fuzzy logic

["John E. Clifford" <pcliffje@crl.com>]

Not too surprisingly, logicians have been involved with
Loglan/Lojban from almost the start and so provisions have been made for
extended logics from early on -- originally just many-valued or
probability-valued bu t fuzzy soon after Zadeh's original paper.  In one
sense, this has been pretty easy, since the *language* of even the most
extreme Wooky logics differ hardly at all from that of a standard logic
and certainly not from such a language modified to be speaka ble in
real-world contexts.  In short, as xorxes has pointed out, lojban is
equipped already to do fuzzy logic -- linguistically at least.

A couple of distinction make the discussion clearer, however.  Zadeh --
and some even madder epigones -- have developed at least three fuzziness
theories: set theory, logic, and arithmetic.  The set theory takes the
range of the characteristic function of a set from the usual {0,1} to
[0,1], from a set with two members to a closed real interval.  But the
underlying logic of this theory is two-valued:  c-set(object)=r gives the
right value or it does not.  The fun comes in figuring out how the values
for derivative sets comes from that for basic sets -- various kinds of
intersections and unions and (worst, since it does not work at all
regularly) subselections (red horses as opposed to things red and horses).
Considering the range of possible members, each set could be seen to have
a characteristic membership gradient (thank you, and) and one of the
developments finding sets related to a given set but with different
gradients, sharper ("very," "extremely," "perfectly" -- this lasted tended
to be almost perpendicular) or flatter ("sorta," "somewhat," "more or
less" and so on).  In lojban these are the tanrus and the lujvos with
_mutce_ and its compounds and opposites (and probably other words as well,
if we need them).

>From this, they moved (almost naturally) to taking the value of the
characteristic function for the object as value of the sentence "object is
a member of set" or even "set object." This undergirds a logic and has its
own definitions of the ususal connectives (strange but no stranger than
those in the probability interpretation and falling well within the
framework laid down by the two-valued system, the least restrictive
logic).  Among the new connectives actually expressed in this system, most
result in sentences which increase or decrease the truth value of
underlying sentence in a fixed way, similar to -- but different from --
the effects of modifying the membership gradient on sets.  The
modificiations are, in both cases, mathematically defined in standard
mathematics.  The metalanguage of the logic is still two-valued, i.e., the
truth value of a sentence either is or is not r, some specific real
number.

The fuzzyists have noted the problem that xorxes keeps pointing out, that
officially the characteristic function value or the truth value is this
very precise number.  To meet this they have developed fuzzy numbers and
the corresponding fuzzy arithmetic.  I haven't fiddled with this much
(it's not logic so I would be less adept at it) but it seems to function
on my grandmother's principle, "Many a mickle maks a muckle," with a
variety of "numbers" not unlike the range of items lojban has in
quantifier set s: "many," "few," "several," and so on, including the
"about n" sort.  These then serve as values for fuzzy characteristic or
truth functions in the latest versions of the earlier theories. The
metatheory to this theory is a fuzzy set theory over ordinary
 numbers, which are more or less in given ranges.
pc >|83