DIGEST OF BUCKLEY, JJ AND SILER, W: A NEW T-NORM. SUBMITTED TO
FUZZY SETS AND SYSTEMS, 1996.
Fuzzy systems theory has been criticized for not obeying all
the laws of classical set theory and classical logic. The
t-norm and t-conorm here presented obey all the laws of the
corresponding classical theory. A somewhat similar theory has
been proposed by Thomas (1994), except that he does not claim
that the distributive property is maintained.
We first propose a source of fuzziness. We suppose that the
truth value > 0 and < 1 of a fuzzy logical statement A is drawn
from a number of underlying (probably implicit) correlated
random variables from a Bernoulli process whose values alpha[i]
are binary, i.e. 0 or 1 with a Bernoulli distribution, and that
the truth value of A is a simple average of these binary
values. (George Klir (1994) proposed a similar process where
the random values are binary opinions of experts as to truth or
falsehood of a statement.) If this is so, then
a = truth(A) = sum(alpha[i] / n
b = truth(B) = sum(beta[i] / n)
r = correlation coefficient(alpha[i], beta[i])
aANDb = truth value(A AND B)
aORb = truth value(A OR B)
sa = standard deviation(alpha) = sqrt(p(alpha)*(1 - p(alpha))
sb = standard deviation(beta) = sqrt(p(beta)*(1 - p(beta))
aANDb = a*b + r*sa*sb
aORb = a + b - a*b - r*sa*sb
rmax = (min(a,b) - a*b) / (sa*sb)
for r = rmax, min(a,b) = a*b + r*sa*sb
rmin = (a*b - max(a+b-1, 0)) / (sa*sb)
for r = rmin, max(a+b-1, 0) = a*b + rmin*sa*sb
Proofs of the following theorems are in the appendices of our
1. maxr = ru, ru <= 1
2. minr = rl, rl >= -1
3. rl <= r <= ru
1. aANDb = a*b + r*sa*sb = a*b + cov(a,b)
2. aORb = a + b - a*b - r*sa*sb = a + b - a*b - cov(a,b)
1. If r = ru, aANDb = min(a,b) and aORb = max(a,b)
2. If r = 0, aANDb = a*b and aORb = a+b-ab
3. If r = rl, aANDb = max(a+b-1, 0) and aORb = max(a+b, 1)
We now suppose that this basic process is inaccessible to us,
but that we do have a history of a number of instances of the
truths of statement A and statement B. Now, given a value of r,
the correlation coefficient between a, the truth values of A,
and b, the truth values of B, the t-norm and t-conorm
appropriate to this history, T (t-norm) and C (t-conorm) are
defined for [a, b] on S, a restricted subset of [0,1]x[0,1].
(The 5 parts of this theorem define the subset S of
possible for r = 1, 0 < r < 1, r = 0, -1 < r < 0 and r = -1.)
Given a value of r, it may be that not all (a,b) combinations
are possible; e.g. a = .25, b = .75 is not possible for r = 1
in the binary process described above.)
1. (Shows that for 0 < r < 1 and (a,b) in S,
ab < T(a,b) <= min(a,b).)
2. (Shows that for -1 < r < 0 and (a,b) in S,
max(a+b-1) <= T(A,B) < ab)
3. (Shows that for -1 <= r <= 1 and (a,b) in S,
max(a+b-1, 0) <= T(a,b) <= min(a,b) and
max(a,b) <= C(a,b) <= min(a+b, 1).
Theorem 6. Shows that T is a t-norm and C is a t-conorm on S.
1. A AND A = A, r is 1.
2. A OR 0 = 0, any r.
3. A OR X = A, any r.
4. A AND NOT-A = 0, r is -1.
5. A OR A = A, r is 1.
6. A OR X = A, any r.
7. A OR 0 = A, any r.
8. A OR NOT-A = X, r is -1.
9. NOT-(A AND B) = NOT-A OR NOT-B, any r.
10. NOT-(A AND B) = NOT-A AND NOT-B, any r.
11. A OR (A AND B) = A, any appropriate r.
12. A AND (A OR B) = A, any appropriate r.
13. A AND (B OR C) = (A AND B) OR (A AND C), any appropriate
14. A OR (B AND C) = (A OR B) AND (A OR C), any appropriate
Klir, GJ (1994). Multivalued logics vesus model logics:
alternate frameworks for uncertainty modelling. In: Advances in
Fuzzy Theory and Technology, Vol II: 3-47. Duke University
Press, Durham, NC.
Thomas, SF (1994). Fuzzy Logic and Probability. ACG Press,