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

A fuzzy number A and its complement NOT A.

Figure 4.1

The intersection of A and NOT A is not everywhere zero, as we would expect

from the laws of classical logic, but has two sharp peaks with m(2) and m(6)

being 0.5. Similarly, the union A OR NOT A is not everywhere one but has two

sharp notches, shown in Figure 4.3.

If, however, we use T L rather than TM , we obtain A > Ac ¼ 1 and not Figure 4.2.

To evaluate A < Ac use C L , then A < Ac ¼ X and we eliminate the notches in

Figure 4.3. An occasion when correlation logic is of theoretical importance was

caused by a paper by Elkan (1994), which offered a proof that fuzzy logic was

only valid for crisp propositions. This paper depended on the fact that standard

fuzzy logic, and indeed all multivalued logic, fail to ful¬ll the laws of excluded

middle and non-contradiction. When the appropriate logic is used for combining

A and NOT A the excluded middle and non-contradiction laws are obeyed, and

Elkan™s proof fails.

A fuzzy number A, Ac and (A AND Ac) using standard min“max fuzzy logic Tm .

Figure 4.2

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65

4.3 APPROXIMATE REASONING

Figure 4.3 A fuzzy number A, its complement Ac, and (A OR Ac) using standard min “ max

fuzzy logic TM .

4.3 APPROXIMATE REASONING

Approximate reasoning is the term usually used to refer to fuzzy logical inference

employing the generalized fuzzy modus ponens, a fuzzy version of the classical

modus ponens discussed in Section 3.1. (Here, we are using “approximate reason-

ing” in a strict technical sense; the term is also used sometimes in a less technical

sense, to mean reasoning under conditions of uncertainty.)

The classical modus ponens is

if A then B (4:28)

which can be read if proposition A is true, then infer that proposition B is true. The

modus ponens itself is a proposition, sometimes written as “A implies B” or

“A ! B”, where “implies” is a logical operator with A and B as operands whose

truth table given in Chapter 3, Table 3.2. The modus ponens is an important tool

in classical logic for inferring one proposition from another, and has been used

for that purpose for roughly 2000 years.

The fuzzy version of the modus ponens, the generalized modus ponens, has been

formulated as:

If X is A then Y is B

from X ¼ A0 (4:29)

infer that Y ¼ B0

in which A and A0 are fuzzy sets de¬ned on the same universe, and B and B0 are also

fuzzy sets de¬ned on the same universe, which may be different from the universe

on which A and A0 are de¬ned. In fuzzy control, usually the membership functions

of fuzzy sets are de¬ned on the real line, and hence are fuzzy numbers.

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

Calculation of B0 from A, B, and A0 is straightforward. First, a fuzzy implication

operator is chosen; implication operators are discussed in Section 3.1. The

implication A(x) ! B(y) de¬nes a fuzzy relation A ! B between A and B. Next,

B0 is calculated by composing A0 with A ! B, following the procedure in

Section 3.4.2:

B0 ¼ A0 (A ! B) (4:30)

The fuzzy conclusion B0 is computed using the compositional rule of inference

B0 ¼ A0 oR (Sections 3.4.1 “ 3.4.2). This expression de¬nes the membership function

for the fuzzy conclusion B0 . The compositional rule of inference is valid for all fuzzy

sets; they do not have to be fuzzy numbers. A and A0 must be fuzzy subsets of the

same universal set X, and B and B0 must be fuzzy subsets of a universal set Y, which

may or may not be the same as X. Let us go through the details of the compositional

rule of inference for discrete fuzzy sets. Let

0:3 0:7 1:0

A¼ , ,

x1 x2 x3

(4:31)

0:5 1:0 0:6

B¼ , ,

y1 y2 y3

and

1:0 0:6 0:3

0

A¼ , , (4:32)

x1 x2 x3

Choose the implication operator T(x, y) ¼ min(1, 1 2 x þ y), called the Lukasiewicz

implication operator, in equation (3.8). Then, R(x, y) is shown in (4.33).

2 3

1 1 0:5

R ¼ 4 0:8 1 15 (4:33)

0:6 1 0:6

We are now ready to obtain B0 from A0 oR. Using T ¼ Tm , we see

B0 ( y1 ) ¼ max(min(1, 1), min(0:6, 0:8), min(0:3, 0:6)) ¼ 1

B0 ( y2 ) ¼ max{min(1, 1), min(0:6, 1), min(0:3, 1)} ¼ 1 (4:34)

B0 ( y3 ) ¼ max(min(1, 0:5), min(0:6, 1), min(0:3, 0:6)) ¼ 0:6

and

1:0 1:0 0:6

B0 ¼ , , (4:35)

y 1 y2 y3

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67

4.3 APPROXIMATE REASONING

The fuzzy sets A and B in approximate reasoning are usually fuzzy numbers (or mem-

bership functions de¬ned on the real line).

Approximate reasoning using the generalized modus ponens has been proposed

for fuzzy inference from if“ then rules. Consider the fuzzy if “ then rule

If x is Big, then y is Slow (4:36)

Where Big is de¬ned by fuzzy number A and Slow is speci¬ed by fuzzy number

B. Now suppose we are presented with a new piece of information about Big in

the form of fuzzy set A0 . That is, we are given that x ¼ A0 and this new fuzzy

number does not have to equal A. Figure 4.4 shows an example of fuzzy numbers