in which the truth values of A, B, and C are, respectively, 0.6, 0.8, and 0.3. Evaluat-

ing from left to right, we have

tv(A) ¼ 0:6

tv(A AND B) ¼ min(0:6, 0:8) ¼ 0:6

(4:58)

tv(NOT C) ¼ 1 À 0:3 ¼ 0:7

tv(A AND B OR NOT C) ¼ min(0:6, 0:7) ¼ 0:6

If parentheses are permitted in propositions, we evaluate as in (4.41) within each pair

of parentheses, from inmost to outermost as conventional in language compilers.

In the theory of fuzzy logic, Klir and Yuan (1995, p. 220 ff.) de¬ne four types of

fuzzy propositions. The ¬rst type is a proposition of the form given above in (4.34);

the second type is a proposition of the form in (4.35).

The third type of proposition is the generalized modus ponens discussed in

Section 4.3:

if X is A then Y is B (4:59)

and the fourth type is the same except that a hedge is introduced:

if X is A then Y is very B (4:60)

While in logic the modus ponens is indeed a proposition, as we have seen in

Section 4.3 such propositions are not usually useful in a fuzzy expert system. In

the theory of fuzzy logic, Klir and Yuan (1995, p. 220 ff.) de¬ne four types of

fuzzy propositions. The ¬rst type is of the form

temperature is high (4:61)

where temperature is a single-valued attribute, and high is a membership function.

The truth value of this proposition is the grade of membership of temperature in

high.

The second type is of the form

temperature is very high (4:62)

where “very” is a modi¬er, a hedge of the type that modi¬es a truth values such as,

for example, by replacing the truth value by its square root.

The third type of proposition is the generalized modus ponens discussed in

Section 4.3:

if X is A then Y is B (4:63)

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

and the fourth type is the same except that a hedge is introduced:

if X is A then Y is very B (4:64)

While in logic the modus ponens is indeed a proposition, as we have seen in

Section 4.3 we do not ¬nd such propositions useful in a fuzzy expert system.

4.8 QUESTIONS

Using the property of associativity, derive equations (4.15) “ (4.19) for n ¼ 3.

4.1

Show that any t-norm will give the AND table, Table 3.1, if x and y are only

4.2

0 or 1.

Show that tv(P OR Q) ¼ C(tv(P),tv(Q)) for any t-conorm.

4.3

Verify that the laws of non-contradiction and excluded middle hold if you use

4.4

TL and CL .

Elkan™s proof that the Zadehian logic fails except for crisp truth values (0 or 1)

4.5

is based on the a set of two logical propositions, P and Q, which are equivalent

for classical logic:

P: NOT(A AND NOT B)

Q:B OR(NOT A AND NOT B)

Show that these propositions are not equivalent using Zadehian logic (4.1) and

(4.7), but are equivalent using correlation logic. Use the Zadehian min“max

logic as a default, but use the bounded logic when appropriate. Hint: Alternatively,

reformulate Q using the distributive law to isolate B OR NOT B. Then, Try to get

an analytic solution. Alternatively, calculate truth tables for P and Q using truth

values of A ¼ 0.25 and B ¼ 0.5, ¬rst using Zadehian min“max logic, and then

using min“max logic as a default and bounded logic where appropriate.

Consider the fuzzy numbers A, A0 , and B, using Figure Question 4.6:

4.6

Using the theory of approximate reasoning and the fuzzy modus ponens,

calculate B0 from equation (4.30), B0 ¼ A0 (A ! B) for selected values of

x. Hint: Write a computer program for this purpose. Select x values of

1,2,3, . . . , 16. Calculate A, Apr, B, the matrix A ! B. Then use min “ max

composition of A0 with the matrix (A ! B) to obtain the vector B0 .

Consider two ways of de¬ning the dispersion of a triangular fuzzy number:

4.7

U ¼ ,central value., ,absolute dispersion.,

a. Dispersion of

,relative dispersion., where the net dispersion ¼

q¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬

,absolute dispersion.2 þ (,central value. Á ,relative dispersion.)2

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83

4.8 QUESTIONS

Fuzzy numbers A, A0 , and B.

Figure Question 4.6

b. Dispersion of U ¼ ,hedge. Á ,central value., where the hedge “nearly”

represents the decimal number 0.05 or 5%, “about” represents 0.1,

“roughly” represents 0.25, and “crudely” represents 0.5.

What are the advantages and disadvantages of these two methods?

Using the fuzzy numbers A and A0 de¬ned in Problem 4.4, add A and A0 by

4.8

a. The extension principle.

b. Alpha cuts.

c. Interval arithmetic.

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5 Combining Uncertainties

5.1 GENERALIZING AND AND OR OPERATORS

This chapter will deal with the problem of combining truth values. In Chapter 3, we

dealt with de¬nitions of the AND, OR, and NOT operators for multivalued logics,

and pointed out that many de¬nitions of these operators can be de¬ned, which

reduce to the classical de¬nitions for crisp logic for crisp truth values. In Chapter 4,

we presented the mathematics of a family of fuzzy logics that obey the classical laws

of Excluded Middle and Non-Contradiction. In this chapter, we will present a

general treatment of combining truth values, with the objective of calculating the

truth value of rule antecedents. In particular, we discuss use of prior association

between operands as a guide to selecting which fuzzy logical operators to use.

The concept of truth-functional operators is based on the idea that the result of

applying these operators will yield a value that depends only on the values of the

operands. As we have seen in Chapter 3, any of the three de¬nitions of AND and

OR are truth functional, and may all give different results; however, we are not

given any basis on which to make a choice among the available operators. Certainly,

the Zadehian min “max operators are used more often than any other, for a variety of

reasons, including some rather nice mathematical properties, and in the real world