: >equivocation. But when we put ourselves into the metalanguage,
: >we know, eg., that there are x such that both mu[TALL](x) > 0 and
: >mu[NOT TALL](x) > 0, as matters of usage, and so we make the
: >mistake of thinking that a statement such as "x is tall" is
: >somehow equivocal. It is not; it is merely fuzzy. It is
: >the same mistake that has led many fuzzy theorists into the
: >trap of thinking that the laws of excluded middle and
: >the of contradiction should not apply to fuzzy. But we
: .....
: >derision. The resolution starts with
: >recognizing that we may have mu[TALL AND NOT TALL](x) = 0
: >for *all* x, to recognize, in the metalanguage, the absurdity
: >of the object-language expression "tall and not tall", in accordance
: >with the law of contradiction, even if, now in the
: >metalanguage, both mu[TALL](x) > 0 and mu[NOT TALL](x) > 0
: >for some x, to recognize the fuzziness of the object
: >langugage terms "tall" and its complement "not tall".
: Suppose x=tallness. We need something like not-tallness.
: For the moment let that be 1-x (i.e. not-tallness) so that
: 1 would be complete tallness and 0 complete-not-tallness (ie
: shortness).
: Suppose x=1/2 then 1-x=1/2. Now we have both
: mu(1/2) > 0 and mu(1-1/2) > 0
: and
: mu([1/2][1-1/2])=mu(1/4)=0.
: Does this mean that we want mu(y)=0 for y <1/4) or that
: we want it zero only in the case of the compound
: proposition?
The general law is of the form (Thomas, 1995; p.117):
{ (1-t).a.b + t.min[a,b], t >= 0
a AND b = {
{ (1+t).a.b - t.max[0,a+b-1], t < 0
where a and b represent the membership functions of two
fuzzy sets A and B ranging on the same universe of
discourse, and -1 <= t <= 1 is a semantic consistency
coefficient that depends only on the membership functions
a and b. t satisfies the requirement that if B = not A,
t = -1. In that case, we have
a AND b = max[0,a+b-1].
Putting b = 1 - a (negation law), we have
a AND ~a = max[0,a+1-a-1] = 0
everywhere, satisfying the law of contradiction.
The law of excluded middle is similarly satisfied by
a companion law for union. The semantic consistency
coefficient t is postulated to be a function of the
correlation coefficient between the two membership
functions.
As Ellen Hisdal has pointed out in a contribution
to this thread, and also in various of her papers
which others have cited, when the membership
functions are identified with probabilities of
word usage, it is possible to *derive* appropriate
rules of combination, rather than to
postulate them directly. It is in this fashion that the
general law mentioned above is derived. As will
be evident, there is in fact an infinity of laws,
depending upon the value of t, but three cases
are of interest, and yield results already quite familiar
from the literature:
a AND b = a.b (t=0 -- semantic independence)
a AND b = min[a,b] (t=1 -- positive semantic consistency)
a AND b = max[0,a+b-1] (t=-1 -- negative semantic consistency)
There are corresponding rules for union, as follows:
a OR b = a + b - a.b (t=0)
a OR b = max[a,b] (t=1)
a OR b = min[1,a+b] (t=-1)
It should be noted that the min-max rules (t=1) and
the bounded-sum rules (t=-1) are intertwined through the
negation postulate and the correlation coefficient: they
are flip sides of the same coin, so to speak, not
separate, arbitrary laws.
: --
: Regards, Mark
Regards,
S. F. Thomas