Let us back up and ask what properties the classical AND, OR, and NOT oper-

ators posses. Among the most basic are the Law of the Excluded Middle and Law of

Non-Contradiction. These can be quite simply formulated:

Excluded middle: P AND NOT P ¼ false ¼ 0 (5:1)

Non-Contradiction: P OR NOT P ¼ true ¼ 1 (5:2)

Unfortunately, all the de¬nitions for the AND, OR, and NOT operators except the

bounded sum fail to obey these laws. While this can be disturbing to some

people, many fuzzy mathematicians seem to regard it as a virtue. In Chapter 4,

we developed the mathematical theory for a family of fuzzy logics that does obey

these laws; in this chapter, we discuss the origin of this development and its utility.

Fuzzy Expert Systems and Fuzzy Reasoning, By William Siler and James J. Buckley

ISBN 0-471-38859-9 Copyright # 2005 John Wiley & Sons, Inc.

85

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86 COMBINING UNCERTAINTIES

5.1.1 Correlation Logic: A Family of Fuzzy Logical Operators that

Obeys Excluded Middle and Non-Contradiction Laws

We felt that the inability to make a rational choice of logical operators and the

failure to obey the laws of Excluded Middle and Non-Contradiction creates a

rather unsatisfactory situation. Spurred by a paper by Ruspini (1982), we investi-

gated what effect prior associations between the operands of the logical operators

might have on the choice of a proper set of operators. We adopted a model for fuzzi-

ness presented by Klir and Yuan (1996, pp 283ff). In this model, a number of persons

were asked whether a proposition were true, and were restricted to true (1) or false

(0) responses. The truth value of the proposition was considered to be the average of

all responses. As the number of observers increases without limit, the granularity of

our estimate of the truth value vanishes; the estimate of the truth value becomes the

probability that an observer would say that the proposition is true. The following

treatment is taken from Buckley and Siler (1998b, 1999).

We extend Klir and Yuan™s model just above to two propositions simultaneously

presented, and ¬nd the probability that an observer would report that P is true; that P

AND Q is true; that P OR Q is true; and the association between the individual

reports of the truth of P and of Q. Table 5.1 gives a sample of such reports.

We found that if the truth values of P and Q were positively correlated as strongly

as possible, the Zadehian AND/OR operators were correct in predicting the mean

value for P AND Q and P OR Q. If the individual P and Q truth values were maxi-

mally negatively correlated, the bounded sum operators gave correct results. If the

individual P and Q truth values were uncorrelated, the product-sum operators gave

correct results, as would be expected from elementary probability theory. In

Table 5.1, the correlation coef¬cient is 0, and the truth values obtained for P

AND Q and OP OR Q are those we would expect from probability, assuming

independence.

This approach yielded a family of logical operators with a single parameter; the

prior correlation coef¬cient between the operands. We concluded that a rational

TABLE 5.1 Sample Table of Observer Reports of Truth

of Two Propositions, P and Qa

Observer P Q P AND Q P OR Q

1 0 1 0 1

2 1 0 0 1

3 1 1 1 1

4 1 1 1 1

5 0 0 0 0

6 0 0 0 0

7 0 1 0 1

8 1 0 0 1

Average 0.500 0.500 0.250 0.750

a

Correlation P and Q ¼ 0.

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87

5.1 GENERALIZING AND AND OR OPERATORS

choice among logic operators could be based on information regarding such associ-

ations. The family of AND/OR operators returns the truth values of P AND Q and P

OR Q given the truth values of P(a) and Q(b) and a single parameter, r, the corre-

lation coef¬cient between prior values of a and b.

We ¬rst place a restriction on the maximum and minimum permissible values of

the parameter r, ru, and rl, respectively, and from these restrictions derive a working

value for r, r 0 . The reason for this is that the values of a and b may make some values

for r impossible. For example, if a ¼ 0.2 and b ¼ 0.6 it is impossible for these values

to be perfectly correlated. If the speci¬ed r is less than rl, then the formulas will use

the bounded sum operators; if the speci¬ed r ¼ 0, the formulas will use the sum-

product operators; if the speci¬ed r is greater than ru, the formulas will use the

Zadehian max “min operators. We ¬rst present the formulas for this family of

AND/OR operators, then present some numerical examples of their performance.

Since we are interested in implementing these operators on a computer, we will

present them as statements in BASIC.

ru ¼ (min(a, b) À a Ã b)= SQR(a Ã (1 À a) Ã b Ã (1 À b))

rl ¼ (max(a þ b À 1,0) À a Ã b)= SQR(a Ã (1 À a) Ã b Ã (1 À b))

if r . ru then r 0 ¼ ru else r 0 ¼ r

(5:3)

if r , rl then r 0 ¼ rl else r 0 ¼ r

aANDb ¼ a Ã b þ r Ã SQR(a Ã (1 À a) Ã b Ã (1 À b))

aORb ¼ a þ b À a Ã b À p Ã SQR(a Ã (1 À a) Ã b Ã (1 À b))

Table 5.2 gives some typical results of applying these formulas.

Of course, this family of operators is not truth-functional, since other information

is required besides the values of the operands. In many cases, that information is

lacking and the Zadehian operators are a good default for expert systems, provided

that we do not combine A and NOT A. They are the mathematical equivalent of the

“a chain is no stronger than its weakest link” common sense reasoning. Further, any

other multivalued logic limits the complexity of the rule antecedents that can be

used: If several clauses are ANDed together, the resulting truth value tends to

drift down to 0; if they are ORd together, the resulting truth value tends to drift up to 1.

TABLE 5.2 Examples of Application of Logic Operator Family

Using the Prior Correlation Coef¬cient

r0

r a b aANDb aORb Resulting Logic

21 20.612

0.2 0.6 0 0.8 Bounded

0 0.2 0.6 0 0.12 0.68 Product

0.2 0.6 0 0.2 1 Max “min

þ1

21 20.612

0.4 0.8 0.2 1 Bounded

0 0.4 0.8 0 0.32 0.88 Product

0.4 0.8 0.408 0.4 0.8 Max “min

þ1

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88 COMBINING UNCERTAINTIES

There are two circumstances under which there is no question as to prior associ-

ations. If P and NOT P are the operands, they are maximally negatively associated; if

P and P are the operands, they are maximally positively correlated. If we want to use

multivalued logic, and wish to retain the laws of Excluded Middle and Non-

Contradiction, we can use any multivalued logic we please unless the equivalent

of P and NOT P or P AND P appear in the same proposition. We can save Excluded

Middle and Non-Contradiction by using the Zadehian max “ min logic when com-

bining P and P, and by switching to the bounded-sum operator pair when combining

P and NOT P. (We might require rearranging the proposition to bring P and P

together, and to bring P and NOT P together.)

In most cases, we will not have prior information available on which to base a

choice of which logical operators we should employ. However, we can choose a

default set of operators. Let us assume that our observers share a common back-

ground. It would seem that in this case, they would tend to agree more than to dis-