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

paper.

Theorem 1:

1. maxr = ru, ru <= 1

2. minr = rl, rl >= -1

3. rl <= r <= ru

Theorem 2:

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)

Theorem 3:

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].

Theorem 4.

(The 5 parts of this theorem define the subset S of

[0,1]x[0,1]

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.)

Theorem 5.

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.

Theorem 7.

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

r.

14. A OR (B AND C) = (A OR B) AND (A OR C), any appropriate

r.

References:

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,

Wichita, KS.