A Measure of Closeness of Weak Implication to Strict Implication

A MEASURE OF CLOSENESS OF WEAK IMPLICATION TO STRICT IMPLICATION

Masanori B. Okamoto Faculty of Economics Hiroshima University Higashisenda-machi Hiroshima City 730,Japan

The concept of weak implication introduced by R.Boudon is relevant to approximate reasoning which is a subject of the study in AI. The weak implication can be interpreted by the fuzzy material implication which is a binary fuzzy relation. A measure of closeness of weak implication to strict implication is defined as the maximum of distances within the given fuzzy relation,but constructed in the 2x2 contigency table. This measure agrees with the absolute value of Boudon's index f of closeness of weak implication to strict implication under some conditions.

l. WEAK IMPLICATION

1.1 Approximate Reasoning

In real-world domain in which reasoning is often judgemental and inexact,it is required to be able to say that " A su­ ggests B" or "C and D tend to rule out E". As an application of AI to real­ world problem a model of approximate implication has been already used,with a certainty factor which indicates the strength of implication,in the knowled­ ge-based medical consultation system in [2). On the other hand,it seems that we have no knowledge-based system in any field of social science as an applicat­ ion of AI. However approximate implica­ tion was discussed firstly by R.Boudon on sociological implication in [1). He criticized the transitive implication on the approximate reasoning that H. Zetterberg have done in [7) and he pro­ posed the concept of "weak implication" against strict implication. He also in­ troduced some measure of closeness of weak implication to strict implication with a certain probabilistic measure.

1.2 ~ Implication ~Fuzzy Implicat10n.

R.Boudon's weak implication is fuzzy implication in a sense,because it is implication that has fuzzy truth value.

Membership function which defines a fu­ zzy subset is different from probabili­ ty as indicated in [3),thus it may be preferable to give another measure of closeness of weak implication to strict implication by the use of the fuzzy im­ plication rather than probabilistic measure. This paper shows that weak im­ plication can be well expressed with a binary fuzzy relation which gives the fuzzy material implication on the fuzzy linguistic variable given by L.A.Zadeh in [4), [5), [6) and another measure of closeness of weak implication to strict implication is newly defined by the ma­ ximum of some distances between of fuz­ zy binary relation which is fuzzy mate­ rial implication.

2. FUZZY RELATION MATRIX OF WEAK IMPLICATION

2.1 Construction of Fuzzy Relation Mat­ rix

We now construct a fuzzy relation matr­ ix for a given 2x2 contigency table,in which the observed frequencies are sh­ own under classification with the two attributes A and B. HOwever it actually occurs that the classification is not proper because of ill-defined of A and B or not-fitted in practice for that criterion of classification. Thus it is more natural to regard A,B (or ~,g) as fuzzy subsets of possible universe of discourse X and Y,respectively. We ass­ ume that normalized frequencies in rows in the 2x2 contigency table (Table 1) are values of membership function of fuzzy subset A or ~;this table may Table 1. Contigency table normali­ zed in rows.

expess a fuzzy relation R in AxB C XxY. This procedure intends to determine the grade of fuzzy subsets by empirical data.

2.2 Fuzzy Material Implication

The extension of the notion of material implication to fuzzy subsets is given by L.A.Zadeh in [6] as follows,

IF A THEN B ELSE C = AxB + I AxC

where U,V are two possible different universe of discourse;A,B and C are fu­ zzy subsets of U,V and V,respectively. The material implication is a binary fuzzy relation in UxV and here we simp­ ly denote them A ~ · B. From the assump­ tion that the normalized 2x2 contigency table is a fuzzy relation matrix,the fuzzy material implication is obtained for the given 2x2 contigency table. For example,we obtain fuzzy implication for a not strict but strong implication as follows;fuzzy subsets A=[l/1,0.2/2],8= [l/l,0/2],relation matrix

The several examples of fuzzy implicat­ ion from strict to very weak implicati­ on are shown in Table 2.

Table 2. Examples of fuzzy implica­ tion.

STRICT IMPLICATION

3. MEASURE OF CLOSENESS TO STRICT IMPLICATION

3.1 Definition of Measure of Closeness

We now give the definition of a measure of closeness of weak implication throu­ gh fuzzy material implication.

The measure d(A~B) of closeness of we­ ak implication to strict implication is defined by

3.2 Examples of Measure of Closeness

The values of d(A~B) are given for examples cited above in Table 3. Furth-

Table 3. Values of d(A~B)

er we obtain values of d(A~B ) for sev­ eral examples in sociology given by 8oudon in [l];d(A~B )=0.5 for the con­ servation of religious tradition,d(A~ 8)=0.31 for the marriage system in anci ­ ent societies and the names called of mothers and mother's sisters,d(A~ 8 )= 0.20 for the effect of TV performance of political candidate to audiences. The values of d in Boudon's counter example to Zetterberg's implication can be written as d(A~B )=0.2,d(B~ C)=0.3, and d(A~ C )=0.1 for fuzzy implication

[ 0. 5 0. 5]

and Ai>C = 0 . 6 0 . 4 . . However we have

d ( (A~ B ) 0 (B~ C ) ) =0. 2f- d (A~ C ) =0 .1

1. BOUDON'S INDEX AND CLSOSENESS ----- MEASURE

4 . 1 Relation between Index f and Measure d(A~ B)

Boudon's index of closeness to strict implication is defined by

f=nll/nl.-n21/n2. or f=nll/n.l-nl21n.2 with the notation in Table !,from which we obtain the following fuzzy subsets A and B when the maximum of components of relation matrix is n 11 ;n~.;

A= [ (nn/n 1 .) /1, (n21;n 2 .) /2],

B=[(n11/n 1 .l/l, (n 12 ;n 1 .)/2]

and

Noticing that I (n11/nl )-(n22/n2 ) I

I (n12/n1.)- (n2l/n2 .) I ;

·

d (A~ B) =max ( I (nll;n 1 ) · (n22/n2 ) I, If ll

It is,however,plausible to assume under meaningful weak implication that

although another ill-conditioned case (n21/n2.l > (l/2) > ln2 2 /n2 ) occurs for a very weak condition. In'fact,d(A=l> B) have the same values as lfl values for previous examples. Therefore we conclude that in the fuzzy relation matrix dedu­ ced by the 2x2 contigency table,the absolute value of Boudon's index lfl is equal to our d(A~B ) under those conditions.

Similar procedure can be done for the pxq contigency table adding zero element to get square matrix. The grade of fuzzy subsets should be chosen to contain the maximum element of the matrix. The dis­ tances may be defined as follows;

5. CONCLUSION

In the previous section we assumed that the 2x2 contigency ·table expresses a fu­ zzy relation after the normalization in rows. We must ask what kind of justifi­ cation there are on this procedure,which gives actually frequency probabilities of A and B in Table 1. The question com­ es to the problem of the determination of the grade of fuzzy subset by empiri­ cal data. Some attempt to solve this problem has been done by S.Watanabe in [8] already. For generalized fuzzy set, a possible extension of the present idea could use his grade of implication 8 instead of our dj (A~ Bj) .