true, resulting in individually larger or smaller estimates as to the truth of P and of Q,

but if the observer who is more strict says that P is true, it seems likely that the less

strict observer would also say that P is true. (Our formulation provides for precisely

this contingency.) So it seems likely that the default correlation should be þ1, yield-

ing the Zadehian operators; indeed, many years of practice have shown that this

choice works.

5.2 COMBINING SINGLE TRUTH VALUES

The most important calculation of truth values takes place when the truth value of a

rule™s antecedent is evaluated. The basic unit of an antecedent is a clause. Several

types of clauses are shown in Table 5.3.

TABLE 5.3 Types of Clauses in a Rule Antecedent

Test Performed Example Truth Value

A. Test of truth value of member size is Small Grade of membership of Small in

of discrete fuzzy set discrete fuzzy set size

age , 35

B. Test of attribute value against Truth value of age AND truth

a literal value of comparison

age , ,A1.

C. Test of attribute value against Truth value of attribute AND

a previously de¬ned variable truth value of comparison

AND truth value of variable

,A1.

age.cf . 0

D. Test of attribute™s truth value Truth value of comparison

against a literal

age.cf . ,X.

E. Test of attribute™s truth value Truth value of comparison AND

truth value of variable ,X.

against a previously de¬ned

variable

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5.3 COMBINING FUZZY NUMBERS AND MEMBERSHIP FUNCTIONS

A. The truth value of “size is Small” is simply the grade of membership of Small in

discrete fuzzy set size. If this is 0.645, then the truth value of the clause is also

0.645.

B. Say the value of age is 32, and its truth value is 0.925. The truth value of the data

comparison, 32 ¼ fuzzy 35, might be 0.550. The truth value of the fuzzy 35

itself is 1.0 by default, since it is a literal. The truth value of the clause is then

min(0.925, 0.550, 1.0) ¼ 0.550.

C. As before, the truth value of age is 0.925. Say the variable ,A1. has pre-

viously been assigned the value 35 and truth value 0.495. The truth value of

the comparison would be 0.550 (as before). The truth value of the comparison

value is not 0.495. The truth value of the clause is then min(0.925, 0.550,

0.495) ¼ 0.495.

D. The attribute age.cf (the truth value of age) is 0.925 from B. just above. The

truth value of age.cf is 1.0. If age.cf is not 0, the truth value of the comparison

is 1. The truth value of the literal comparison value 0 is 1.0 by default. The truth

value of the clause is then min(1.0, 1, 1.0) ¼ 0.

E. Age.cf is 0.925 as above, and its truth value is 1.0. Say that ,X. has value 0.5

and truth value 0.75. The truth value of the Boolean comparison is 1. The truth

value of the clause is then min(1, 1, 0.75) ¼ 0.75.

Antecedent clauses referring to the same declared data element are grouped

together into a pattern. We list such a pattern, with the truth values of its individual

clauses:

(in Data x <3 AND (y < 0 OR NOT size is Small))

(tv 1) (tv 0.8) (tv 0.23)

or

tv = (1 AND (0.8 OR NOT 0.23))

= (1 AND (0.8 OR 0.77))

= (1 AND 0.8)

= 0.8

5.3 COMBINING FUZZY NUMBERS AND

MEMBERSHIP FUNCTIONS

Fuzzy numbers and membership functions are fuzzy sets, and hence may be

combined logically. If Small, Medium, and Large are declared to be fuzzy

numbers, we might ask IF (pro¬t is Small OR Medium), or perhaps

IF (pro¬t is NOT Large), both requiring logical operations on fuzzy

numbers. Another example of combining fuzzy numbers logically is the combi-

ning a fuzzy (,2) and a fuzzy 2 in order to make the approximate comparison

x˜<=2.

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90 COMBINING UNCERTAINTIES

Figure 5.1 Membership functions Small and Medium.

Suppose we wish to combine the membership functions Medium and Large. We

de¬ne these two functions in Figure 5.1.

Assume that we have adopted the Zadehian AND (min) and OR (max) as

defaults. Figure 5.2 shows the fuzzy number that results from ORing Small and

Medium using the max OR operator.

The notch in Figure 5.2 is quite counterintuitive. To eliminate the notch, we intro-

duce here the notion of semantic inconsistency, ¬rst proposed by Thomas (1995). It

occurs because we are combining P and NOT P with an inappropriate logic. The

Zadehian max “ min logic holds if the operands are positively associated as strongly

as possible. However, the membership functions for Small and Medium are not so

associated; they are semantically inconsistent, a notion introduced by Thomas

(1995), and the use of min“ max logic here is probably invalid.

The proper logic is the correlation logic de¬ned in Section 4.2.2. We simply cal-

culate the cross-correlation between the fuzzy numbers or weighted membership

functions being combined, over the area of overlap. We then OR the memberships

with the formulas in (5.3), using the cross-correlation as the default correlation

coef¬cient.

Membership function Small OR Medium using min “ max logic.

Figure 5.2

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5.3 COMBINING FUZZY NUMBERS AND MEMBERSHIP FUNCTIONS

Weighted membership functions Small and Medium.

Figure 5.3

Suppose that our basic membership functions are those shown in Figure 5.1, and

that rules have established the truth value of Small as 0.75 and that of Medium as

0.25. Then the functions we must combine are shown in Figure 5.3.

The correlation coef¬cient calculated over the area of overlap is 21. Using cor-

relation logic, the function Small OR Large is shown in Figure 5.4, which seems

intuitively much more acceptable than the functions ORd with max “ min logic,

shown in Figure 5.5.

Once we begin to apply set-theoretic operations to fuzzy numbers, the failure to

obey the laws of Excluded Middle and Non-Contradiction can give quite counter-

intuitive results.

There may be restrictions on when the bounded operators can be fruitfully used;