A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms

Fuzzy Sets and Systems 11 (1983) 115-134 North-Holland

115

A GENERAL CONCEPT OF FUZZY NEGATIONS AND IMPLICATIONS t-NORMS AND t-CONORMS Siegfried

CONNEtX’IVES, BASED ON

WEBER

Fachbereich Mathematik, Johannes Gutenberg-Uniuersitlit,

Maim, Federal Republic of Germany

Received August 1982 Revised November 1982 All known connectives ‘aod’l’or’ for fuzzy sets or some classes can be introduced as t-noms/t-cooorms, where Liog’s representation theorem is used as a basic tool, and which is illustrated by various known and new examples (Section 2). Given a strict negation function and one connective, the other can be constructed, so that the corresponding De Morgan law is valid. In case of given Archimedean connectives, there can be constructed negation functions (Section 3). Given a non-strict Archimedean connective, a negation function and the other connective can be constructed, so that in addition to the De Morgan laws, the excluded middle law and the law of non-contradiction are valid, i.e. the negation function results to be complementary (Section 4). In function of connectives and negation three types of fuzzy implication operators are introduced, which include almost all known implications, and that of type I using ‘and’ only; of type II using ‘or’/‘non’; of type III using ‘and’/‘non’. In the non-strict Archimedean cases the formulas become particularly lucid (Section 5). Finally the different types are compared with respect to some logical properties (Section 6). Keywords: Fuzzy connectives, Negation, De Morgan, Archimedean.

Complement,

Implication,

t-norm,

1. Introduction Since Zadeh [17] introduced min(cp, $I),

mdcp,

$1,

fuzzy sets and suggested to use 1 - cp

(1)

as intersection, union, negation, many authors have given justifications for this setting. Others, e.g. [6], [5], [7], [16], proposed other connectives or negations resp. Recently a few papers appeared, considering instead of (1) the use of cpnh

CPU+,

n(cp),

(2)

where n is a t-norm, u a t-conorm, n any negation function and the operations are done pointwise; for more references see e.g. [8]. In the present paper I will discuss which of the classical laws of distributivity, De Morgan, excluded middle and non-contradiction can be preserved, given various constructions and examples. Of special importmce are the Archimedean 0165-0114/83/$3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)

116

S. Weber

t-norms and t-conorms, anb

i.e. those that can be written [9] as respectively

=f’-“(f(a>+f(b))

and

sub

= g’-“(g(a)+g(b))

(3)

with so-called additive generators f, g and its pseudo-inverses f’-“, g’-“. Furthermore I give various constructions of implication operators a + b for a, b E [0, 11, using t-norms, t-conorms and negations. In Section 2 I give definitions and representation theorems concerning tnorms and t-conorms, completed by various examples for non-Archimedean, strict Archimedean (S) and non-strict Archimedean (NS) cases. The only properties which cannot be preserved, when working with (one of) the connectives in (2) other than in (l), are (one of) the laws of distributivity, absorption and idempotency. Section 3 contains the constructions of De Morgan-connectives for strict negation functions. While this work was under preparation Dombi published a paper [3] in which he independently arrives at our results for the cases (S). In most of the literature the conjugate connectives are treated, i.e. those with n(a)=a”:=

l-u.

(4)

But in general these connectives do not hold the laws of excluded middle and non-contradiction, as can be seen even for Zadeh’s connectives (1). Therefore in Section 4 I present a construction which fulfils also these laws, and so I will call them complementary connectives. This is possible for cases (NS). The rest of the paper was motivated by Bandler and Kohout [l], where they consider the implication operators most used in the literature. In Section 5 I present the construction of three types, which contain the known ones. In case (NS) the first two types coincide and lead to a-,b=bua’

(5)

and the third type to aAb=bva’

(6)

n(a) = a’ = a:,

(7)

where is the complement of a based on I-I from Section 4 and where v stands for max. The representations (5) and (6) permit the interpretation of “if a then b” as “b or non a”. In Section 6 I compare the three types concerning contrapositive symmetry and contradiction. Remarks on crispness and fuzziness conclude the paper.

2. Fuzzy conmctives The following

induced by t-norms and t-conomw

operations

in [0, l] will be fundamental

for the present paper.

2.1. Definition. (a) A function n : [0, l]x[O, l] + [0, l] will be called t-norm iff (i) I-I is non-decreasing in each argument,

Fuzzy connectiues, negationsandimplications

117

(ii) 17 is commutative, (iii) n is associative, (iv) II has 1 as unit. A t-norm will be called Archimedean iff (v) n is continuous, (vi) n(a, a) C a for all a E (0, 1). An Archimedean t-norm will be called stict iff (vii) n is (strictly) increasing in (0, 1) X (0, 1). (b) A function u : [O, l]x[O, l] + [O, l] will be called t-conorm or Archimedean or strict iff u has the same properties as in part (a) with the modifications: (iv) u has 0 as unit, (vi) ~(a, a)> a for all a E (0, 1). We remark that for t-norms and ‘t-conorms (vii) always implies (vi). In the following I will state the representation theorems in the form I need; for proofs and more details see [9], [ll], [12]. 2.2. Theorem. (a) A function n:[O, l]x[O, l]+ [0, l] is an Archimedeun t-norm 8 There exists a decreasing and continuous function f: [0, l] + [0, m] with f(1) = 0 so that anb =f’-“(f(u)+f(b)), where f-l)

is the pseudo-inverse off,

f(-l)(y) := f-‘(y) i0

dejined by

if Y E10,f(O>l, if Y E[f(o), 4.

Moreover, n is strict if f(0) = co. (b) A function u : [O, l]x[O, l] + [0, l] is an Archimedeun t-conorm ifl There exists an increasing and continuous function g: [0, l] -+ [0, co] with g(0) = 0 so rhat au b = g’-“(g(u) where g’-”

+ g(b)),

. the pseudo-inverse of g, defined by IS

g'-"(y)

:= 1

g-‘(y) if Y ED, dl)l, 1 if Y ECdl), ~1.

Moreover, u is strict $7 g(1) = 03.

2.3. Remark. The functions f and g are called additive generators (briefly: generators) of n and u, respectively. They are unique except for multiplication with positive numbers. In the non-strict case we will call the additive generator with f(0) = 1 respectively g(1) = 1 the normed generator. Note that

f’-‘Yf (x)>= 5 and analogously

for g.

f W’YyN = fink

f NW,

S. We&r

118

2.4. Remark.

As fuzzy sets can be seen as functions

cp:n-[0,

l]

from any non-empty connectives

set 0 to [0, 11, a t-norm

defined pointwise. For simplicity the operations in [0, 1-J. 2.5. Example.

and t-conorm

u induce

I will present examples and results in terms of

Consider

q,(a,b):=

a b {0

n,

n

ifb=l, if a=l,

i&u,

a ifb=O, b if a = 0,

b):=

otherwise,

{ 1 otherwise.

is the weakest t-norm, u, is the strongest t-conorm.

They are not continuous.

2.6. Example. Consider a A b = min(a, b), a v b = max(a, b). A is the strongest t-norm, v is the weakest t-conorm. They are continuous, but not Archimedean. They are proposed as connectives for fuzzy sets by Zadeh [17] and the most frequently used in the literature. 2.7. Exa&ple. Consider ~(a, strict with generators

b) := a * b, ~,-,(a, b) : = a + b - a . b. Q, and u0 are

f(x) = -1n x for no with f-‘(y) = exp(-y}, g(x) = -In(l-x) for u. with g-‘(y) = 1 -exp(-y}. These connectives were already mentioned in [l] and conSidered later by various authors, e.g. Goguen [6] has treated the ‘soft’ intersection r!+ 2.8 Example. KJu,

Hamacher b):=

[7] proposed to use n = H, and u = Hz for y > 0, where

u-b y+(l-y).(u+b-ub)

define a family of strict t-norms with generators f,(x) =tln

vc(t-r)x

with f;‘(y)

= l-(~~~~~x~~~

. y),

and where H;(u,

b) :=

u+b-ub-(l-y)ub 1-(1-y)ub

define a family of strict t-conorms

with generators with g;‘(y) =

1- exp(-39) 1 - (l-

r)exp(--ryl

’

Fuzzy connectives, negations and implications

The H, decrease from the strongest Ho to the weakest IL= connectives always found by forming the corresponding limits, with f;l(y)

119

n, the extreme e.g.

1 =1+y’

including HI =nc,. The HlfS increase from the weakest Hz to the strongest Hz= L$, including the Einstein operator G, which gives the addition theorem for tanh. 2.9. Example.

Dombi

[3] proposed connectives n and u given by the generators and

f(x) = (yr

g(x) = (&)

for A > 0. Also these are strict. 2.10. JZxample. Consider n,(a, b):=max(a+b-LO), n, and LJ, are non-strict with normed generators

~,(a,

b):=min(a+b,

1).

f(x) = l-x for n, with f’-“(y) = max(l- y, 0), g(x) = x for u, with g’-“(y) = min(y, 1). These connectives are often cahed ‘bold’ and have been treated by many authors, e.g. Giles [5].

2.11. Example. Yager [16] proposed to use n = C, and u = D,, for p Z=1 (it can be done for p > 0), where q(u,

b) := 1 -min([(l-

define a family of non-strict

u)~ + (lArchimedean

with f;‘(y)

fpb)=o-x)p

b)P]“P, 1)

t-norms with normed generator

= 1- y’lp for y s 1,

and where D,,(a, b):=min([aP+bp]l’P, define a family of t-conorms gpw=xp

1)

with normed

generator

with g,‘(y) = Y”~ for y S 1.

The C, increase from the weakest C,, = n,,, to the strongest C,= A, including C1 = nm. The Dp decrease from the strongest Do = u, to the weakest D, = v, including D1 = u,. 2.12. Example. I suggest alternatively also non-strict Archimedean, where WA(a, b):=max

u+b-l+Aub l+A

the use of n = WA and u = VA for A > -1,

90

120

S. We&r

with normed generator

fhW = l-

ln(l+h * x) ln(l+h)

with fi’(y)

= i [(‘l + A)lmY - 11 for y < 1,

and where U, (a, b) : = min(a + b + Aab, 1) with normed generator a(x) =

ln(l+A * x) ln( 1 + A)

with g;l(y)=i[(l+A)Y-l]

for ~61.

Here both families increase, the W, from the weakest W-r = & to the strongest W,= no, and the U, from the weakest U-r = u. to the strongest U, = Q, including also n, = W. and u, = U,. These connectives appear to be complementary in the sense of Section 4. The lJ, appeared already as ‘addition-rule’ for Sugeno’s A-fuzzy measures [13]. In a proceeding paper [14] I used U,, and WA as special ‘decomposable rules’. Now we pass to the discussion of laws for connectives. In this section we look at the laws of distributivity: un(buc)

= (anb)u(anc),

au(bnc)=(aub)n(auc),

(Dil) (Di2)

absorption : (anb)ua

= a,

(Abl)

- (aub)nu

= a,

(AW

idempotkncy: aua=a,

(IU

ana=a,

uw

and give the relations

in:

2.13. Theorem. (a) (Dil) j (Abl) j (Mu) + (u = v), (b) (Di2) j (Ab2) 3 (Idn) j (n = A). Proof. (a) (Dil) with c = 1 leads to (Abl). (Abl) with b = 1 leads to (Mu). andb~aimpliesthata=(auO)~(aub)~(aua)=a,i.e.(aub)=a,sou=v. (b> BY ~@sy.

(Mu)

This result tells us that it is impossible to fulfil the mentioned laws except for n= A or u= v. In this sense, what now follows can be called a non- or semi-distributive theory for connectives.

Fuzzy connectiues, negations

3. De Morgan

and implications

121

connecfives

In view of the following sections I will distinguish various types of negations, including the different concepts which exist in the literature. 3.1. Definition. A function n: [0, l] --* [0, l] will be called negation (function) iff (i) n(O) = 1, n(1) = 0, (ii) n is non-increasing. A negation will be called strict iff (iii) n is decreasing, (iv) n is continuous. A strict negation will be called involution iff (v) n(n(a)) = a for all a. Clearly, for any negation n, (v) implies (iii), n-l = n. Now we can look at the laws of De Morgan: n(aub)

= n(a)nn(b),

(DMl)

n(anb)

= n(a)un(b).

(DM2)

For involutions these two laws are equivalent. For strict negations necessary, and we are led to the construction of:

this is not

3.2. Theorem. Let n be any strict negation. (a) To any t-norm n (continuous) define (sub):=

n-‘(n(a)nn(b)).

Then: (i) u is a t-conorm (continuous), which fuZf;Zs (DMl), (ii) u is Archimedean (strict) with generator g =f 0 n, if n is Archimedean (strict) with f, and g(1) =f(O). (b) To any t-conorm u (continuous) define (anb):=

n-‘(n(a>un(b>>.

Then: (i) n is a t-norm (continuous), which fulfils (DM2), (ii) n is Archimedean (strict) with generator f = g 0 n, if u is Archimedean (strict) with g, and f(0) = g(1).

J?roof. (i) Trivial. (ii) For part (a), (au b) = n-l 0 f’-“[(f

0 n)(a) +(f 0 n)(b)].

Note that any strict negation can also be seen as the normed generator of a t-norm, and composed with another generator of a r-norm (t-conorm) gives a generator of a t-conorm (t-norm). Therefore the following construction requires an Archimedean

case.

122

S. Weber

3.3. Theorem. To an Archimedean t-norm dean t-conorm I-I with generator g define jql) : = f-1) 0 g for f(O) s g(l), q2) : =

0f

g’-1’

n with generator f and an Archime-

for f(O) a g(l).

Then: (i) nCl,, nC2)are continuous negations, and nCl, decreasing in (0, g-‘(f(O))) only, nt2) decreasing in (f-‘(g(l)), 1) only. (ii) n(k) fu@k (Dbfk) and is unique besides taking other generators. (iii) If furthermore f(0) = g(l), then: nil,, nt2) are strict negations, n(,, = n&, and ql, = nC2)9 nC1) involution

e

Proof. (i) Trivial. For (ii) n(,,(a u b) = f”’ 0 g 0 g’-“(da) = f’k(4 I 0 =f’-“[f

nC2) involution.

+ g(b))

+ g(b))

if g(a) + 0)
= n&)nn&), where variously we have used that f(0)
0 n-‘(n(a)>+

g 0 n-‘(n@))l = f’-“LMa))

+fb@Nl,

i.e. g OJI-l is as f a generator for n, which implies that n = f’-“(A * g) for some A > 0. The proof for q2) follows by analogy. (iii) Now ql) and nC2)are strict, therefore in the formulas appear the inverses f-’ and g-’ respectively, which immediately gives the rest. 3.4. Remark. Without 3.3 would give ndl)

the restrictions

= f-‘Ml))

n,,,(O) = g-V(O))

>0

about f(0) and g(l),

the constructions

of

for g(l)
C 1 for f(O) < g(1).

The results of 3.2 and 3.3 for strict connectives can be found also in [3], the roles of a generator and its inverse interchanged. 3.5. Notation. (a) n, u will be called De Morgan-connectiues with respect to n if one of the De Morgan laws is valid, and strong if both are valid. I will use the notations u = n”, n = un. (b) (Strong) De Morgan-connectives with respect to n(a) = a* : = 1 -a will be called conjugate connectiues, and the notations u = n*, n = u* are used.

Fuzzy connectiues, negations and implications

3.6. Example. All pairs from Examples clearly f(x) = g(l-x) must be. 3.7. Example.

123

2.4 t; 2.11 are conjugate

The pair of connectives from Example

connectives;

2.12 leads to the involution

n*(u) := n(l)hb) = n(2)kb) =z, which is an example

of a complement,

treated in Section 4.

3.8. Example. n= WA, u= u, are both non-strict Archimedean. normed generators, we are lead to the strict negations q2) = fA from 2.12. q1, = f2 3.9. Example.

Taken

the

n = WA, u = no are mixed with f(0)
n(l,(a)=

4. Complementary

~[(l+*)l+‘n’l-L’-l]

if a
0

if a 2 1-e-l.

conmxtives

For Boolean connectives, besides all the laws mentioned, tary laws are valid, excluded middle: aLJn(a)=

1,

also the complemen(EM)

non-conrradiction: ann(a)=O.

(NC>

These are the only laws which are not fulfilled by Zadeh’s connectives A, v with respect to n(u)= u *. The bold connectives n,, u,, however, fulfil them with respect to the same negation. Therefore, what will be exposed in this section can be considered as theory of (non-distributive) complementary connectives, generalizing that for the bold ones. In [14] I have introduced this concept in the context of decomposable measures. This is done in an analogous manner to the construction of pseudo complements in the Brouwerian lattice [O, 11, see e.g. [2], the first one used already by Goguen [6] for certain operations including also no. 4.1. De6nition.

(a) For any t-norm n define

(b+a):=sup(z

1nnzZb},

ak:= (0+-a).

(b) For any t-conorm

u define

(b-a):=inf{y 1auyab}, aL:= (l-a).

124

Clearly,

S. Weber

a:, and aL are negations

4.2. Remark.

(in the sense of 3.1).

(a) Let n be any strict t-norm,

ak= a’h = a’us=

then

1 if a =O, 0 ifa>

I

is the weakest negation and will be denoted by n,.,. (b) For any strict t-conorm u, aL= a’” = a’rlr=

1 if a
I

is the strongest negation and will be denoted by n,. (c) In the strict cases the De Morgan constructions from 3.2 do not apply with respect to a& or aL, because they fail to be strict negations. 4.3. Theorem. (a) Let n and u be any non-strict Archimedean t-norm with generator f, respectively r-conorm with generator g. Then (3 QA = f-‘(f(O) -f(a)) and C = g-‘k(l) - g(d), (ii) a& and aL are involutions. (b) Let n be any non-strict Archimedean t-norm with generator f, and u=n’ the corresponding De Morgan-t-conorm with respect to n(a) = a,&. Then (iii) u has generator g(a) = f(O)-f(a), (iv) aL = a:, = : a’, (v) (EM) and (NC) hold. Proof.

[14] or (a) directly and (b) as corollary

from Section 3.

4.4. Notation. In the non-strict Archimedean cases I will call a:, (or aL> the complement to a with respect to n (or u), n’ (or u’) the complementary t-conorm to n (or t-norm to u), n and n’ complementary connectives. 4.5. Remark. In the non-strict Archimedean cases: I?(,) := (n’)* is a t-norm with f(,)(a) = f (0)- f (1 - a), 19~~):= (n*)’ with different f&u) = f (1- a’).

is a t-norm

4.6. Jhample (Connectives alternative to Yager’s). Yager’s C, with fp induces Yager’s 0, = Cz with g, = ft, and this induces &jP = 0; with f(z)*(x) = g;(x) = gp(x’). So I propose as alternative connectives A = DL, u = D, which are complementary with respect to t.- aD. t = (l- (p)l’P, apeand where ?(a, b) = [max(ap + bP - 1, O)]“’ with normed

generator

fp(x) = 1 - xp.

Fuzzy connectives, negations and implications

125

This t-norm Q already has appeared in [12]. Note, that C,,,, has a more complicated form. 4.7. Example (continuation respect to

of 2.12 and 3.7). WA and U, are complementary

with

l-a a;=l+*.a.

But if one prefers to use a* instead of a;, then also we can find connectives, which fulfill (EM) and (NC): For A 3 0 we have U, au,,,, therefore E . 4.8. Remark. Let us have any non-strict

Archimedean t-conorm u as one connective. Having no idea of what could/should be the other and the negation, then can/shall be recommended the use of n(a) = a; and n = u’ from this section. On the other hand, having already an idea of n (or n), then can/shall be recommended the use of one of n(,, and q2) (or n = u”) from Section 3.

5. Fuzzy implication operators

This and the following section were motivated by the paper by Bandler and Kohout [l], where they listed and discussed the following first six implications, already known from the current literature and listed in ‘order of fuzziness’, see [l]. The seventh implication has been added by Willmot [15]. For references and more details see [l], [15] and the highly recommended overview paper by Gaines

r41. 5.1. Detinition.

a*lb= a+,b=

1 0

ifa
1

ifasb,

0 otherwise, a+,b=

1 if a G b, 1 b otherwise,

a-,,b=min(b/a, a+,b=min(l-a+b,

l),

where b/0:=1, l),

a +6 b = max(b, 1 - a), a +7 b = max(min(a,

b), 1- a).

126

S. Weber

I will present and discuss three types of implication, which contain these special implications, as well as others, which could be reasonable too. 5.2. Notation. A fuzzy implication operator (briefly: implication) -will be defined as a binary operation on [0, 11, which extends the Boolean implication, i.e. a+b 0 1 0 1 1 ---I-- 1 0 1 and which can be expressed as function of connectives n, u and negation n. As order between connectives, negations and implications we consider the order between functions. I will restrict the discussion to the three following 5.3. Dehition.

types.

Type I:

ay+=b:=sup(z)

allz~b}=(b+u).

Type II:

Type III: a?b:=

n(ann(anb)>.

5.4. Special cases of type I: aTb=

1 if a-Cl, b otherwise,

aTb=a+,b, aTb=aB4b, a-pb=a+,b. 5.5. Remark. The family of implications especially we have

.where 2

7

of type I is non-increasing

is the strongest one between the implications

in n,

of type I and also

between the implications + with 1 + b = b, and where +a is the weakest one of type I. Furthermore, for a < b always a 7 b = 1.

Fuzzy connectives,

5.6. Theorem.

negations and implications

127

(a) Let II be an Archimedean t-norm with generator f, then:

aFb=f-‘(f(b)-f(a))

fora>b.

(b) Let furthermore n be non-strict and b’= b;; then a?b

= (b’na)‘.

Proof. (a) For all .z>f-‘(f(O)-f(a)) there is (anz)Sbezcf-‘(f(b)-f(a)). Taking the supremum over all such z gives the result. (b) BY 4.3, (b’na)’ = f-‘(f

which is a? Clearly,

(0) -f

0 f’-“(f

(0) -f(b)

+f (a)))

b by part (a).

in the strict case part (b) does not remain valid.

5.7. Special cases of type II: a&b

=

1 if a<1 or b>O, I 0 otherwise, 1 if a=0 or b>O, 0 otherwise,

a”v-b= u.

a&b=a+lb, a>b=a+,b.

5.8. Remark.

The family of implications

+

of type II is non-decreasing

in u

when n is fixed, and is non-decreasing in n when u is fixed, but nothing obvious when n depends on u. Especially we have L>-,>A.>L>-*, u, also -*1>+=2,

but +i

L-43

is

”

and +5. e.g., are incomparable.

Clearly,

is the absolutely strongest implication and h2 is the weakest one of type II, still weaker than +3. Also for type II, a S b implies ash = 1.

S. Weber

128

5.9. Theorem. negation;

‘(a) Let u be an Archimedean

t-conorm,

with generator g.and n any

then:

ash

= n(g-‘(g(a)-

(b) Let furthermore

for a > b.

g(b)))

n be strict and n = U” the corresponding

De Morgan

t-nom

then : ash

= n(b)Fn(a).

(c) L.et furthermore u be non-strict, t-norm ; then :

n(a) = aL and n = u’

the corresponding

complementary

a+b=bua’=aFb.

Proof. (a) Similar to 5.6(a). (b) By part (a) for a > b, = f-‘(f (n(a))-

ash

f (n(b)))

with f = g 0 n-’

by 5:6(a).

= n(b)Tn(a)

For part (c), a+b

= b’cr,a’

by

part

(b)

by 5.6(b)

=(anb’)‘=a~b

= bua’. 5.10. Remark.

For stiict u and n(a) = aA = n,(a), instead of 5.9(c) there is

5.11. Special cases of type III: a$b=a*6b, a+b=a--t,b, a>b=a+,b.

5.12. Remark. Nothing

is obvious about the order of implications

s

i& incomparable

,to G

= 3;

G

is incomparable

both to d6 and ++,

of type III, e.g.

;

Fuzzy connectives, negations and impkaiions

129

but on the other hand e.g. -5’+%>+1,

5.13. Theorem. a>

(a) Let n be any involution

and u=n”;

then:

b = n(a)u(anb).

(b) Ler furthermore n be non-strict Archimedean,

n(a) = aA and. u G n’; then:

a*b=(bva’)=(b’xa)‘. Proof.

(a) Trivial.

For part (b),

b = a’u(anb)

a*

by part (a)

= g’-“(g(a’)+g(l)-

g 0 g’-“(g(a’)+g(b’))) by f(x) = gb’) = g(l) - g(x)

= b I a’

if da’> + db’) s g(l),

if g(a’)+g(b’)ag(l)

if a’sb, if a’==b = bva’=(br\a)‘. = b I a’

We will finish this section by illustrating tive examples, and include a comparison. 5.14. Example

(continuation

some of the results for two representa-

of 2.8). By 5.6(a),

a4b=b+(r-Ob(l-a) a+(y-,l)b’(l-a) H,

for “b*

Directly or by 5.5 it is seen that the family weakest ab ab+a-b’

azb=min to the strongest I

\

(0

of 7

increases in y frdm the

1

ifb=Oanda>O,

which is the strongest one with 1 “* b = b and a + 0 = 0 for all a > 0, but whibh is ; still weaker than H,-=rr,-, see 5.5.

130

S. Weber

Simple calculations -5>+>*

give that * .>H,=-s4>.

HZ

but for y > 2 all the H,-

are incomparable

5.15. Example (continuation

-=+ W-1

to + 5, taking b 5 ll(y - 1) and a > b.

of 2.12 and 4.7). By 5.9(c), 1-a+b+A

aTb=a*b=

The family of w,

* *>+>ws, HO

-b

l+h*a

for a>b,

decreases in A from the strongest

mv

to the weakest -=-4, W,

including

w,-=+~.

On the other hand, by 5.13(b), 1-U l+Aa’

a*b=bv-

5.16. Remark. For all (y - 1) (1 +A) s 1, --+>-, WA

H,

and for the other pairs of y, A the corresponding

implications

are incomparable.

Proof. For b
e

aTb>aTb

l>b(y-l)(l+A).

The right inequality is valid for all b, if (y - l)(l+ A) G 1. On the other hand, if F;i)Jll+ A)> 1, there can be found b < 1 with 12 b(y - l)(l +A) and C.-Jwith The tirst result can be illustrated: ->->

W-1

WA

yy=+w,”

->+

H I+I,~L+*,

H>T1+1,~1+*1

for all -l
CO,

HZ

for ail O
6. Logical comparison of the implication operators

For short and a more lucid illustration, I will display here some results in form of tables. We use the abbreviations (S) for strict, (NS) for non-strict Archimedean.

Fuzzy connectives, negations and implications

131

6.1. If one side is crisp, we have the values of Table 1. Table

tYPeI

I

O..-b...l

azb

a

a’

1

0

type

a&+b

011

a,!, and b’ = b;)

1. (a’ =

II

0111

1

a

da)

1

I

0

b

ash

O...b-.-l

1

III 0 . . . be..1 1

011

n(b’)

Note, that for (NS) the following a*l=l

type

I

and

1

a

1

1

n(a)

n(annW)

0 n(n@N

1

are valid also for type III:

l*b=b.

6.2. Here consider Table 2. Table

I II

1 1

III

1 avn(a)

forn(S)orn=A,n=n, for n=A, n involution for n(NS), n(a) = a,L,

ava’

Note, that for I/II

2

a + a is a ‘strong tautology’.

6.3. In the case of Contrapositive Table Type

n(b) --, n(a)

I

a:b

for n(NS),

a&pb

fern(S)

a+b

for

II

a-b aT III

a*b

symmetry

we have Table 3.

3

n(a) = aA n=n,

orn=A,

u(NS),

n(a)=a’

foru(S)oru=v,

b for n = u”, forn(NS),(S)orn=A,

n=n, n involution

n(a)=&

Note, that in cases (NS) always the contrapositive

symmetry

is ‘strong’.

S. Weber

132

6.4. Defining (ut,b):=(u-+b)rl(b-+a): we can consider contradiction

as shown in Table 4.

Table 4

I

0

forn(S)orn=A, for n(NS), n=u’,

n=n,

(a’ua’)n(aua)

Fata’)n(aua)

foru(S)oru=v, for u(NS), n = u’,

n=n, n(a) = a’

0 a A n(a)

for n(NS), (S) or n = A, for n=A, n involution

II III

n(a)=a’

n(a)

=

aA

Clearly, this list is far from complete, but the other cases can be treated rapidly, e.g.: Let n be (S) or (NS) and n an involution, then by 5.6(a),

Taking

e.g. n=~

and n(u) = a*, we obtain, see [l]:

u~~u*=min

(

“,&). U

6.5. Towards a measure of fuzziness. In 6.2 has already appeared what can be used to define crispness and fuzziness respectively, but I prefer to take as measures of ctispness of a: C(u):=u+u, and of fuzziness of a: F(u) := n(C(u)). For n=

A, $~)=a*

we have, see [l],

C(u)=uv(l-a). For n(NS),

n(u) = aA we get

C(a) = f-W

mWK9 -fW, fb)l>,

and especially C(u)=)l-24

for n= n,.

6.6. Final remarks. For the non-strict Archimedean cases the results of Sections 4 and 5 suggest three reasonable combinations, shown in Table 5.

connect&s, negations and implications

Fuzzy

133

Table 5 and (1) n (2) m (3) A

or

non a

if a then b

u v u

a’ a’=aA a’=aA

bus’, type I/II bva’typeIII (b’Aa)‘, type III

Especially for classical and bold connectives the combinations can be suggested (compare with [5]).

shown in Table 6

Table 6 (1) % (2)

n,,,

(3) A

u,

*

‘5

v

*

-6

u,

*

-6

Finally, we can express n, u, n by + (and n), as is done in [lo] for combination (1). Clearly, also formulas from 5.3, 5.6(b), 5.9(c) generalize those in

cm I hope that the present investigation can throw more light on a structure which. we will impose onto a problem, and can be helpful for selecting adequate connectives, negation and implication.

References [1] W. Bandler and L. Kohout, Fuzzy power sets and fuzzy implication operators, Fuzzy Sets and Systems 4 (1980) 13-30. [2] G. Birkhoff, Lattice Theory, (American Mathematical Society Colloquium Publications Vol. XXV (AMS Providence, RI, 1960). [3] J. Dombi, A general class of fuzzy operators, the De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators, Fuzzy Sets and Systems 8 (1982) 149-163. [4] B.R. Gaines, Foundations of fuzzy reasoning, Int. J. Man-Machine Studies 8 (1976) 623-668. [5] R. Giles, cukasiewicz logic and fuzzy set theory, Int. J. Man-Machine Studies 8 (1976) 313-327. [6] J.A. Goguen, The logic of inexact concepts, Synthese 19 (1969) 325-373. [7] H. Hamacher, iiber logische Agregationen nicht bin& explizierter Entscheidungskriterien (Rita G. Fischer Verlag, 1978). [S] E.P. Klement, Construction of fuzzy a-algebras using triangular norms, J. Math. Anal. Appl. 85 (1982) 543-565. [9] C.H. Ling, Representation of associative functions, Publ. Math. Debrecen 12 (1965) 189-212. [lo] M. Mizumoto and K. Tanaka, Fuzzy sets and their operations, Information and Control 48 (1981) 30-48. [ll] B. Schweizer and A. Sklar, Associative functions and statistical triangle inequalities, Publ. Math. Debrecen 8 (1961) 169-186. [12] B. Schweizer and A. Sklar, Associative functions and abstract semigroups, Publ. Math. Debrecen 10 (1963) 69-81. [13] M. Sugeno, Theory of fuzzy integrals and its applications, Thesis, Tokyo Institute of Technology (1974). [14] S. Weber, I-decomposable measures and integrals for Archimedean t-conorms I, J. Math. Anal. Appl. (to appear).

134 [15] [16] [17]

S. Weber R. Wilhnott, Two fuzzier implication operators in the theory of fuzzy power Systems 4 (1980) 31-36. R.R. Yager, On a general class of fuzzy connectives, Fuzzy Sets and Systems L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-353.

sets, Fuzzy 4 (1980)

Sets and

235-242.