Connectives and fuzziness for classical effects

ELSEVIER Fuzzy Sets and Systems 106 (1999) 247-254

ZZY sets and systems

Connectives and fuzziness for classical effects

Stanley Gudder* Department of Mathematics and Computer Science, University of Denver. Denver, CO 80208. USA

Received April 1997; received in revised form July 1997

Abstract

This article investigates various constructs on a set of classical effects (measurable fuzzy sets or fuzzy events). We begin by studying the properties of a connective M and its dual connective U which seem to have been neglected in the literature. The importance of these connectives for fuzzy probability theory is pointed out. We then introduce and investigate the properties of a commutator which can be employed to define a degree of relative fuzziness. A special case of this commutator defines a degree of fuzziness. The expectation of the commutator provides a relative entropy whose properties are delineated and the expectation of the special case gives an entropy function that has been previously studied as a measure of fuzziness. © 1999 Elsevier Science B.V. All rights reserved.

1. Introduction define

Let (f2,~1) be a measurable space and let 8 = g ( f 2 , ~ ¢ ) be the set o f measurable functions in [0, 1] a. Thus, f E ~ if and only if f is a random variable on f2 with values in [0, 1 ] and we call f a classical effect. Our terminology stems from quantum mechanics where the analogous objects are called (quantum) effects [1, 9, 12]. Various authors call the elements of d ~ measurable fuzzy sets or fuzzy events. We shall only need the measurability o f f E [0, 1] ~ when we integrate with a probability measure and much of this work applies to a general fuzzy set theory.

We now introduce various operations on g. For f E d ~, we define f ' = 1 - f E ~. This negation or complementation seems to be universally accepted. For f , g E o ~, we write f_ l_g i f f + g ~ < l and we

* Tel.: +1 303 871 2436; fax: +1 303 871 2254.

f ® g = f + g if f . L g,

f a g = min(f , g),

f V g = max( f , g),

fng=&,

f t d g = f + g - f g .

The partial operation O is analogous to the one used in quantum effect algebras [5, 7, 8]. The operations A and V are commonly employed by most authors in the field. Although M and U sometimes appear, they are largely neglected in the literature. However, as emphasized very early by Zadeh [14, 15], these connectives are the appropriate ones in fuzzy probability theory. This is because the natural definition for the probability that f E g occurs is its expectation E ( f ) . Moreover, the natural definition for f and g to be independent effects is that they be independent as random

0165-0114/99/$ - see front matter (~ 1999 Elsevier Science B.V. All rights reserved. Pll: S0165-0114(97)00258-3

248 S. Gudder/Fuzzy Sets and Systems 106 (1999) 247-254

variables. Then if f , g E d ~ are independent, we have the important and basic law E ( f n g)=E(f )E(g) which fails for f A g. Without this law, one can hardly do any probability theory. An attempt can be made to reverse this process by defining f, g E ~ to be independent if E ( f A g)= E( f )E(g ) but it appears that this does not hold for the effects in most practical applications.

In this article we shall take ~q and its dual connective LJ as our basic operations. We shall study the properties of these operations and sometimes compare them with the properties of A and V. We can already see that they both share the valuation property.

f V g + f A g = f +g,

f U g + f M g = f +g.

In contrast, the following law of logic:

f V g : f V ( f ' A g ) = g V ( g ' A f ) ,

does not hold but it does hold for M and U in the interesting sense that

f u g = f O ( f ' n g ) = g ® ( g ' fq f ) .

We shall next study the properties of the commutator

[f, g] =fg2 _ gf2 : fg(g _ f ) .

Since [ f ,g] can be negative, it may not belong to d ~. Notice, however, that if f ~ g, then [ f , g] E e and we call [ f ,g] the degree of fuzziness o f f relative to g. There are circumstances in which the universal effect is some g E # that is not the crisp effect 1 [8]. In this case, it is of interest to compare the fuzziness of various effects relative to g. When g = 1 we have the special case [ f , 1] = f f ' = f f7 f ' which is called the degree of fuzziness o f f . The reader will observe that the basic laws of classical logic (or set theory) hold for fl and U to within a commutator or sum of commutators.

If f ~ g, the entropy of f relative to g is defined as H a ( f ) = E([ f , g]). We shall show that H a ( f ) has the desired properties of a relative entropy, and thus, provides a measure of fuzziness o f f relative to g. The special case H ( f ) = Hz( f ) is simply the entropy of f and this is a standard entropy studied by various authors [4, 6, 10, 11]. Moreover, H ( f ) is a standard measure of fuzziness for f . Some of our results

involve straightforward (but occasionally, tedious) calculations. In such cases, the proofs are usually omitted.

2. The basic connectives

Our first result summarizes some of the properties of the basic operation U.

Lemma2.1. (i) f U g = g U f . (ii) ( f u g) U h = f U ( g U h ) . (iii) f u r ' = l - I f , I]. (iv) f u f = f + [ f , 1]. (v) f U 0 = f . (vi) f u 1=1. (vii) f u g = 0 implies f = g = 0 . (viii) f<~g, h<<. i imply f Uh<~gUi. (ix) ( f U g ) ' : f ' rig'. (x) ( f 71 g)' = f ' u g'.

Dual properties hold for M. For example, f 1~ f~ = [ f , 1]. This formula and Lemma 2.1(iii) and (iv) illustrate how the basic laws of logic hold to within a commutator. As usual we call f E ~ crisp i f f = f2 or equivalently if [f , 1] = 0. The following lemma gives some of the many characterizations of crispness.

I.emma 2.2. The following statements are equivalent. (i) f is crisp. (ii) f = f O f . (iii) f U f ' = 1. (iv) f ~ f '=O. (v) [f, f U f]=O.

Proof. That ( i ) - ( iv) are equivalent follows from Lemma 2.1. To prove that (i) and (v) are equivalent, we have

[ f , f U f ] = f 5 _ 3f4 + 2f3 = f 3 ( f _ 2 ) ( f - 1).

Hence, [ f , f U f ] = 0 if and only if f(co) = 0 or 1 for all o9Ef2. []

The next lemma considers the distributive laws.

Lemma 2.3. (i) f M ( g u h ) = ( f n g ) u ( f n h ) - gh[f, 1]. (ii) f U ( g l - V h ) = ( f u g ) v l ( f u h ) + g'h'[f, 1]. (iii) I f f is crisp, then f M (g U h)= ( f Mg) U ( f Mh) and f U ( g M h ) = ( f Ug) l-q ( f U h ) .

Notice that U satisfies the following inclusion- exclusion conditions that are important in probability theory.

f u g = f + g - f g ,

f U g U h = f + g + h - f g - f h - oh +fgh.

S. Gudderl Fuzzy Sets and Systems 106 (1999) 247-254 249

It follows by induction that

f l U . . . U f , '

= E f i -- Z f i L + f i f j fk

+ - . . + ( - 1 y + ' f l f 2 . . . f . ,

where the ' indicates that the summation is over dis- tinct increasing indices. We also have the following interesting condition for the U of n equal terms:

f U . . . U f = ( f ' . . . f ' ) ' = 1 - (1 - f)n.

We say that f , g satisfy the MV-condition if

( f U g') ' u f = ( g U f ' ) ' Ug.

The MV-condition is the main axiom for classical multi-valued logic and is satisfied when U and ~ are replaced by V and A. To check if this condition is satisfied, we have

( f U g ' ) ' U f

= ( f ' R g) U f = f ' g + f - f f ' g

= f + g - 2 f g + f2g.

By symmetry, we have

(g U f ' ) ' U g = f + 9 - 2fg + g2f.

Hence,

( f Ug')' U f =(gU f ' ) ' U g - [f,g],

so f , g satisfies the MV-condition if and only if [ f , 9] = O. In this sense, [ f , g] gives a measure o f the deviation from multi-valued logic.

If f ~< g, we now characterize the elements h E that satisfy f U h = g. When f ~< g, we define ( g \ f ) s and ( g \ f ) l by

g((o) - f ( ~ ) (g \ f )~(o) = 1 - f ( o ) i f f(~0) # 1,

0 if f ( m ) = 1,

g((o) -- f ( m ) ( g \ f ) l ( a ) ) = 1 -- f(~o) if f ( ~ o ) ~ 1,

1 i f f ( ( o ) = 1.

Notice that ( f \ O ) s = ( f \O) t = f and i fg is crisp, then (g \ f )~ and ( g \ f ) l are crisp.

(ii) h = ( g \ f ) l is the largest element of ~ such that f u h=g. (iii) f II h = g if and only if ( g \ f ) s <~ h ~ ( g \ f ) t .

ProoL (i) The statement f U h = .q is equivalent to h(1 - f ) = g - f which is equivalent to

g(~o) - f ( m ) h(a 0 = whenever f ( m ) ¢ 1.

l - f ( o )

Since ( 9 \ f ) s satisfies this condition, we have f U (g\ f )~ =g. Suppose that f U h = g . Then f(co) ¢ 1 implies h(m) = (9\ f )~(m) and f (co) = 1 implies (g\fL.(co) = 0 ~<h(co). Hence, ( 9 \ f ) s <~ h. The proof of ( i i ) is similar and (iii) follows from (i) and (ii). []

Corollary 2.5. There exists an h E ~ such that f U h = g i f and only iff<~ g.

For f E g, we define f ~ E 8 by

1 i f f ( e J ) = 0 , f ~ ( ° o ) = ( l \ f ' ) t ~ ( ~ ° ) = 0 i f f ( o g ) # 0 .

It is well known that ~ satisfies the following Brouwer-Zadeh conditions [2, 3]: f<<,f-~, f<<-9 implies g~ ~< f ~ , f A f ~ = O, f ~ ' = f ~ % More- over, f ~ ~ is the smallest crisp effect that satisfies f ~< f ~ ~ and f ' ~ is the largest crisp effect that satisfies f ' ~ ~< f . The next two results give relationships between ~ and [-1, U.

Theorem 2.6. (i) f ~ and f ' ~ are the largest effects that satisfy f n f ~ = 0 and f U f ' ~ = f, respectively. (ii) f ~ ~ and f ' ~ ~ are the smallest effects that satisfy f N f ~ ~ = f and f tl f ' ~ ~ = 1, respectively.

Theorem 2.7. (i) ( f u g )~= f ~ • g~ = f ~ Ag% (ii) ( f I-I g ) ~ = f ~ U . q ~ = f ~ V g ~. (iii) f ~ ~ U g - ~ = f ~ ~ V g~~ is the smallest crisp effect that dominates both f and 9, (iv) f ' - N g '~ = , f '~ A g'~ is the largest crisp effect that is dominated by both f and 9.

For simplicity of notation, in the next theorem we denote ( g \ f ) , by g \ f . This result shows that (g , 0, 1, \ ) is "almost" a D-poset [5].

T he or e m 2.4. Let f, g E g with f <<,g. (i) h = ( o \ f ) , is the smallest element of 8 such that f u h = 9.

Theorem 2.8. (i) I f f <~g,h and g \ f = h \ f , then g=h. (ii) I f f<~g<~h, then h\g<~h\f. (iii) I f

250 S. GudderlFuzzy Sets and Systems 106 (1999) 247-254

f <~ g <~ h, then

[ (h \ f ) \ (h \g ) ] (~o) = ( g \ f ) ( o J )

i f and only i f one o f the following conditions holds: h(oJ) # 1, f (og) = g(~o), f (og) = 1, g(~o) = 1.

Conversely, suppose that h(co) = 1, f (co) # g(co) and f (o) ) , g(co) ¢ 1. Then again, [ (h \ f ) \ (hkg)](co) = O, but we have

( g \ f ) ( o ) = g ( o ) - f (¢o) _~ O. [] 1 - f ( o )

Proof. (i) If f ~< g, h and g \ f = h \ f , then

g = f M ( g \ f ) = f U ( h \ f ) = h,

(ii) If f ( c o ) = 1, then g(co)= 1 so that (h \g) (co)= 0 = (h \ f ) (co) . I f f ( o ) ) # 1 and g(co) = 1, then (h \g ) ( co )= O<~(h\f)(co). Now, suppose that f (co) , g(co) # 1. By taking the derivative, we see that the function F ( x ) = ( a - x) / ( l - x ) on the interval x E [o, a), 0 < a ~< 1, is nonnegative and decreasing. Hence,

h(oJ) - g(oJ) ~< h(o~) - f (og)

1 - g ( ~ o ) 1 - f ( c o ) '

so that (h\g)(co) < . ( h \ f ) ( o ) . (iii) If h(oJ) # 1, then f(og), g (o) , (h\g)(~o) ¢ 1 and an algebraic computa- tion gives

[ (h \ f ) \ (h \g ) ] (¢o)

= ( h \ f ) ( o ) - (h\g)(co)

1 - ( h \ g ) ( ~ o )

(h(og) - f (co) ) (h(o~) - g(oJ))

(1 - f ( c o ) ) (1 - g ( o ) )

1 - ( h ( o ) ) - g ( o ) ) )

(1 - g ( c o ) )

= g(¢o) - f (oJ ) = ( g \ f ) ( o J ) . 1 - f ( ~ o )

If f (co) = 1, then g(~o) = h(a 0 = 1 and

[ (h \ f ) \ (h \g ) ] (o9) = 0 = (g \ f ) (¢o) .

If f ( ~ o ) ¢ 1 and 9 ( a 0 = 1, then h(~o)= 1. Hence, (h\g)(~o) = 0 so that

[ (h \ f ) \ (h \g ) ] ( co ) = (h \ f ) ( co ) = 1 = (g \ f ) (o ) ) .

Assuming that one of the four conditions holds, the last case to be considered is h(co) = 1, f (o) ) , g(to) ~ 1 and f (co) = g(o)). But in this case we have

[ ( h \ f ) \ ( h \ g ) ] ( o ) ) = 0 = (g \ f ) (co) .

We now compare M and LI with the operations A and V.

Lemma2 .9 . (i) f l ~ "'" U f , = l / f and only i f f t V ' " V f , = l . (ii) f t U . . . U f ~ = l and f~ + • .. + f , <~ 1 i f and only i f f i is crisp, i = 1, . . . , n, and f l + . . . + f n = 1.

Proof. ( i ) I t is clear that the following statements are equivalent: (a) f l U - . . t_l f , = 1, (b) 1 = ( f~M. . -Vl f ' n ) t= 1 - - ( 1 - f l ) ' " ( 1 - - f n ) , ( C ) ( 1 - f l ) • " ( 1 - L ) = 0 , (d) f l V ' " V f ~ = l . (ii) This follows from (i). []

For f , g E d', define the following three sets:

S ( f ) = {~OE(2: f ( ~ o ) ¢ 0 } ,

S , ( f ) = {ogEl2: f ( o J ) = 1},

S ( f , g ) = {o E £2: f(~o)~<g(og)}.

T h e o r e m 2.10. The following statements are equivalent. (i) f M g = f Ag. (ii) S ( f ) N S ( f , g ) C _ S I ( g ) and S(g) n S(g, f ) C_ St ( f ) . (iii) f It g = f v g.

Proof. Suppose that (i) holds. If 0 < f(co)~< g(o)), then f ( o J ) g ( c o ) = f ( c o ) so o)ESl(g). Similarly, if 0 < g(co)<~f(co), then o ) E S l ( f ) . Hence, (ii) holds. Suppose that (ii) holds and f (co) ~< g(co). If f (co) # 0 , then by (ii) we have g(co)= 1. Hence,

( f t_l g)(co) = 1 = ( f V g) (o ) .

If f ( o ) ) = 0, we have

( f U g)(og) = g(o~) = ( f V g)(og).

If g(o)~< f(co) , we proceed in a similar way. Hence, (iii) holds. Finally, suppose that (iii) holds. If f (co) <~ g(co), then

f (oJ) + g ( o ) - f(oJ)g(oJ)

= ( f U g ) ( o ) = ( f V g ) ( o ) = g(oJ).

5L GudderlFuzzy Sets and Systems 106 (1999) 24~254 251

Hence,

( f M g)(co) = f(co)g(co) = f (co) = ( f A g)(co).

By symmetry, if g(co) ~< f (co) , then

( f 71 g)(co) = f(co)g(co) = g(co) = ( f A 9)(09).

Hence, (i) holds. []

It is obvious that if f and g are both crisp, then f gl g = f A 9 and f Idg = f V g. However, this is not quite so obvious if just f is crisp.

Theorem 2.11. The following statements are equivalent, (i) f is crisp. (ii) f M 9 : f Ag for all 9 E o z . ( i i i ) f t 3 g : f V g J b r a l l g @ &

Proof. If f is crisp and 0 < f(co)<.~g(co), then g(co)= 1 so mESt(g) . Also, if 0 < 9(co)<-~f(co), then f ( c o ) = 1 so coESl ( f ) . Hence, Condition (ii) of Theorem 2.10 holds so that f M 9 = f A g and f u g = f v g. Conversely, if f M 9 = f A g for all g E ~, then f 2 = f A f = f so f is crisp. Also, if f U g = f V g for all g E ~, then

2 f - f 2 = f tA f = f V f = f ,

so f is crisp. []

We now give an example which shows that it is possible for f M g = f A g and f Idg = f V g when f and g are both fuzzy. Let f2 --- [0, 1 ] and let a¢ be the a-algebra of Borel sets on f2. Define the functions f , g E g(f2, .~') by

1 if coE [0, ½], f ( c o ) = ½ ifcoE(½,1] ,

1 if coE(½, 1], g(co) = ½ if COE [0, ½].

Then f and g are both fuzzy but Condition (ii) of Theorem 2.10 holds.

3. The commutator

Theorem 3.1. (i) [ f , g ] = - [g,f]. (ii) l f f is crisp, then [fg, h] : f[g,h]. (iii) f[9,h] + 9[h, f ] + h [ f ,g] =0 . (iv) [ f , g ]= f [1 ,g ] + g[ f , l ] . (v) [f ,g] + [ f ' , g ' ] = [ f , 1] - [g, 1]. (vi) I f f<~g<~h, then [ f , g] ~< [ f , h]. (vii) [ [ f , g], h] + [[h, f ] , g] + [[g, h], f ] =h[ f , g ] 2 + g[f ,h] z + f[g,h] 2.

If f~< g, we say that f is crisp relative to g if f (co) : g(co) for all coES( f ) . Notice that for g = 1, this reduces to the usual notion of crispness.

Theorem 3.2. (i) [ f , g ] : 0 i f and only i f f ( c o ) = g(co) for all co E S ( f ) M S(g). (ii) I f f <~ g, then f is crisp relative to g if and only i f [ f ,g] =0.

Proof. ( i ) N o t i c e that [ f , g ] = 0 if and only if [fg(g - f ) ] (co) --- 0 for all co E f2. But this last condition holds if and only if f (co)=g(co) for all t o E S ( f ) A S(9). (ii) If f~< g, then S ( f ) C S(g) so that S ( f ) ( 3 S ( g ) : S ( f ) . The result now follows from (i). []

Corollary 3.3. if' f A g = 0 or if f and g are both crisp, then [ f , g] = O.

We denote the complement of a set A by A' and the indicator function of A by In.

I l i f an d Theorem 3.4. (i) I[f,g]l ~ and [[f,9]] : a only if there exists an A E d such that f =IA + ½1A,

1 and g = I A, + 2 I A. (ii) I f f <-%9, then I f , g] <~ 93 / 4 and [ f , 9 ] = 93/4 if and only if f = 9/2.

Proof. (i) We have

t[f ,g][ = Ifg(g - f ) [ ~<f(1 - f ) .

The function F : [0, 1 ] ~ II~ given by F(x) -=- x( 1 - x) satisfies F(x)<~¼ and attains its maximum at

1 and the unique point x = ½. Hence, I[f,g]l < I[f,9]l (co)= ¼ if and only if f ( c o ) = ½, g(co)= 1

1 The result now follows. or f ( c o ) = 1, g(co)= ~. (ii) Let 0 < a ~ 1. The function F : [0,a] ---* I~ given by F(x) = ax(a - x ) attains its maximum at the unique point a/2. Since F ( a / 2 ) : a3/4 we have F(x)<~a3/4. The result now follows.

We begin with some of the basic properties of the commutator.

Corollary 3.5. [ f , 1] < ¼ and [ f , 1] = ¼ i f and only i f

f=½


We conclude that y/2 is the fuzziest effect relative to g and that f = ½ is the unique fuzziest effect.

The next result characterizes the effects that are commutators.

Theorem 3.6. (i) I f aE[O, 1], then [af, g]>~a[f,g]. (ii) I f f Ug<~h, then [ f ug , h]<<.[f,h] I_1 [g,h].

Proof . (i) Since aE[0 , 1], we have

Theorem 3.8. There exists a g E ~ such that f = [g, 1] /f and only if f<.¼. Moreover, i f f < l , then there exists a unique gl >>- ½ and a unique g2 <~ ½ such that f = [gl, 1] = [g2, 1].

[af, g] =afg(g - af)>~afg(g - f ) = a [ f , g ] .

(i i) Since f , g ~< f IA g ~< h ~< 1, we have

2h + 2fg<.2 + f g +fg<~2 + f h + gh.

Also, since f + g - f g = f O g<~h, we have

2 f + 2g<~2h + 2fg<~2 + fh + gh.

Hence,

h(h 2 - gh - fit + fg) <~ h + fg - gh - fh

<~ h + f g + 2 - 2 f - 2g

and we conclude that

fghe(h - f ) ( h - g) = fgh2(h 2 - gh - fh + f9)

<~ fgh( h + f9 + 2 - 2 f - 2g).

We then obtain that

[ f U g, h] = ( f + g - f 9 ) h2 - ( f + 9 - f 9 ) 2h

= [ f ,h ] + [g,h] - f o h ( h + f9 + 2 - 2 f - 29)

~< [ f , h ] + [g,h] - fgh2(h - f ) ( h - g)

= [f, h] u [g, h]. []

Corol la ry3 .7 . (i) I f ftAg<~h, then [ f o g , h]<. I f , h] + [g,h]. (ii) I f f i g and f ® g ~ h , then [ f , h ] A_ [g,h] and I f @ g,h]<<.[f,h] ~ [g,h].

Proof . (i) This follows from Theorem 3.6(ii) and the fact that [ f , hi U [g, h] ~< [ f , h] + [g, h]. (ii) Since [ f , h] + [g, h] ~< f + g, we have [ f , h] A_ [g, h]. More- over, we have that

[ f ~ g,h] = ( f + g)h 2 - ( f + g)2h

= [ f , h ] + [g,h] - 2fgh

~< [ f , h ] @ [g,h]. []

Proof . I f f = [g, 1] for some g E ¢, then it follows from Corollary 3.5 that f~< ¼. Conversely, assume

that f -.~ a and let

g(o)) = 1 4- ½ [1 - 4f(co)] '/2 .

Then

a(~o) - g2(~o)

= i ' + 1 [1 - 4f(o9)11/2 41

__14 [ l - - 4f(co)] T ½ [1 - 4 f ( m ) ] 1/2

= f ( m ) .

For the second statement, let

[1 - 4f(o9)] U2 , 01((-0) = ½ +

g2(m) = ½ - ½ [1 - 4f(co)] '/2 . []

The following result gives a relationship between the commutator and three-valued logics.

Theorem 3.9. (i) [ f , f ' ] = 0 i f and only i f the values o f f are contained in the set {0, ½, 1 }.

(ii) [If , f / ] [ ~ 4 and [ [ f , f ' ] [ = @ i fandonly i f there exists an A E d such that

Proof . ( i) Note that

[ f , f ] = f ( 1 - f ) 2 _ (1 - f ) f 2

= 2 f 3 -- 3 f 2 + f

= f ( f - 1 ) ( 2 f - 1).

Hence, [ f , f ' ] (~o) = 0 if and only if f (~o) E {0, ½, 1 }. (ii) The polynomial 2x 3 - 3x 2 + x has an extremum

on [0, 1] if and only if 6x 2 - 6x + 1 = 0 or x = ½ zk 6 ~ .

The maximum @ is at x = ½ - 6"~ and the minimum

_ x/g is at x = ½ + 4 " The result now follows. [] 18

S. GudderlFuzzy Sets and Systems 106 (1999) 247-254 253

Theorem 3.10. (i) [ f , h] = [g, h] i f and only if for all 09 E S(h) either f(co) = g(09) or f(o~) = h(09) -9(09). (ii) O<~[f,h]<~[g,h] if and only if for all 09ES(h) either f(09)<~g(09)<~ ½h(09) or h(09)>/ f(09)>~g(09) >1½h(09).

Proof . (i) Suppose that for every 09ES(h) either f (09) =9(09) or f ( m ) = h(09) - g(09). I f 09q[S(h) or if f (09 )=g(09 ) , then clearly, [f,h](~o)=[g,h](09). Assume that ooES(h) and f ( 0 9 ) = h(09) - g(09). We then have

[ f , h](09) = f(09)h(09) [h(09) - f(09)]

= [h(09) - g(09)]h(09)g(09) =- [g, h](09).

Conversely, suppose that [ f , h] = [g, hi and 09 E S(h). Then by the previous calculation we have

f(09) [h(09) - f(09)] = g(eg) [h(09) - 9(09)]-

Hence,

f (09) [h(09) - f (09) - g(09)]

= g(09) [h(o~) - f (09) - g(09)],

so that

[f(09) - g(09)] [h(w) - f (09) - 0(09)] = O.

It follows that either f (09) = g(09) or f (09) = h(09) - g(09).

(ii) Suppose that for every ~oES(h) either f (09) ~< g(09)<-.. ½h(09) or h(09)>l f(09)>l-g(09)>t. ½h(09). I f 09 q~ S(h ) then clearly,

O<.[f,h](09)<[g,h](aQ,

so suppose that 09ES(h). A necessary and sufficient condition for [f,h](09),[g,h](m)>~O is that f(09), g(~o)~< h(og) so we shall assume this condition. For 0 < a ~< 1, the function F : [0, a] ---, R defined by F(x) =x(a - x) satisfies F(b)<.F(c) i f and only if b<~c<<. ½a or b>~e>~ ½a. Letting a = h(~o) we have

af(09)[a - f(og)] = [ f , h](09)

~< [g, h](09) = ag(09) [a - 0(09)]

if and only if f ( e 8 ~< g(¢o) ~< ½h(~) or f (09) >7 g(09) >~ ½(09). The result now follows. []

I f f , g ~< h, then Theorem 3. lO(i) gives a necessary and sufficient condition for f and g to have the same

degree of fuzziness relative to h and Theorem 3.1 0(ii) gives a necessary and sufficient condition for g to be fuzzier than f (or f to be crisper than g) relative to h. For f ~< h, we define the relative complement o f f in h by f h = h - f . Of course, we then have f l = f~. Theorem 3.10(i) states that f and 9 have the same degree of fuzziness relative to h if and only if for all 09ES(h) either f (09)= g(09) or f ( 0 9 ) = gh(09). The following corollary gives similar characterizations of fuzziness (as opposed to relative fuzziness).

Corollary 3.11. (i) [ f , 1] = [g, 1] i f and only if there ex&ts an A E d such that f = g l A + g'IA,. (ii) [ f , 1]~<[g, 1] i f and only if there exists an A E . ~ such that f(09)<.g(09)<.½ for all 09EA and f(09) >~ g(09) >~ ½ for all 09 E A'.

The partial order in Corollary 3.1l(i i) has been extensively employed in the past [4,10,13]. The last result of this section shows that f ~-+ [ f , h] is a valuation that is invariant under relative complementation.

Theorem 3.12. (i) [ f , h ] = [ h - f ,h] . (ii) [ f v g , h] + [ f A y , h]=[f ih] + [g,h].

Proof. (i) We have

[ f , h ] = f h ( h - f ) = ( h - f ) h f = [h - f,h].

(ii) Since f V g + f A . q = f + 9 and ( f V 9 ) 2 + ( f A 9) 2 = f 2 _1_ 92 we have

[ f V 9,h] + [ f A y , h]

= ( f Vg)h 2 - h ( f V g ) 2 + ( f A g ) h 2 - h ( f A g ) 2

= ( f + g)h 2 - h ( f 2 + y 2 ) = [ f , h ] + [g,h]. U]

4. Relative entropy

Our previous discussion applies not only to measurable fuzzy sets but to arbitrary fuzzy sets as well. We now introduce a probability measure # on (g2,~&) and for f Eg(f2, s¢) we define the probability of f to be its expectation E ( f ) = f f d p . Notice that f ~ E ( f ) is a probability measure in the sense that 0 < ~ E ( f ) ~ < l and if f J-g, then E ( f ® g ) = E ( f ) + E(g). More- over, f ~-* E ( f ) is countably additive in the sense


that if jr,. is an increasing sequence of effects, then by the monotone convergence theorem, we have

lim E(j~) = E(lim j~ ).

Also, if f = l A is crisp, then E ( f ) = p ( A ) so for crisp effects E reduces to the usual probability.

I f f~< h, we call H h ( f ) = E ( [ f , h]) the entropy of f relative to h. Notice that Hh(f) gives a measure of the fuzziness (more precisely, the average fuzziness) o f f relative to h. The next theorem shows that Hh has the desired properties of a relative entropy. The proof follows from results in Section 3.

Lemma 4.3. E(l[f, g]l) ~< 2[E(f2)]I/2[E(g2)]I/2.

Proof. Applying Schwarz's inequality, we have

E(l[f,g][)

= f lfg2-gf2ld~<~ f fg2d~+ /gf2d,

<. 2 /fgd.<~2 ( / fZ du) l/2 ( / v2 d~) '/2

= 2[E(f2)]I/Z[E(92)] 1/2. []

Corollary 4.4. H q ( f ) ~< 2[E(fZ)] 1/2 [E(g2)] 1/2.

Theorem 4.1. (i) H h ( f ) = 0 if and only if f is crisp relative to h almost everywhere. (ii) Hh(f) is' a maximum if and only if f = ½h almost everywhere. ( i i i ) I f for almost every toED we have either f(co) <~g(co) <~ ½h(o9) or f(co) >'g(co) >>- ½h(co), then Hh(f)<~Hh(g). (iv) Hh(f )=Hh( fh) . ( v )Hh( f v g ) + H h ( f Ag)=Hh(f )+Hh(g) . (vi) I f a E [0, 1 ], then Hh (a f ) >>- aHh ( f ) . (vii) I f f I g, then Hh( f ® 9) <~Hh(f) + Hh(Y).

For f C g , we call H ( f ) = H l ( f ) = E ( [ f , 1]), the entropy of f . The entropy function H has been investigated in the literature and is an example of a measure of fuzziness [4,6,10,11,13]. From Theorem 3. l(iv) we see that Hh is an increasing function of h and H gives the highest value. Conditions ( i ) - (v) in the following corollary have been previously demonstrated in the literature.

Corollary 4.2. (i) H ( f ) = 0 if and only if f is crisp almost everywhere. (ii) H ( f ) is a maximum

i almost everywhere. (iii) I f for if and only i f f = 1 almost every co E f2 we have either f(co) <~ g(co) <~

or f(co) >f g(co) >- ½, then H ( f ) <~ H(g). (iv) H ( f ) = H ( f ' ) . (v) H( f V g) + n ( f A g) = H ( f ) +H(g). (vi) Ifa C [0, 1], then H(af) >>. a l l ( f ) . (vii) l f f A_ g, then H ( f q3 g) <~ H ( f ) + H(g).

Finally, we present a curious result that resembles the Heisenberg uncertainty principle.

References

[I] P. Busch, P. Lahti, P. Mittlestaedt, The Quantum Theory of Measurement, Springer, Berlin, 1991.

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[3] G. Cattaneo, G. Nistic6, Brouwer-Zadeh posets and three- valued Lukasiewicz posets, Fuzzy Sets and Systems 33 (1989) 165 -190.

[4] A. de Luca, S. Termini, A definition of a non-probabilistic entropy in the setting of fuzzy set theory, Inform. and Control 20 (1972) 301-312.

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[6] B.R. Ebanks, On measure of fuzziness and their representations, J. Math. Anal. Appl. 94 (1983) 24-37.

[7] D. Foulis, M.K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994) 1331-1352.

[8] S. Gudder, E-test spaces, effect algebras, Found. Phys. 27 (1997) 287-304.

[9] A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam, 1982.

[10] J. Knopfmacher, On measures of fuzziness, J. Math. Anal. Appl. 49 (1975) 529-534.

[11] S.G. Loo, Measures of fuzziness, Cybernetica 20 (1977) 201-210.

[12] G. Ludwig, Foundations of Quantum Mechanics I, Springer, Berlin, 1983.

[13] E. Roventa, On the degree of fuzziness for a fuzzy set, Fuzzy Sets and Systems 36 (1990) 259-264.

[14] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 328 -353.

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Connectives and fuzziness for classical effects