on many-valued logics, fuzzy sets, fuzzy logics and their applications

Fuzzy Sets and Systems 1 (1978) 129-149. ©North-Holland Publishing Company

O N M A N Y - V A L U E D L O G I C S , F U Z Z Y SETS, F U Z Z Y L O G I C S A N D T H E I R A P P L I C A T I O N S

Heinz J. SKALA Fachbereich Wirtschaftswissen-schaften, Gesamthochschule Paderborn, Paderborn, Federal Republic of Germany

Received April 1977 Revised July 1977

This paper gives a survey of some aspects of many-valued logics and the theory of fuzzy sets and fuzzy reasoning, as advocated in particular by Zadeh. It starts with a short discussion of the development of many-valued logics and its philosophical background. In particular, the systems of Lukasiewicz and their algebraic models are presented. In connection with the famous Arrow paradoxon, Boolean valued and fuzzy social orderings are discussed. After some remarks on inference, fuzzy sets are introduced and it is shown that their definition is sound if some acceptable rationality requirements are demanded. Deformable prototypes are suggested in order to obtain the numerical values of the membership function for some applications. Finally, a recent paper of Bellman and Zadeh on a fuzzy logic, where the truth values themselves are fuzzy, is reviewed.

I. Introduction

Tracing back the history of logic we realize that Jan Lukasiewicz and Emil Post in the early 1920s must be given credit for introducing many-valued logics in a systematic way.

The interest in many-valued logics originated from philosophical problems resulting from admitting statements which are neither true nor false. Since 1931 when Wajsberg succeeded in axiomatizing Lukasiewicz's three-valued logic we observe an ever increasing mass of new findings in various areas. The reader interested in questions of axiomatization and quantification theory of many-valued logics should consult Rosser and Turquette's book [37]. In this book we also find the first remark about many- valued set theory. Apart from pure logic where, in connection with many-valued logics, such interesting questions as semantic interpretation, Gentzen's method, the application of many-valued systems to intuitionistic and modal logic have been studied, we find also important applications in set theory, switching theory, quantum mechanics and most recently in theoretical economics. For more information concerning the theory of many-valued logics the interested reader should co~lsult the excellent survey by Rescher [34]. Also Ackermann [1] and Zinov'ev [52] are recommended. Some applications of many-valued logics can be found in the Proceedings of the 6th International Symposium on Multiple-Valued Logic, Utah, IEEE 76 CH 1111-4 C. We have already mentioned that Rosser and Turquette [37] posed the problem of creating a set theory based on a many-valued logic. With the aim of avoiding paradoxes in set theory, many-valued logics have been applied especially by Chang i-8] and Skolem [42]. Completely different was the motivation of Zadeh, who introduced in 1965 the notion of a fuzzy set. He took as given a predetermined set theory and characterized

129 D

130 H.J. S kala

fuzzy sets by functions from some ordinary set A into the unit interval [0, 1], i.e. he allowed for a continuum of grades of membership, thus providing "a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership rather than the presence of random variables" (Zadeh [48]~.

Up to now there have been published some 500 papers in the field of fuzzy sets and fuzzy systems. We only cite Kaufmann's book [ 18] as it is an excellent introduction to the theory of fuzzy sets and their applications.

One should clearly distinguish between many-valued logics and fuzzy logics. The latter, although shortly mentioned by Menges and Skala [25], have been createdand studied by Bellman and Zadeh [4].

The main feature of fuzzy logics is that they aim at providing a model for approximate rather than precise reasoning. Thus characteristically we find fuzzy truth- values and approximate rules of inference. Although Bellman and Zadeh use a "classical" logic (i.e. Lukasiewicz' logic with truth-values in the real interval [0, 1]) as base logic we do not think that this should be typical. Hopefully when getting more insight into our handling of vague concepts we are able to create an "empirical logic", adequate for the imprecise situations of our life without appealing to the very precise results of "classical" logic.

There are many reasons why economists and social scientists became increasingly interested in many-valued and fuzzy systems. The most important one is that concrete problems in economics and in the social sciences often do not have a well-defi~ed structure. For example it is not unusual that we have definitions of properties according to which certain objects can only partially be ascribed these properties. It is our point of view that this type of vagueness is an inherent property of social sciences and we must try to handle vague concepts appropriately. For example, we have to create a decision theory on the basis of illdefined classes of events or inexact concepts. Sometimes probability distributions may be used for representing such concepts. However, this is not alway~ the most satisfactory way, e.g. the manipulations allowed in probability theory might be implausible.

As we shall shortly discuss Arrow's paradoxon in the sequel let us for instance indicate where vagueness occurs in the problem of social choice. First of all it is common experience that different strengths of preference exist at the individual level. One way to represent the situation is by fuzzy orderings. However, this idea is in principle not new to economists, who have long since tried to use numbers representing people's attitudes to get social orderings. The crucial questions in the social choice problem are obviously measurement and interpretation. Up to now there is no way accepted h,, ,h,~ .'..ajo:':-.: ,.; ~ . . . . ; * . . . . . co,,vm,s~s. Thus we shall concentrate on a simpler ,roblem. Let ~.i:~e inmvidual orderings be non-fuzzy and look at what consequences a

change in the logical basis has. This can be interpreted as disturbing the original problem by admitting fuzzy social orderings. At the present state of affairs, the co ocept of a social ordering must be considered to be a vague one as an immense number of social decisions are made without referring to a definite social ordering. It should be noted in connection with inexact concepts that they call for short deduction chains. The length of deduction chains also play a crucial role in Selten's theory of restricted rationality (oral communication). In our opinion both vagueness and restricted

On many-valued logics, fuzzy sets, fuzzy logics and their applications 131

rationality are very near to some ideas put forward by the ultra-intuitionists, especially by Yessenin-Volpin. Unfortunately, to our knowledge, no investigations have been made in this direction.

Another reason for being interested in many-valued and fuzzy logics is, that they provide a linguistic means for disturbing problems. Looking at the disturbed problem it might turn out that it shows a different qualitative behaviour. We shall indicate that Arrow's paradox is a nice example in this respect. The situation seems to be quite similar in principle to problems discussed by Thom [47] and others.

2. Many-valued logics 2.1. Some informal remarks

If we assume that one of the main roots of logic is the reality of ordinary languages, many-valuedness arises quite naturally. Remember for example the amusing dialog between Ros,~:~tte and Turquer [37] about the question whether the janitor is in a certain room when entering the door or not. At this point we should also like to remark that, in particular, three-valued logic has been used in connection with the assignment of truth values to future contingencies [22, 30], and modal propositions [22].

Although we shall not enter into a discussion of the law of the excluded middle (for this see [34]), we refer the interested reader to a paper of Fitch [ 15] which seems to be of great philosophical significance. It is well known that Tarski showed that the truth concept with respect to a formal language cannot be defined in that language itself. If one is interested in a universal metalanguage for philosophical problems this result of Tarski seems to be disappointing as we must have at our disposal a chain of metalanguages ending up with a natural language at the top. Fitch argues that Tarski's criterion for the definition of truth is unnecessarily restrictive if it is applied to forma~ systems in which there are propositions which do not satisfy the law of the excluded n~iddle. He comes up with a considerable rich language that can define its own truth.

Before going into some details we mention two more fields in which many-valued logic has been successfully applied. In the field of quantum physics one finds two schools of thought. According to Destouches-F6vrier [ 11 ] the logic of the microcosmos is essentially three-valued whereas Reichenbach [33] used three-valued logic as a convenient tool in order to create an appropriate language of the microcosmos. We are more sympathetic to the second point of view, as in everyday life (macrocosmos) we may encounter quite similar problems to those of microphysics. Consider, for example, competing investment policies. Once a certain policy has been agreed upon the others are no longer testable (i.e. verified or falsified as good ones) as the truth value of empirical propositions depends on specific situation and time. As we cannot go backwards in time they must be given an indefinite truth value.

The second field of application is in pure mathematics. The construction of Boolean- valued models (i.e. models with truth values in a complete Boolean algebra) of set theory has turned out to be a very effective tool for providing independence proofs {see for exa:,ple [-39, 37]). It is well known that the construction of Boolean-valued models is equivalent to the method of forcing invented by Cohen in order to prove the independence of the Continuum Hypothesis. However, as in the paper of Scott [39], Boolean-valued models often have a probabilistic setting which is quite well understood.

132 H.J. Skala

Under the heading "Probability Logic" Krauss [21] studied models with truth values in a probability space. A paradoxical example in the theory of stochastic processes, given in the dissertation of Krauss, induced Eisele [12] to study repre- sentations of stochastic processes in the framework of Boolean-valued models. Forcing, or equivalently Boolean-valued models, can be used in order to provide a model of set-theory in which every set of reals is Lebesgue-measurable [44]. Interestingly enough this model has an immediate application in mathematical economics. With its help one can show [41] that the visible dictator in Arrow's paradoxon reappears in the in~nite case.

2.2. The many-valued logics of Lukasiewicz There are four main reasons why we are going to present the family of

Lukasiewiczian systems in some detail:

(1) As already mentioned the nondenumerably infinite system of Lukasiewicz is used as base logic in the fuzzy logics of Bellman and Zadeh.

(2) A language appropriate to social choice based on the uncountable infinite- valued logic of Lukasiewicz will be used in the sequel in order to discuss one aspect of Arrow's paradoxon.

(3) Scott [39] has suggested to replace many truth values by many (two valued) valuations. In doing so the idea arises to consider the family of Lukasiewiczian systems as calculi of error.

(4) In his dissertation Maydole [24] discussed at length the general principle of comprehension in naive set theory which is well known to lead to paradoxes in two- valued logic. He was able to prove that Curry-like paradoxes arise in the families of Bochvarian, G/Sdelian, Postian and Standard Sequences Logics as well as in probability logic and Reichenbach's 3-valued quantum logics. Only for the nondenumerably infinite logic of Lukasiewicz he was not able to provide such a proof. Thus the general principle of comprehension could be saved for set theories based on the last mentioned logic.

Let us start by introducing a well known many-valued logic, i.e. Lukasiewicz' three- valued system, by means of truth tables. We take negation ( -1 ) and implication (~ ) as primitive and write:

1 1/2 0

0 1 1/2 1/2

1 0

p ~ q

1 1/2 o

1/2 o 1 1 1/2 1 1 1

(From the truth-values 1, 1/2, 0 only 1 is designated1.)

If only 1 is designated the usual tautologies of Lukasiewicz' three-valued logic are obtained. Designating 1 and 1/2 a revised system is obtained in which many of the classical tautologies which do not hold in Lukasiewicz' original logic are valid.


For the sake of simplicity the following abbreviations are introduced:

p v q for (p--,q)--,q,

p A q for 7 ( - T p v 7q) ,

p~--~q for ( p~q )A (q--~p).

As already mentioned Wajsberg (in 1931) succeeded in giving an axiomatization of this logic by the following schemes:

(1) p-~(q-~p),

(2) (p~q) -~( (q -~r )~(p~r ) ) ,

(3) ( T p ~ 7q)~(q-~p) ,

(4) ( (p~ 7 p ) ~ p ) ~ p ,

together with modus ponens and the rule of substitution as inference rules. It is very well known that classical logic has an interpretation in Boolean algebras.

It was the aim of Moisil [-26-28] to introduce algebraic structures (three-valued Luke.~iewicz algebras) playing a similar role for the three-valued logic of Lukasiwewicz as Boolean algebras do for classical logic.

Before introducing the notion of a three-valued Lukasiewicz algebra we need the following

Definition 2.1. If (A, 1, v , A ) is a distribution lattice with a last element 1 and 7 is a unary operation defined on A such that:

(1) 7 7 x = x

and

(2) 7 ( x v y ) - T x ^ - - 3 3 ,

then (A, 1, 7 , v , A ) is called a de Morgan algebra. ( 7 1 = 0 is the first element of A). If it holds that

(3) XA 7 x < y v 7y ,

we call <A, 1, 7 , v , ^ ) a Kleene algebra.

Definition 2.2. If <A, 1, 7 , v , A > is a Kleene algebra and It is a unary operator, the possibility operator, defined on A such that

(4)

(5)

(6)

then (A, 1,

- l x v / , x = 1,

XA --IX-- 7X A pX,

/a(x A y ) < p x ^ lay,

7 , p, v , A ) is called a three valued Lukasiewicz algebra.

134 H.J. Skala

The following properties of three-valued Lukasiewicz algebras are well known

(7)

(8)

(9)

(lO)

(ll)

(12)

#/IX = #X,

#(x v y ) = # x v #y,

#(x A y ) = # x A #y,

X<#X,

#0=0 ,

#X = X iff X is a Boolean element of A (i.e. i[there exists an element -1 x e A such that x ^ --1 x = 0 and x v --I x = l ) .

If we denote by B the subalgebra of all Boolean elements of A, then for a2! x e A we have

(13) # X = A { b e B ' x < - b }

and dually

(14) vx= v { b e B ' b < x } .

v has been called necessity operator by Moisil [26, 28]. He also stated the following determination principle"

(15) x < y iff I~x<luy and vx<vy.

For more details the interested reader should consult [9, 10]. One should note that the matrix for the three-valued logic of Lukasiewicz is just a

three-valued Lukasiewicz algebra with three elements. Thus the connectives of possibility and necessity can be introduced by the following truth tables

#p vp

1 1 1/2 1 0 0

or by defining #p= -1 p--.p and vp= --1#'-I p. If we form in the usual way the algebra of formulas of a tHree-valued system and then

reduce by the relation =defined by p - q iffboth p--,q and q--.p get the truth value 1, i.e. forming the corresponding Lindenbaum algebra [32], it can be shown that the Lindenbaum algebra is a three-valued Lukasiewicz algebra.

For more information about the family of Lukasiewiczian systems the reader should consult e.g. [34].

On many.valued logics, fuzzy sets, fuzzy logics and their applications 135

2.3. Many-valued logic.~ and Arrow's paradoxon

We only shortly restate Arrow's axioms. Background information and thorough discussions may be found in [2, 7, 14, 19,41].

(1) card (V) = n > 2. (There are at least two individuals.)

(2) card (X)>3. (There are at least three alternatives.)

(3) To every preference profile (>-1,-. . , >', ) there is associated a social ordering

(4) If in a profile a>-b , i= 1,...,n, then a>..~b. (Pareto-efficiency.) !

? t t (5) If ( , , - 1 , . . . , > - , ) = ( > - ~ , . . . , > - , ) on {a,b}, then > - ~ = > G on {a,b}. (Independence.)

(6) There is no v ~ V such that, for all a, b ~ X, and all possible profiies, if a >-, b then a>-~b. (No dictator.)

Arrow's Theorem. For finite V (1) through (5) imply the eximence of a dictator, i.e. (1) through (6) are inconsistent.

We are now going to discuss the problem of social choice within a Boolean system and within a many-valued Lukasiewiczian logic.

Boolean systems are oased on a philosophy of possible Worlds, i.e. every proposition has a "basic" truth value with respect to each possible world we W. See [34]. Introducing ~p]] for the truth value of p we have the following truth rules"

~.p~ = {w ~ w. lpl - 1}, 2

~.Tp~=W-~.p~,

~p v q]] = ~p]]w~q]],

liP ̂ q]] = [[p]]c~l[q]],

~P~q]] = ~P]l =[[q~ = ~ -1p~w~q]].

Passing to a first order language, i.e. introducing quantification we must interpret individval variables as ranging over the elements a in some given domain A.

Thus for quantified formulae we have"

~3 xp(x )]]= U ~p(a )]], aEA

~Vxp(x)]] = ('] ~p(a)]]. a ~ A

Note. From a formal point of view every complete Boolean algebra is an adequate truth value structure. Obviously every power set algebra is complete.

2The "basic" truth value of a proposition p with respect to a possible world w e W is denoted by [Plw- The basic truth values are assumed to include 1 (true) and 0 (false).

136 H.J. Skala

In the social choice problem we take as the set of possible worlds the set V and get thus the following valuations of the atomic formulas in our problem:

[[a~.t~b~ = { v ~ V : a>'-vb} , 3

[[a.~ ab'~ = {v~ V" a~.vb}.

By this we get a Boolean valued model of Arrow's axioms (1) through (6). In particular ~-~ is a Boolean valued weak ordering. Consider for example

a~-~b^b~-Gc~a>-~c .

For any a, b, c ~ X this proposition must get the truth value V (identified with the Boolean l) as else there would exist at least one v~ V such that a~-,,b ^ b>-L. c but -qa~-, c.

Thus we can state:

Theorem 2.3. For finite V Arrow's axioms (1) through (6) are consistent if we only admit Boolean-valued ,social orderings.

We shall now replace Boolean truth values by values in the interval [0, 1] in a Lukasiewiczian fashion and define for x~, x i e X :

ai~) = else, , b~) = else.

In the appropriate first order language for our problem we start with the valuation of the atomic formulas

[Vxi~-~x~ = 1/card (V) ~ ai~), 4

~x,-~ t;x~ = l/card (V)y~ bi'), u

and extend it as follows:

lip v = m a x (lipS, ),

A = m i n (lipS, )

3This means nothing else as: the proposition that society prefers a to b gets as truth value the ,~et of all people in the society preferring a to b. Obviously ~ is no longer a binary relation in the usual sense as > c :X × X---,~(V).

4In the terminology olZadeh [48] we have introdaced a fuzzy relation ~6 onX which is characterized by a membership function #~ . :X × X~[O, 1].


~3xp(x )~_ =sup ~p(a )~, a~X

~_Vxp(x )~ = inf ~.p(a )]l, aeX

[ [P~q~={l if ~.p]]<~.q~, -~p~+~q]] if ~_p ]] > ff_ q~ .

The situation here is not so straightforward as in the Boolean case. Consider for example the voting paradox:

X1 ~" 1 X2, X2 ~>" 1 X3 a n d thus xl > 1 x3,

X2~"2X3, X3~2X 1 and thus X2~ 'zXI ,

X 3 ~ " 3 X 1 , X I ~ ' 3 X 2 andthus X3~'3X 2 .

~-X 1 ~" GX2~ -- 2/3, ~x2 > ~x3]] = 2/3, ~X3 ~ 'G X1 ~ -" 2/3.

Thus we get

[IX 1 ~ ' G X 2 A X 2 ~" GX3 "--+X 1 > G X 3 ~ -" 2/3

and not as desired the designated value 1. However, one may show that some sort of transitivity still remains. To do so we

introduce a new connector ® with the following truth rule:

~.p®q~ = max (0, [[p_~ + [[q~ - 1 }.3

It can now be shown that

~.Xi>- aXjQXj>- aXh--+Xi>., aXh~= 1

and we are able to derive a possibility result similar as in the Boolean case. For more details see !-41].

The above results show that if we allow disturbances in the social ordering (i.e. Boolean valued or fuzzy social orderings) then Arrow's dictator disappears.

The same can be shown if we disturb the individual orderings in a similar way. This was advocated by Zadeh. Thus the social choice problem in the sense of Arrow shows a sort of inherent structural instability.

2.4. Remarks on inference

Remen,,ber the most important rule of inference, i.e. modus ponens. For the sake of

5This definition occurs in a natural way when treating Arrow's problem. My research assistant Mr. Manfred Kraft has pointed out to me that Klaua [20] also used this definition in a completely different context.

138 H.J. Skala

convenience we shall adopt the following notation:

P

P~q

Let us start with some remarks about probabilistic inference. We follow here the paper of Suppes [45]. 6

According to him the inference form

P(p)>__o~,

P (qlp ) >-_ fl P(q)>-~fl

(1)

is the most natural generalization of the classical scheme. The proof of (1) is a trivial application of the theorem of total probability

Plq ) = P(q [ P )P(P )+ P(q I P)P(P)

by which also the following special case of (1) is immediately verified:

p(p)= 1,

P(q[P)=~ P{q)=~,

(1')

Of course (1) cannot be called in question. We only may ask whether probabilities and conditional probabilities are the right entities into which we can translate a specific empirical problem. As for example people cannot check the coherence of very complex betting systems one may question additivity. This has indeed been done by various authors.

Let us now consider another natural inference form. Remember that we have defined the Boolean counterpart of material implication p--,,q by p v q. Analogous to (1) we get:

P(p)>__ot, P(p-~q)> fl

P(q)>o~ + f l - 1 (2)

As Suppes has pointed out a simple application of the general addition law of probabilities proves (2).

6The main aim of S,~pes' paper was to give a serious critique of the principle of total evidence.


An empirical justification for either one of the two forms can only be given if we are able to decide whether in a certain situation it is more natural to think in conditional probabilitiesor in the Boolean counterpart of material implication. We cannot go into these important questions, but only try to get more feeling when comparing (1) and (2). To do so let us consider the following special case

P ( p ) > - l - e , P(q lp)>=l-e P(q)>=(1-e) 2' (3a)

P(p)> l - e ,

P(p-,q)>= 1 - e

P(q)-> 1 - 2 a (3b)

(Obviously ( 1 - e ) Z > l - 2 e for e>0.) The following theorem is now easily verified:

Theorem 2.4. / f P(pi)> 1 - e for each of the premises p~,l < i<n , and q is logically implied by these premises then p(q)> 1-he .

After having gained some feeling of the probabilistic case we shortly discuss Lukasiewicz' system with truth-values in [0, 1]:

[p]] > 1 - e,

[[p~q[~ = 1

~q]l >= l - e (4)

This result is easily verified by simple arithmetic using the truth-rules stated in Section 2.3. It simply states that the application of modus ponens preserves the truth value of the premise. If q is logically implied by the premises Pl,-.., P, then the smallest truth value of the Pi is decisive for the truth value of q. In case the truth value of the implication is smaUer than 1, then a result similar to (3b) holds. In any case, whether we apply a probability logic or a Lukasiewiczian one when treating inexact concept, long deduction chains may have fatal consequences.

3. Fuzzy sets

3.1. Introducing fuzzy sets

It was Zadeh 1"48] who introduced in an ad hoc way the notion of a fuzzy set. For our further discussion let us denote by X a nonempty set (the universe). It is obvious that a subset A of X may be identified with the corresponding characteristic function fA (x) defined by

fA(x)=l if x e A ,

fA(x)=0 if x e A .

140 " H . J . S k a l a

Associating with each point x e X a real numberfA(x) in the interval [0, 1] we ha~e introduced a generalized characteristic function or membership function in the sense of Zadeh [48].fA(X) may be considered as truth value or degree of acceptance which x~e are prepared to assign to the proposition "x is a member of A". As long as only atomic propositions are involved we feel rather free to use our subjective evaluations. But what shall we do if connectives like "and", "or" and "not" are involved.

It has been shown by Bellman and Giertz [37] that there are rather natural consistency requirements showing that Zadeh's ad hoc definitions, which we are now prepared to state, are in some sense natural.

Definition 3.1. Let X be a nonempty set and let the membership functions take valwes in the interval [0, 1]. The empty fuzz), set 0 is defined by f0(x)=0, for all x ~X.

The complement A t of a fuzzy set A is defined by

fA'(X)=I--fA(X), for all x~X.

We say that the fuzzy set A is included in the fuzzy set B and write A c B, if

JA(X)<fn(x), for all x~X.

The union AwB of two fuzzy sets is defined by

f~.~Blx)=max(fA(x),fn(x)), for all x~X.

The intersection Ac~B of two fuzzy sets is defined by

f4~B(x)=min(fA(x),fa(x)), for all x~X.

It should be observed, and has been proved by Zadeh [48], that for a given X the system of fuzzy sets has all the nice properties we know from the system of ordinary subsets of X with the following two exceptions"

AuAc=X,

AnAl= ¢.

To see this let, for example, fA(X)= I/2~ then fa,_,a~(x)=max(1/2, 1 - 1/2) and J4~A':IX) =min(1/2, 1-- 1/2).

Note the difference between fuzziness and probability! If we want the notion of fuzziness to be useful to model vague situations

appropriately, we must give convincing reasons why it should have the above mentioned properties. We are thus in a situation quite similar to the subjectivlstic interpretation of probability theory where consistency requirements also play a fundamental role. Because of its instructiveness let us follow the discussion given by Bellman and Giertz [3]. Again we restrict ourselves to the truth-value set [0, 1] and denote by fa(x) the (subjective) degree of acceptance (truth-value) associated with the

On many-valued logics, fuzzy sets, fi¢zz)" logics and their applications 141

atomic statement .'..'eA. We shall write as usual ~p~ to denote the truth value of a statement p.

Starting with ~x ~ A~=fA(x ), ~x ~ B~ =fn(x) the question arises how to define in a rational way [Ix ~ A ^ x ~ BIT, ~x ~ A v x ~ B~ and ~ ~ A]].

Neglecting for the moment the definition of [Ix ~ A~ we have to impose restrictions on the functions g and h:

~x ~ A A x ~ B~ =h(F,x ~ A~, ~x ~ B~),

F,x ~ A v x ~ B-I = g(~x ~ A~, fix ~ B~ ).

Theorem 3.2. l f the followin O conditions (i)-(vii)are imposed on the.[imctions g: [0, 1] x [0,1] -~ [0,1] and h: [0, 1] x I -0,1]~[0,1] , then g(~x~A?~, ~ x e B ~ ) = m a x (~x~A~, ~x~B~) and h(~xeA~, I ]x~B~)=min (~xeA~, ~xeB~).

(Writing down the seven condition.s we use the abbreviations a, b, c, [br ~x ~ A ~. ~x ~ B~, EXECS.)

(i) h(a, (ii) h(a,

g(a,

(iii) h(a, g(a,

(iv) h(a,

(v) h(a,

(vi) h(a,

(vii) h(a,

b )=h(b ,a) ;g(a ,b)=g(b ,a) (symmetry),

h(b,c))=h(h(a,b),c)

g (b ,c ) ) -g(g(a ,b) ,c ) (associativity),

g(b,c))=g(h(a,b),h(b,c))

h(b,c))=h(g(a,b) ,g(b,c)) (distributive laws),

b) and g(a, b) are continuous and non-decreasing in a,

a) and g(a,a) are strictly increasing in a,

b) <min In, b 7, g(a, b )> max (a, b),

a ) - l ,g (0 ,0 ) -0 .

The reader should convince himself that tile above conditions are acceptable rationality requirements. A detailed discussion (also including negation) and a proof,~f the theorem may bc found in Bellman and Giertz [3].

Let us shert!y indicate some generalizations by asking the question" In which structure should the generalized characteristic functions take their values? As the real interval [0, 1] seems sometimes to be too special, a natural choice would be at least a partial ordering. If we are interested in more specific questions than order theoretic ones, the partial ordering must have some additional structure, e.g. in case one is interested in the maximum and minimum of fuzzy sets a complete lattice is necessary. It was Goguen [16] who investigated in some detail L-(lattice) fuzzy sets.

In order to show that L-fuzziness is a very familiar phenomena let us look at multicomponent criteria which we suppose to constitute a partial ordering ~6:= (C,

);C 1 ~ C 2 means that if cl is satisfied this implies that c2 is also satisfied. In case that neither cl <c2 nor c 2 <c l we have two independent criteria. It is nov¢ well known that if L is a lattice and c# is a partial ordering then L': is also a lattice, tMereover if L is distributive or modular so is L~.) If we suppose that every criteria in % can

~" denotes the either be satisfied (1) or not (0) then the greatest element of [0, 1, maximum degree of satisfaction (all criteria satisfied).

142 H..I. Skala

The interpretation for the other elements (truth-values) of {0, 1 }~ should be obvious. In some cases it is also possible to reduce a complex criterion with truth values in a

lattice to a partially ordered set of yes-no criteria. This is a consequence of the following

Theorem 3.3. Let L be any distributive lattice of finite length, then there exists a partial ordering ~ such that L= {0, 1 }~. (For the proof of an even stronger theorem and related facts the reader ,should consult Birkhoff [5].)

We finish this section by mentioning a further generalization, i.e. fuzzy membership mapped onto intervals (see [17]). It should be clear that interval numbers in the sense of Moore [29J or triplex numbers may be used when we are interested in giving error bounds for calculations in which inaccurate generalized characteristic functions are involved. Also in fuzzy reasoning interval magnitudes and similar co.qcepts can very usefully be applied. Although from a structural point of view the theory developed by Grattan--Guiness is not very satisfactory it should have its relevance for empirical applications.

3.2. Vague events

Although we cannot give even a sample of research activities in which the ideas of Zadeh have been fruitfully applied we should like to state at least one example. (The interested reader is referred to [51] where a broad spectrum of topics related to the theory of fuzzy sets is presented.)

It was again Zadeh [49] who brought together probability theory and fuzzy sets. In everyday expcrience as well as in the social sciences and even in physics [31] there are situations in which an event is a fuzzy rather than a classical set. Reasons for this are, for example, the imprecision of the used language, measurement procedures etc.

Let ( R ' , ~ , P ) be a classical probability space, i.e. R" is the Euclidean n-space, ~ is the or-field of Borel sets in R n and P is a probability measure.

If A e M then

P(A )= j'a dP,

or in the terminology of characteristic functions

P(A )= j'¢/~A(x)dP = E(UA).

(We use #a for two-valued i.e. classical, characteristic functions.) 'Fhis is easily generalized when A is a fuzzy event in R" characterized by a Borel-

measurable characteristic function ~ • R"--,[0, 1]. The probability of the fuzzy event A is then

P(A )= ~R"fA (x)dP = E ( f A ).

As fA is Borel-measurable the above Lebesque-Stieltjes integral exists. It has been shown by Zadeh [49] that many of the usual properties have a fuzzy

counterpart. Introducing for example the product AB of two fuzzy sets A and B by fan(x)=fa(x)fn(x), for all x, we can give an appropriate definition of independence.

On many-t:alued logics, fuzzy sets, fuzzy logics and their applications 143

Unlike in the classical case where independence is defined via intersection we have: Let A and B two fuzzy events in <R",~,P) then A and B are called independent if

P(AB)=P(A)P(B).

The concept of the conditional probability is now defined by

P(AIB) --P(AB-------~) P(B)>O. P(B) '

Many of the basic notions in probability theory, e.g. mean, variance, entropy, etc., can now be introduced.

One should not confuse probability measures of fuzzy events with the notior, ,~f a fuzzy measure as introduced by Terano and Sugeno [46]. Their definitions have not hing to do with a fuzzy event structure. They call a set function g on ,~, where.~ is a Borel field of a set X, a fuzzy measure if:

Is) g(O)=0, g(X)=l.

(21 If A.B~:~ and A=B, then glA)<g(B).

(3) If A ,6M and {A,} is monotone, then lim,_., g(Fn)=g(lim,,., F,,).

Which means fuzzy measures are simply set functions having the property of monotonicity but not necessarily ofadditivity. Such measures have also been discussed by Fine [13] and Skala 140]. There is some evidence that it would be worthwhile combining the two approaches. Ofcourse the empirically meaningful property (21 has a straightforward interpretation if A and B are fuzzy sets: ~;uppose someone chooses an x ~ X not knowing which one. He may guess that x belongs to A. This is the first type of uncertainty considered bv Terano and Sugeno.

Now, if the person is informed which x he has chosen it still need not be completely clear to him that x belongs certainly to A or belongs certainly not to A. (x may only gradually belong to A; second degree of uncertainty.)

Note. In the case of fuzzy sets A c B means that the carrier of A is a subset of the carrier o r b and every point ofthe carrier of A belongs at least to the same degree to B as to A.

3.3. How to get the numerical values of tlle membership function--an application

In a review article on pattern recognition Bremermann [6] discusses at some length the method of deformable prototypes. The idea is quite simple. We have a prototype. which can be deformed by manipulating parameters Pl, . . . , P,. Given an object it is tried to deform the prototype such that a maximal matching is obtained. Pr~,vided that [[ II is a suitable norm measuring the dissimilarity between object and prototype we can introduce a matching function:

m (object; P 1,..., P, ) - ] l o b j e c t - P rot otype (p l , . . . , P,)ll.

144 H.J. Skala

This idea has been implemented by Hodges, a scholar of Bremermann, to character recognition. As, for example, exactly the same characters can be matched by the A and H prototype, matching functions in general will not be good measures of similarity between characters and prototypes. Thus Hodges suggested thinking of prototypes as consisting of rigid bars held together by elastic springs, such that in the springs' relaxed state the ideal position is obtained. It is now natural to introduce a distortion function d(p l , . . . , p,,) taking as value the distortion energy associated with the parameter vector (Pl,..., P,,)- The measure of similarity is considered to depend on the matching function as well as on the distortion function. Hodges used the following function for his problem:

sim(object)= min (m(object;pl . . . . , p , , ) + w d (Pl,...,P,,)), Pl . . . . . Pn

where w is a weighting constant. The results obtained using a global optimization algorithm developed by Bremermann have been reported to be quite satisfactory. In her thesis Fuzzy Sets and their Application to Medical Diagnosis Pattern Recognition, Merle Albin has exploited similar considerations.

The reason why we have briefly discussed the method of deformable prototypes, is that it is hoped to use such methods for economic problems. Moreover the method of deformable prototypes is a very effective way to determine the numerical values of the membership functions in question. This problem is central to the fuzzy set approach to empirical problems and has often been discussed, although not very satisfactorily, in the literature.

As the prototype perfectly ma!ches itself and sire (prototype) = 0 this function cannot be used as membership function, laowever, as Brcmermann proposes

.liprolotype ] (object) = 1 sire (object)

max sim

will do. Another central problem for the application of the fuzzy set approach to empirical

problems is the calculation of the membership function for combined fuzzy statements. Empirical results in this direction have been obtained by R6dder 1-35].

4. Fuzzy logic

The most recent development in the field of vagueness is the creation of fuzzy logic. Before going into details let us state the most important features a fuzzy logic has according to Bellman and Zadeh [-4]:

(1) The truth-values themselves are fuzzy. (2) The truth-values are assumed to be generated by a grammar. (3) The used linguistic truth-values can be given a meaning by semantic rules. (4) Fuzzy logic is local, i.e. connectives and truth-values have a variable meaning. (5) The rules of inference are approximate rather than exact.

The stated characteristic features show that fuzzy logic will serve as a (more or less) formal language ill the field Jf approximate reasoning.

On many-valued Iogics,.fitzzy sets, Juzzy logics and their applications 145

If this is so we cannot expect fuzzy logic to present itself as a unique system: rather, it must be thought of as a collection of systems.

Let us now start with a discussion ofsome essential points in the, as yet, quite "'fuzzy" theory of fuzzy logic. The first step is to choose some base logic. Lukasiewicz' [0, 1]- valued logic would be a typical example of a base logic. In this case the truth-values of the fuzzy logic would be fuzzy subsets of the unit interval. (If we want to introduce designated values these would be also fuzzy subsets of [0, 1].)To the fuzzy subsets,71"the unit interval one can assign labels of the form, true, very true, more or less true, false etc. Thus introducing linguistic truth-values. This convention gives us automatically a meaning of the linguistic truth-values, namely fuzzy subsets of [0. 1]. Moreover Bellman and Zadeh assume that the [countable) set of linguistic truth-values true, very true, not very true, more or less true, false . . . . can be produced by a grammar. This means for example that tbr every truth-value a syntax tree can be given (see in this connection also Zadeh [50]).

We are now ready to consider fuzzy propositions of the simplest form, namely ".\- is F " or more explicitly "'x is a member of F" .

Unlike the classical situation, F is a frizzy subset of a universe ofdiscourse U. For the sake of simplicity Zadeh uses relational assignment equations instead of the original propositions. Consider for example the proposition "'Betty is young" where young is a fuzzy subset of the reals. This translates into

R (Age(Betty)) = young.

Note that Age is an attribute of Betty which is implied by young. Of course the use of relational assignment equations carries oxcr to n-ary fuzzy relations. For example the filzzy proposition "'Betty is much younger than Henry" translates into

R(Age(Betty), Age(Henry)) = much younger.

Again the linguistic value "much younger" is represented by a binary fuzzy relation on the reals ( = a ff~zzy subset of the two fold Cartesian product of the reals).

Bellman and Zadeh study in some detail four translation rules:

(1) The modiJier rule. This asserts that ifm is a modifier such as not. very. more or less etc. the translation of a fuzzy proposition of the form x is mF is expressed by

x is m F ~ R ( A ( x ) ) = m F .

Note. A ( x ) is an implied attribute o fx and m is interpreted as an operator v,'hich transforms the fuzzy set F into the fuzzy set inF.

Example. x is not F is expressed by

xis not F-- ,x is F " ~ R ( A ( x ) ) = F "

where F c is the complement of F, i.e. i~., .= 1 --l~x..

146 H.J. S kala

(2) The formation of composite fuzzy propositions by connectives such as conjunction, disjunction etc. Here we have to consider noninteractive and interactive situations. For example the rationale for identifying noninteractive conjunction with set intersection is given by the theorem of Bellman and Giertz stated in Section 3.1.

(3) The use of fuzzy quantifiers. The propositions have the general form "Qx are F *' where Q is a quantifier such as most, some, few, very few etc. One way to handle such quantifiers is by using a generalized cardinality concept. For example let F be a fuzzy subset of a finite universe of discourse U={u~,...,u,} then we may define the cardinality of F by

card F=J~-(ul )+"" +fF(u,,).

0'i tuft denotes the grade of membership of u i in F.) Now consider the proposition "most Swedes are tall". Let sl, .... s, be a population of

Swedes and let f , i = 1,..., n, be the grade of membership of si in the fuzzy set tall. The generalized proportion of Swedes who are tall may be expressed by

L + " + L 9

n

and th~s we may translate "most Swedes are tall" into

R(:/] + " ' + f ' ) = m ° s t n

where most is a fuzzy subset of [0, 1].

(4) Truth functional modifications. Consider a fuzzy proposition p, e.g. Betty is young. By a truth-functional modification we may arrive at a proposition p' ,~vhich reads: Betty is young is very true. Such situations may be handled by ti~e translation rule

x is F is z ~ x is F',

where r is a linguistic truth value and F' is a fuzzy set appropriately related to F and r.

Let us now consider a typical inference in fuzzy logic. Assume that p denotes the fuzzy proposition "x is small" and that q denotes the fuzzy proposition "x and y are approximately equal" where the universe is U = I 1,2, 3, 4}. We may define"

number

small: degree of membership

1 2 3 4

1 0.6 0.2 0


approximately equal"

1 2 3 4

1 0.5 0 0 0.5 1 0.5 0 0 0.5 1 0.5 0 0 0.5 1

and give the translations R(x) = small and R(x, y) = approximately equal. Denoting by o the composition of fuzzy relations [50] we get R(y)=R(x~R(x,y)=small approximately equal. In order to state R(y) explicitly we may form the max-min product of the above relation matrices which results in:

number

degree

1 2 3 4

1 0.6 0.5 0.2

Now this result can be approximated by a linguistic value of),. This value might be for example "more or less small".

We hope that the reader has gained some feeling with respect to the aim of and the tools used in fuzzy logic.

As fuzzy logic is still in its infancy there are up to now no deep theoretical results of its fragments. There is however some hope to obtain a compactness theorem for a fragment of fuzzy logic if the usual definition of a compact fuzzy topological space is redefined in ~uch a manner that Tychonoff's theorem holds. However, much more interesting fol the application of fuzzy logic as a basis for the formation of theories ill the social sciences, in particular as a basis for a theory of restricted rationality, is the empirical verification of some crucial assumptions.

Acknowledgements

The author is indebted to the referees for various helpful comments. Alter this paper was written Professor Zimmermann kindly provided the author with copies of the following two papers:

B.R. Gaines, Foundations of fuzzy reasoning, Int. J. Man-Machine Studies 8, (19761 623-668.

B.R. Gaines, L.J. Kohout, The fuzzy decade: A bibliography of fuzzy systems and closely related topics, Int. J. Man-Machine Studies 9 (1977) 168.

Both articles are highly informative and are strongly recommended to the interested reader.

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