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