole.  So the "beauty" (better "ugly-beautiful") spectrum, if scaled
somehow, need not just look at values for the set "beautiful" or
even just for that and "ugly,"  but may use any convenient factors
that play a role (typically in both sets and points in between).  The
simplest would be a categorical scale based on the two pole sets:
ugly, beautiful, and neither (which could, of course, be fuzzy or
not).  And the "neither" area might be subdivided  ("homely, plain,
attractive" comes to mind)  and so on.  Or, of course, we might
skip the descriptive labels and just go for slices of the spectrum (I
skip over the implausibility of there being a well-defined spectrum
in this case, but we often operate as though we thought there were
and it works linguistically) numbered from left to right, as it were.
Of course, the polar sets might be subdivided, too, taking "very
ugly" and "sorta beautiful" as separate slices (I suspect that even
9's on the traditional scale are 1s on the characteristic of
"beautiful," though maybe not on "very beautiful" -- I am sure that
1s on the traditonal scale are still thought to be comfortably inside
"ugly.")

These comments also apply to Goran's (? djer's?) complaint that
_da je'axiny melbi_  would not imply _da melbi_, as it seems it
ought.   At least some of the time, _melbi_ is being used for the
whole scale and _je'axiny_ (or _ja'axiny_, for that matter) for a
slice or point on that scale, at other times _melbi_ is the proper
polar set and _je'axiny_ is a value (range) of its characteristic
function -- or maybe even a related set with a derived
characteristic function.  In the first use, _je'axiny melbi_ need not
be _melbi_ but might be all the way down into _to'e melbi_ or in
the middle or....  In the second, it would be in _melbi_
(presumably scoring n on the characteristic function indeed -- if I
understand THIS intended use of subscripts, which is not a favorite
of mine).  On the third, its relation to _melbi_ would be
determined by the derivation of its characteristic function -- "very",
for example is entirely in the base function, "somewhat" tends to
pick up some from the 0 range, and so on.  At least in some of the
fuzzy discussions these three notions (at least) have been
thoroughly muddled together.  Or, to be fair to the linguistic
situation in English, where the same words are often used for all of
them indifferently, have not been adequately sorted out.  Lojban
seems to have the resources to already in hand to do the sorting in
a systematic way but this has not been carried out and adhered to
consistently.

stivn keeps insisting that natural languages use fuzzy logic a lot
and that Lojban doesn't.  The problem with this line is that all the
evidence that natural languages use fuzzy concepts, etc., can
equally well be taken as evidence that Lojban does too, since
Lojban matches natural language usages very well indeed.  The
fact is that the basic form of fuzzy logic (and, indeed, all the
Wooky logics out of slash-L-sub-inf) LOOK the same in terms of
grammar and vocabulary; you can't generally tell by looking what
kind of logic you are up against at the elementary level.  As you
get deeper into it, patterns start to emerge that will indicate what
the game is.  Insofar as any natural langauge has features of the
sort that indicate fuzziness, Lojban can match them and, often,
beat a particular language out by a large degree (since it has tricks
drawn from several langauges). On the other hand, insofar as
Lojban vocabulary is more carefully defined than most natural
languages, some cases of clear rejection of fuzziness may emerge.
For example, the fact that the predicate for "is a member of" does
not have a place for a value in [0,1], nor does the predicate for "is
true" (but see the discussion above about the difference between
polar sets and spectra), indicates that these notions have not been
thought of fuzzily.  However, for most concepts, no such overt
indications are appropriate, and thus their absence leaves Lojban
as open to fuzzy interpretations as any other language.   Of the
most obvious fuzzy patterns in logic, Lojban has good facilities for
the derived sets: forming compounds with expressions for "very,"
"slightly," "extremely," and the like, which can be (and have been
for English) correlated with various derivative characteristic
functions.  Not so obviously fuzzy (but here often muddled in),
Lojban also has a system for a five-way (I think I've counted)
lopping up a spectrum defined by one of its poles, the NAhE set.
Finer granularity is presumably available by compounds or the
apparently now acceptable devise of ordinal subscripting (hawk
ptui). (I do not know quite what to say about cases where the
spectrum does not run from pole to pole but just from one class to
another.  Partly, there is a problem about what can lie between
some pairs of classes and partly it is a matter of expressing the
general patterns of dividng up the space.  On the latter, I suppose
that the ordinal words would be of some use, since they specify at
least one terminus and might be jiggled to express the other as
well.  That still leaves granularity and scale-type, however,
unexpressed. As noted, this does not seem to me to be a fuzzy
matter per se.)  I am not really convinced that the device of
dividing characteristic values of a set into smaller sets is ever
actually used (as opposed to the two notions above), but, if it is,
then I do not see a Lojban device ready-made for it.  I do think,
however, that it can be handled simply by using tanru with
_meliny_ (assuming that _me_ has not gotten so kakked up as no
longer to be usable in its original basic way), maybe with a fuzzy
n, if wanted.

If we add a place to the structure of "is a member of" or just agree that
the standard "extra place" flag with it is to stand for a fuzzy value,
then we seem to have fuzzy set theory -- basic fuzzy logic -- in hand.
Aside wanting to say what the fuzzy truth value of a sentence is -- which
seems to be again just a matter of extending the place structure of a
gismu, there seems to have been little discussion of fuzzy propositional
or quantifier logic.  And, indeed, there has not been a lot of a very
specific nature anywhere.  If we did come down to issue of which fuzzy
connective each of our given connectives should be, I suppose we could
handle that quite unnoticed.  The problem of expressing some of the other
connectives, if we decide we need them, might be more difficult (I would
not want to get into devices that would have to affect the whole range of
connectives at all the various levels that they have come to have).
Happily, there is almost no evidence that languages actually use a variety
of fuzzy connectives in the standard places nor a variety of non-standard
ones.  The actually used fuzzy set seems, if anything, less complex than
the probability set, which has an array of conditionals and disjunctions.
The only question with fuzzies seems to be whether the "and" and "or" are
min and max or product and sum (roughly -- the last two have to be
finagled, of course), so a few spare forms might be handy in case someone
wanted to use both (but we need those already for probability logic, if we
start getting serious about letting Wookies in).  As for fuzzy arithmetic,
that just needs a mark (same class as -/+) to say "fuzzy number,"  since,
so far as I can find, no specifically fuzzy functions are used.  If we
need to mess more in that area, MEX is the easiest place to work, since so
little of it is in a settled state (or, at least, so few of us are sure of
what state it is in and firmly locked into that state).
pc>|83