are combining A and NOT A prior knowledge is not required; we know that A and

NOT A are maximally negatively correlated, and their correlation is 21.) Let

a ¼ tv(A)

b ¼ tv(B)

(4:22)

r ¼ prior correlation coef¬cient between a and b

p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

d ¼ a(1 À a)b(1 À b)

Then T ^ (a, b) and C^ (a, b) are de¬ned by

d ¼ a(1 À a)b(1 À b)

rU ¼ (min(a, b) À ab)=d

rL ¼ (max(a þ b À 1, 0))=d

if r , rL then r0 ¼ rL

(4:23)

else if r . r U then r 0 ¼ r U

else r 0 ¼ r

T ^ (a, b) ¼ a þ b À r0 d

C^ (a, b) ¼ a þ b À ab À r0 d

A speci¬cation of r as 1 is equivalent to specifying the standard min“max fuzzy

logic. It is possible to specify values for a, b, and r that are incompatible. For

example, if a is speci¬ed as 0.4 and b as 0.6, a value for r of 1 is not possible. In

(4.16), rU and rL are the limits of possible values for r given a and b. In the event

that a value of r outside those limits is speci¬ed, the possible value of r(rEff)

nearest the speci¬ed value is used. If a value of r of 1 is speci¬ed, this value of

rEff will always result in the standard Zadehian min“max logic being used, no

matter what values a and b have.

The notion of semantic consistency between fuzzy sets was put forward by

Thomas (1995). Up to now, we have considered proposition with a single truth

value. We may also have to consider combining fuzzy numbers and membership

functions de¬ned on the real line. Let one fuzzy number or membership function

be de¬ned by m1 (x), and the other by m2 (x). In this case, we compute the cross-

correlation coef¬cient using the well-known formula

p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

r ¼ cov(m1 (x), m2 (x))= var(m1 (x)) Á var(m2 (x))

by integrating over the area of overlap only. If no overlap, r ¼ 21.

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62 FUZZY LOGIC, FUZZY SETS, AND FUZZY NUMBERS: II

It was shown that

T^

TL TM (4:24)

C^

CM CL (4:25)

T ^ ¼ T p , C ^ ¼ Cp , if r ¼ 0; T ^ ¼ T L , C^ ¼ C L , if r ¼ À1 (4:26)

and

T ^ ¼ T m , C^ ¼ C m , if r ¼ 1 (4:27)

A computer routine in the C language to calculate a AND b and a OR b using

correlation logic is

//Function CorrLogic - given a, b and default r,

//returns aANDb and aORb

//04-17-2004 WS

#include <math.h>

double min(double x, double y);

double max(double x, double y);

bool CorrLogic (double a, double b, double r, double

* aANDb, double * aORb)

f

double std, ru, r1;

if (a < 0||a > 1||b < 0||b > 1||r < -1||r > 1)

return false;

std ¼ sqrt(a * (1 2 a) * b * (1 2 b));

if (std > 0)

f

ru ¼ (min(a, b) 2 a * b)/std;

rl ¼ (max(a þ b) 2 1, 0) 2 a * b)/std;

if (r < rl)

r ¼ rl;

else if (r > ru)

r ¼ ru;

g

*aANDb ¼ a * b þ r * std;

*aORb ¼ a þ b 2 a * b 2 r * std;

return true;

g

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63

4.2 ALGEBRA OF FUZZY SETS

double min(double x, double y)

f

if (x < y)

return x;

else

return y;

g

double max(double x, double y)

f

if (x > y)

return x;

else

return y;

g

There are two basic cases where it is obvious what to choose for r. To ¬nd

A < A, A > A we must use r ¼ 1 since A and A are maximally positively

correlated. Then, T ¼ T m and C ¼ Cm so that A < A ¼ A, A > A ¼ A. For

A < Ac , A > Ac we must use r ¼ À1 because A and Ac are maximally negatively

correlated. Then, we have T ¼ T L , C ¼ C L for this value of r so that A < Ac ¼

X, A > Ac ¼ 1 and the laws of non-contradiction and excluded middle hold.

We showed that using this new t-norm and t-conorm (correlation logic), and

properly choosing the value of the parameter r, all the basic laws of crisp set

theory also now hold for fuzzy sets, including the laws of excluded middle and

non-contradiction.

We may ask: If we have no knowledge of prior associations between A and B,

what should the default logic be? We suggest, on the basis of nearly 20 years of

experience, that the Zadehian min“ max logic is a desirable default. If we are eval-

uating rules with complex antecedents, with any other logic the truth value of an

antecedent with several clauses ANDed together tend to drift off to zero as the

number of clauses increases; and when aggregating the truth values of a consequent

fuzzy set member by ORing them together, the resulting truth value tends to drift up

to one. The Zadeh logic, unless combining B and NOT B, passes a pragmatic test; it

works, and works well.

4.2.3 Combining Fuzzy Numbers

Since fuzzy numbers are fuzzy sets, we may perform logical operations upon them,

such as A > B, A < B, A > Bc , and so on, when A and B are fuzzy numbers.

Consider the fuzzy number A and its complement NOT A in Figure 4.1.

We now construct the intersection A AND NOT A using the conventional min“

max logic, Tm , shown in Figure 4.2. Because segments of membership functions

coincide in a number of places, the labeling of the graph is a little complicated.