silogismo difuso

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7 5 4 I E E E TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. SMC-15, N O. 6 , N OVEMBER/DECEMBER 19 85

Syllogistic Reasoning in Fuzzy Logic and itsApplication to Usuality and Reasoningwith DispositionsLOFTI A. ZADEH, FELLOW, IEEE

AbstractA fuzzy syllogism in fuzzy logic is defined to be an inferenceschema in which the major premise, the minor premise and the conclusionare propositions containing fuzzy quantifiers. A basic fuzzy syllogism infuzzy logic is the intersection/product syllogism. Several other basicsyllogisms are developed that may be employed as rules of combination ofevidence in expert systems. Among these is the consequent conjunctionsyllogism. Furthermore, we show that syllogistic reasoning in fuzzy logicprovides a basis for reasoning with dispositions; that is, with propositionsthat are preponderantly but not necessarily always true. It is also shownthat the concept of dispositionality is closely related to the notion ofusuality and serves as a basis for what might be called a theory ofusualitya theory which may eventually provide a computational frame-work for commonsense reasoning.

I . INTRODUCTIONFUZZY logic may be viewed as a generalization ofmultivalued logic in that it provides a wider range oftools for dealing with uncertainty and imprecision inknowledge representation, inference, and decision analysis.In particular, fuzzy logic allows the use of a) fuzzy predicates exemplified by small, young, nice, etc; b) fuzzyquantifiers exemplified by most, several, many, few, manymore, etc; c) fuzzy truth values exemplified by quite true,very true, mostly false, etc. d) fuzzy probabilities exemplified by likely, unlikely, not very likely etc; e) fuzzypossibilities exemplified by quite possible, almost impossible, etc; and f) predicate modifiers exemplified by very,more or less, quite, extremely, etc.What matters most about fuzzy logic is its ability to dealwith fuzzy quantifiers as fuzzy numbers which may bemanipulated through the use of fuzzy arithmetic. Thisability depends in an essential way on the existencewithinfuzzy logicof the concept of cardinality or, more generally, the concept of measure of a fuzzy set. Thus if oneaccepts the classical view of Kolmogoroff that probabilitytheory is a branch of measure theory, then, more generally,the theory of fuzzy probabilities may be subsumed withinfuzzy logic. This aspect of fuzzy logic makes it particularlywell-suited for the management of uncertainty in expertsystems (Zadeh [48]). More specifically, by employing a

Manuscript received August 14, 1984; revised July 7, 1985. This research was supported by NSF grant IST-8320416, NASA grant NCC2-275,and DARPA contract N0039-84-C-0089.The author is with the Division of Computer Science, University ofCalifornia, Berkeley, CA 94720, USA.

single framework for the analysis of both probabilistic andpossibilistic uncertainties, fuzzy logic provides a systematicbasis fo r inference from premises which are imprecise,incomplete or not totally reliable. In this way it becomespossible, as is shown in this paper, to derive a set of rulesfor combining evidence through conjunction, disjunction,and chaining. In effect, such rules may be viewed asins tances of syllogistic reasoning in fuzzy logic. However,unlike the rules employed in most of the existing expertsystems, they are not ad hoc in nature.Our concern in this paper is with fuzzy syllogisms of thegeneral form

p(Qi)qJQi) (l.i)r(Q)

in which the first premise p{ Qx) is a fuzzy propositioncontaining a fuzzy quantifier Q x\ the second premise q(Q2)is a fuzzy proposition containing a fuzzy quantifier Q2;and the conclusion r(Q) is a fuzzy proposition containinga fuzzy quantifier Q. For example, the intersection/productsyllogism may be expressed as

QXA 's are B 'sg 2 ( ^ a n d ) ' s a r e C ' s (1.2) , 4 ' s a r e ( a n d C ) ' s

where A, B, and C are labels of fuzzy sets, and the fuzzyquantif ier Q is given by the product of the fuzzy quantifiers Q x an d Q2, i.e.Q = i 2 (1-3)

where denotes the product in fuzzy arithmetic(Ka ufm ann and G upt a [21]) (Fig. I) . 1 It should be notedthat (1.3) may be viewed as an analog of the basic probabilistic identity (Jaynes [19])p(B,C/A)=p(B/A)p(C/A,B). (1.4)

x More generally, a circle around an arithmetic operator represents itsextension to fuzzy operands.

0018-9472/85/1100-0754$01.00 1985 IEEE


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ZADEH: SYLLOGISTIC REASONING IN FUZZY LOGIC 755

1 P r o p o r t i o nFig. 1. Mu ltiplica tion of fuzzy quantif iers./s

most2most

ProportionFig. 2. Repres entation of most an d most2.

A concrete example of the intersection/product syllomost students are youngmost young students are single ( 1 -5)most2 students are young and single

most2 denotes the product of the fuzzy quantifierwith itself (Fig. 2).For concreteness, we shall restrict our attention to sylp, q, and r are propositions of the formp = QxA's a r e a ' s ( 1 . 6 )q= 2 C ' s a r e Z T s ( 1 . 7 )r = 3 ' s a r e F ' s ( 1 . 8 )

A, B, C, D, E, and F are interrelated fuzzyThe interrelations between A, B, C, D, E, and F

=enotes conjunction, and V = denotes disjunction).Intersection/product syllogism :

C = A A B, E = A, F= C A D. (1.9)Chaining syllogism :

C = B, E = A, F=D. (1.10)Consequent conjunction syllogism :A = C = E, F= B A D. (1.11)

Consequent disjunction syllogism :A = C = E, F= BV D. (1.12)

Antecedent conjunction syllogism:B = D = F, E = A A C. (1.13)

Antecedent disjunction syllogism:B = D = F, E = \ C. (1.14)

An important application of syllogistic reasoning in fuzzylogic relates to what may be regarded as reasoning withdispositions [50]. A disposition, as its name suggests, is aproposition which is preponderantly but not necessarilyalways true. To capture this intuitive meaning of a disposition, we define a disposition as a proposition with implicitextremal fuzzy quantifiers, e.g., most, almost all, almostalways, usually, rarely, few, small fraction, etc. This definit ion should be regarded as a dispositional definition in thesense that it may not be true in all cases.Examples of commonplace statements of fact which maybe viewed as dispositions are overeating causes obesity,snow is white, glue is sticky, icy roads are slippery, etc. Anexample of what appears to be a plausible conclusiondrawn from dispositional premises is the followingicy roads a re slipperyslippery roads are dangerous (1-15)icy roads are dangerous.As will be seen in Section III, syllogistic reasoning with

dispositions provides a basis for a formalization of the typeof commonsense reasoning exemplified by (1.15).The importance of the concept of a disposit ion stems

from the fact that what is commonly regarded as commonsense knowledge may be viewed as a collection ofdispositions [49]. For example, the dispositions

birds can flysmall cars are unsafeprofessors are not richstudents are youngwhere there is smoke there is fireSwedes are taller than ItaliansSwedes are blondit takes a little over one hour to drivefrom Berkeley to Stanforda TV set weighs about 50 pounds

may be regarded as a part of commonsense knowledge.The last two examples typify a particularly important typeof disposit ion: a dispositional valuation, that is, a disposition which characterizes the usual value of a variablethrough the implicit fuzzy quantifier usually. Thus uponexplicitation, that is, making explicit the implicit fuzzyquantif iers , the examples in question becomeusually it takes a little over one hour to drive

from Berkeley to Stanfordusually a TV set weighs about 50 pounds.

Dispositional valuations have the general form(usually)(X is F) (1.16)

where (usually) is an implicit fuzzy quantifier; X is avariable which is constrained by the disposit ion; and F isits fuzzy value. As an illustration, expressed in this form,the last example reads 2(usually)(weight(TV) is A B O U T. FIFTY . PO U N D S) .

2 Upp er case symb ols and periods serve to represent compoun d labels offuzzy sets.


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USUALLY

Fig. 3. Representation of ( usually)( X is ABOUT a).

A basic syllogism which governs inference from disposi-t ional valuations is the dispositional modus ponens(usually)(X is F)if X is F then (usually)(Y is G) (1.17)(usually2)(Y is G) ,

where F and G are fuzzy predicates and usually2 is theproduc t of the fuzzy quantifier usually with itself.To a first approximation, the fuzzy quantifier usuallym ay be represented as a fuzzy number of the same form asmost (Fig. 3). More generally, however, usually connotes adependence on the assumption of "normal i ty ." Morespecifically, let Z denote what might be called a conditioning variable whose normal (or regular) value is R, where Rin general is a fuzzy set which is the complement of a set ofexceptions. In terms of the conditioning variable, the disposit ional valuation (usually)(X is F) m ay be expressed as(usually)(X is F) if ( Z is R) then (most X's are F).

(1.18)For example

(usually)(duration(trip(Berkeley, Stanford)) isLITTLE.OVER.ONE. HOUR)

where B ( A, the intersection of B and A, is defined by^ B n ^ ( " ) = M B (" ) ( , e t / . (2.4)Thus , in terms of the membership functions of B and A,the relative sigma-count of B in A is given by

2/Me(M/) ^ , ) coxmt{B/ ) , 7 * ( " ,) (2.5)The concept of a relative sigma-count provides a basisfor interpreting the meaning of propositions of the formp = QA 's are B 's , e.g., most young men are healthy. Morespecifically, th e fuzzy quantifier Q in the proposition QA 's

are B 's may be regarded as a fuzzy characterization of therelative sigma-count of 5 in yl, which entails that thepropos i t ion in question may be translated asQA 's are B 's -> 2 c o u n t ( / , 4 ) is Q. (2.6)

The r ight-hand member of (2.6) implies tha t Q, viewed as afuzzy number, defines the possibility distribution of count(B/A). This may be expressed as the possibilityassignment equation [45]n*=e (2.7)

in which th e variable X is the sigma-count in question and x is its possibility distribution.As was stated earlier, a fuzzy quantifier is a second-orderfuzzy predicate. The interpretation expressed by (2.6) an d(2.7) shows that the evaluation of a fuzzy quantifier may bereduced to t ha t of a first order predicate if Q is interpretedas a fuzzy subset of the real line. Thus let us consider againthe propos i t ion p = QA's are B's, in which A an d B arefuzzy sets in their respective universes of discourse, U an dV\ and Q, regarded as a second-order fuzzy predicate, isassumed to be characterized by its membership functionfiQ(X,Y), with X and Y ranging over the fuzzy subsets ofU and V. Then based on (2.6) and (2.7), we can define


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n the right-hand member of which Q is a unary first-orderfuzzy predicate whose denotation is a fuzzy subset of theunit interval. Consequently, in the proposition QA's are's, Q may be interpreted as a) a second-order fuzzyredicate defined on (/* X P , where U* and F* are thefuzzy power sets of U and V or b) a first-order fuzzyredicate defined on the unit interval [0,1].It is useful to classify fuzzy quantifiers into quantifiersf the first kind, second kind, third kind, etc., dependingn the arity of the second-order fuzzy predicate which theuantif ier represents. Thus, Q is a fuzzy quantifier of theirst kind if it provides a fuzzy characterization of the

of a fuzzy set; Q is of the second kind if ita fuzzy characterization of the relative cardinality

Q is of the third kind if it serves theame role in relation to three fuzzy sets. For example, th euzzy quantifier labeled several is of the first kind; most ismany more in there are many morein B 's than A's in C 's is of the third kind. It should bein terms of this classification, the certaintyin such expert systems as M Y C I N [38]

are fuzzy quantifiers of the thirdThe concept of a fuzzy quantifier gives rise to a number

to syllogistic reasoning,are the concepts of compositionality and

Specifically, consider a fuzzy syllogism of the generali.e.

an example of which is

p(Qi)q{Qi)r(Q). (2.9)

Q may be expressed as a function of Qx and Q2of the denotations of the predicates whichp and q, excluding the trivial case, where Q is

and b) if Qx and Q2 are numericalso is Q. Fur thermore, we shall say that theis weakly com positional if only a) is satisfied, in

if Qx and Q2 are numerical quantifiers, Q mayin order toit is necessary, in general,

in p and q. For example, the syllogismQXA 's ar e B 's 2 ' s a r e C ' s( i 2 M ' s a r e ( a n d C )' s

(2.10)

if B c A.Turning to the concept of robustness, suppose that wea nonfuzzy syllogism of the form

/ ' (al l )g(all)r (al l )

(2.11)

all A's are B 'sall B 's are C 'sal l A's are C ' s .

(2.12)The original syllogism is robust if small perturbations inthe quantif iers in p and q result in a small perturbation inthe quantif ier in r. For example, the syllogism representedby (2.12) is robus t if its validity is preserved when a) thequantif ier all in p and q is replaced by almost all, and b)the quantif ier all in r is replaced by almost almost all. (Inmore concrete terms, this is equivalent to replacing al l in pand q by the fuzzy number 1 , where is a small fuzzyn u m b e r and b) replacing all in r by the fuzzy number1 e.) More generally, a syllogism is selectively robust ifthe above holds for perturbations in either the first or thesecond premise, but not necessarily in both. For example,i t may be shown that the syllogism expressed by (2.12) isselectively robust with respect to per turbat ions in the firstpremise bu t not in the second premise. In fact, the syllogism in question is brittle with respect to per turbat ions inthe second premise in the sense that the slightest perturbation in the quantifier all in q requires the replacement ofthe quantif ier al l in r by the vacuous quantifier none-to-all.

As was stated earlier, the concept of dispositionality isclosely related to that of usuality. In particular , a disposi-t ional valuation expressed as X is F may be interpreted as{usually ){X is F) or equivalently as U(X) = F, whereusually is an implicit fuzzy quantifier, and U{X) denotesthe usual value of X. For example, glue is sticky m ay bein terpreted as (usually)(glue is sticky); lamb is more expensive than beef may be interpreted as (usually)(lamb is moreexpensive than beef); and most students are undergraduatesm ay be interpreted in some contexts as (usually)(moststudents are undergraduates).

To concretize the meaning of the fuzzy quantifier usually, assume that X takes a sequence of values Xv , Xnin a universe of discourse U. Then, usually, in i ts unconditioned sense, may be defined byusually ( X is F) = most X 's are F (2.13)

where F plays the role of a usual value U(X) of X. Animmediate consequence of this definition is that a usualvalue of X is not unique. Rather, given a fuzzy value F, wecan compute the degree to which it satisfies the definition (2.13) by employing the formula

T ^ M O S T ~ ( M F ( * I ) +n + ( ) (2.14)where / A M O S T * S t n e membership function of the fuzzyquantif ier most; is the membership function of F; andthe argument of JU,MOST * S t n e relative sigma-count of then u m b e r of t imes X satisfies the proposition X is F.

More generally, when usually is defined in its conditioned sense via (1.18), the degree of compatibil i ty of Fwith the definit ion of a usual value may be expressed as

T = /A M o s T ( 2 c ou n t( F/ ) )


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in which the relative sigma-count is given by2 c o u n t ( i y # ) = , / = , . , . , .

2 , - / * A ( * )The impor tance of the concept of usuality stems fromthe fact that it underlies almost all of human decisionmak-ing from the t ime of awakening in the morning till retire

ment at night. Thus in deciding on when to get up, we takeinto consideration how long it usually takes to dress, havebreakfast, drive to work, etc. In fact, we could not functionat all without a knowledge of the usual values of thevariables which enter into our daily decisionmaking and anability to employ them in decision analysis. It is of interestto observe that the usual values are usually fuzzy ratherthan crisp. How to manipulate the usual values in thecontext of fuzzy syllogisms will be discussed briefly in thefollowing section.III. FU ZZ Y SYLLOGISMS AND REASO NING WIT H

DISPOSITIONSAs was stated earlier, one of the basic syllogisms in fuzzylogic is the intersection/product syllogism expressed by(1.2).In what follows, we shall employ this syllogism as astarting point for the derivation of other syllogisms whichare of relevance to the important problem of combinat ionof evidence in expert systems.A derivative syllogism of this type is the multiplicativechaining syllogism

QXA 's are B 's 2 ' s a r e C ' s (3.1)> (61 2 ) ^ ' s a r e C 's

in which > (Q1 Q2) should be read as at least Q1 Q2.This syllogism is a special case of the in tersect ion/productsyllogism which results when B c A, i.e./**("/) < (" )> UI u> = 1, (3-2)

For , in this case A B = B, and since B C is containedin C, it follows that(Qi 2 ) ^ ' s a r e ( 5 a n d C)' s =>

> ( l 2 M ' S a r e C " S ( 3 3)(I t is of interest to note that if Q in the proposition QA'sare B 's is interpreted as the degree to which A is containedin B, then the multiplicative chaining syllogism shows that,under the assumption B c A, the fuzzy relation of fuzzy-setcon ta inment is product transitive [43], [48].

If, in addition to assuming that B c A, we assume thatQ1 and Q2 are monotone increasing [3], [47], i.e.> i = i> 2 = 2 , (3-4)

which is true of the fuzzy quantifier most, then> (Qi Qi) = Qi Qi (3-5)

and the multiplicative chaining syllogism becomes (Fig. 5)QXA 's are B 'sg 2 ' s a r e C ' s (3.6)(Q i 2 ) ^ ' s a r e C ' s .

A s an illustration, we shall consider an example in whichthe containment relation B c A holds approximately, as inthe propos i t ionp = most American cars are big. (3-7)Then , if

q = most big cars are expensive (3.8)we may conclude, by employing (3.6), that

r = most2 American cars are expensivewith the unders tanding that most2 is the product of thefuzzy number most with itself [47].It can readily be shown by examples that if no assumptions are made regarding A, B, and C, then the chaininginference schema QxA's a r e a ' s

> 2 ' s a r e C ' s (3.9)QA's ar e C' s .

is not weakly compositional, which is equivalent to sayingthat , in general, Q is the vacuous quantif ier none-to-all.However , if we assume, as done above, that B c A, then itfollows from the intersection/product syllogism that (3.6)becomes weakly compositional, withQ = >(Qi*Qi) (3.10)

and, furthermore, that (3.6) becomes strongly composit ional if Qx and Q2 are monotone increasing.Another important observation relates to the robustnessof the multiplicative chaining syllogism. Specifically, if weassume thatQ l = lO elQ2 = lGe2

where x and e2 are small fuzzy numbers, then it canreadily be verified that, approximatelyQ x Q2 s 1 l 2 (3.11)

which establishes that the multiplicative chaining syllogismis robust. However, in the absence of the assumptionB a A, the inference schema (3.9) is robust only withrespect to per turbat ions in Qv To demonstrate this , assume that Ql = almost all and Q2 = all. Then, from thein tersect ion/product syl logism it follows that Q = >{almost all). On the other hand, if we assume that Ql = alland Q2 = almost all, then Q = none-to-all. Thus , as wasstated earlier, the inference schema (3.9) is brittle withrespect to per turbat ions in the second premise.Th e MPR Chaining Syllogism

In the preceding discussion, we have shown that theassumption B c A leads to a weakly compositional multiplicative chaining syllogism. Another type of assumption


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ZADEH: SYLLOGISTIC REASONING IN FUZZY LOGIC 759which also leads to a weakly compositional syllogism isthat of major premise reversibility (MPR).3 This assumption may be expressed as the semantic equivalence

QXA 's are B 's ( ( ( i 0 2 l ) M ' s a r e C ' s .

(3.13)We shall refer to this syllogism as the MPR chainingsyllogism. It follows at once from (3.13) that the MPRchaining syllogism is weakly compositional and robust. Aconcrete instance of this syllogism is provided by thefollowing example

most American cars are bigmost big cars are heavy (3 \ 0 ( V ) ( 2 most 1) American cars are heavy.

The Consequent Conjunction SyllogismThe consequent conjunction syllogism is an example of abasic syllogism which is not a derivative of the intersection/product syllogism. I ts s tatement may be expressed asfollows:

QXA 's are B 'sg 2 y l ' s a r e C ' s ( 3 . 1 5 ) M ' s a r e ( a n d C ) ' s

whereO 0 ( i 2 1 ) < < 1 2 (3 .16)

From (3.16), it follows at once that the syllogism is weaklycompositional and robust.An illustration of (3.15) is provided by the examplemost students are youngmost students are singleQ students are single and young

here2most < < m ost. (3.17)

his expression for Q follows from (3.16) by noting thatmost (/0) most = most

3 It should be noted that, in the classical theory of syllogisms, theremise in question would be referred to as the minor premise.

and0 ( v ) ( 2 m o s t 1) = 2 mo st 1.

The importance of the consequent conjunction syllogismstems from the fact that it provides a formal basis forcombining rules in an expert system through a conjunctivecombination of hypotheses [38]. However, unlike such rulesin M YC IN and PRO SPEC TOR , the consequent conjunction syllogism is weakly rather than strongly composit ional. Since the combining rules in MYCIN and PROSP E CT O R a r e ad hoc in nature, whereas the consequentconjunction syllogism is not, the validity of strong com-pos i t ional i ty in MYCIN and PROSPECTOR is in need ofjustif ication.The Antecedent Conjunction Syllogism

An issue which plays an important role in the management of uncertainty in expert systems relates to the question of how to combine rules which have the same consequent but different antecedents.

Expressed as an inference schema in fuzzy logic, thequestion may be stated asQXA 's are C 'sg 2 ' s a r e C ' s ( 3 . 1 8 )Q(A a n d ) ' s a r e C ' s

in which Q is the quantifier to be determined as a functionof Q l and Q2.It can readily be shown by examples that, in the absenceof any assumptions about A, B, C, Q l9 and Q2, what canbe said about Q is that it is the vacuous quantifier none-to-all. Thus, to be able to say more, it is necessary to makesome restrictive assumptions which are satisfied, at leastapproximately, in typical s i tuations.

The commonly made assumption in the case of expertsystems [9], is that the items of evidence are conditionallyindependent given the hypothesis . Expressed in terms ofthe relative sigma-counts of A, B, and C, this assumptionmay be written as2 c o u n t ( ^ B/C) = count(A/C) count(B/C).

(3.19)Using this equality, it is easy to show that2 c o u n t ( C / M B) = ^ 2 c o u n t ( C / M ) 2 c o u n t ( C / )

(3.20)where the factor K is given by

Scount (y l ) Scoun t ( i ? )" S c o r n i t i n ) 2 c o u n t ( C ) ' ^ ' '

The presence of this factor has the effect of making theinference schema (3.18) noncompositional. One way ofgetting around the problem is to employinstead of the


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760 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. SMC-15, NO. 6, NOVEMBER/DECEMBER 1985sigma-counta count defined by

2 c o u n t ( i ? )p 2 c o u n t ( )

\ ( / ) =2count ( - ,2?) ( / )

2 c o u n t ( - n / , 4 )

(3.22)

(3.23)

in which iB denotes the negation of B (or, equivalently,the c om plem ent of 2?, if B is interpreted as a fuzzy setwhich represents the denotation of the predicate B) . Thesecounts will be referred to as psigma-counts (with p standing for ratio) and correspond to the odds which are employed in PROSPECTOR. Thus, expressed in words, wehavep 2 c o u n t ( B ) = ratio of B 's to no n-5 's (3.24)

\ ( / ) = ratio of B 's to non -5 's among A 's.(3.25)

In terms of psigma-counts, it can readily be shown thatthe assumption expressed by (3.19) entails the equalityp 2 c o u n t ( C / y l B) = p 2 c o u n t ( C / ^ ) p 2 c o u n t ( C / 5 )

p S c o u n t ( ^ C) . ( 3 . 2 6 )This equality, then, leads to what will be referred to as theantecedent conjunction syllogism

ratio of C 's to non-C 's among A 's is Rxratio of C 's to non-C ' s among B 's is R2ratio of C 's to n o n - C ' s

among (A and 2?)'s is x 5 2 ^ 3(3.27)

whereR, ratio of C 's to non-C's .

It should be noted that this syllogism may be viewed as thefuzzy logic analog of the likelihood ratio combining rule inPROSPECTOR [9] .In the foregoing discussion, we have focused our attention on some of the basic syllogisms in fuzzy logic whichmay be employed as rules of combination of evidence inexpert systems. Another important function which theseand related syllogisms may serve is that of providing abasis for reasoning with dispositions, that is, with propositions in which there are implicit fuzzy quantifiers.The basic idea underlying this application of fuzzy syllogisms is the following. Suppose that we are given twodispositions

(3.28)icy roads a re slipperyslippery roads are dangerous.Can we infer from these dispositions what appears to be aplausible conclusion, namely

icy roads are dangerous!As a first step, we have to restore the suppressed fuzzyquantifiers in the premises. For simplicity, assume that thedesired restoration may be accomplished by prefixing the

dispositions in question with the fuzzy quantifier most, i.e.icy roads are slippery -> most icy roads are slipperyslippery roads are dangerous> most slippery roads are dangerous.

Next, if we assume that the proposition most slipperyroads are dangerous satisfies the major premise reversibilitycondition, i .e. ,most icy roads are slippery

! , , pk, with dispositional premises expressed as (usu-ally)p l, (usually)p2,- , (usually)p k.As a simple illustration, a basic inference rule in fuzzylogic is the entailment principle, which may be expressed as

XisAAoB (3.31)XisBwhere A and B are fuzzy predicates. In plain words, thisrule states that any assertion about X of the form X is Aentails any less specific assertion X is B.

A disposit ional variant of the entailment principle maybe expressed as

(usually)(X is A)AaB (3.32)(usually)(X is B) .


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SYLLOGISTIC REASONING IN FUZZY LOGIC 76 1

1 ProportionFig. 4. Representation of usually and usually2

Q, A'sQ 2 CA(Q , 9

>Q, e

a r ea nd

B'sB) 's

Q2) A'sQ2)

CQ i e Q2)A 's

a rea rea r e

C's(B arC's

A 's are C's- m o n o t o n i e

d O ' s

5. Representation of the intersection/product syllogism and itscorollaries.Another var iant is

XisA(usually)(A c B)(usually)(X is B).

Still another variant is(usually)(X is A)(usually)(A c 5 )( w ^ / / y 2 ) ( X i s ) .

(3.33)

(3.34)

ote that usually2 is less specific than usually (Fig. 4).A special case of the entailment principle which plays animpor tant role in fuzzy logic is the projection rule, whichm ay be stated as follows. Let X and Y be variablesranging over U and V, respectively, and let R be a fuzzyrelation in U X V, which is characterized by its membership function ju (w, u\ u e /, V G V. Then, in symbols,the projection rule may be expressed as

(X,Y)isRX is projfj R (3.35)

where the first premise signifies that X and Y are R-related, and proj^ R denotes the projection of R on U. Themembership function of proj^ R is given by

VPro ]uR(u ) = S U P ^ W , ^ ) . (3.36)Since R is contained in pvo)uR, it follows from (3.32)that

(usually)((X,Y) is R)( usually)( X is Proj^ R). (3.37)

This inference rule may be regarded as a dispositionalversion of the projection rule.The dispositional inference rule expressed by (3.37) maybe applied to the derivation of a dispositional form of the

compo si t ional rule of inference in fuzzy logic [45]XisA(X,Y)isR (3.38)Y is A R

where A R denotes the composition of A and R, which isdefined byPA-R(V) = S U P M (V>A(U) /* /?("> *>)) ( 3 3 9 )

Specifically, assume that the premises in (3.38) are expressed as the dispositional valuations(usually)(X is A) (3.40)

an d(usually)((X, Y) is R) (3.41)

in which the fuzzy quantifier usually is the same in the twopremises in the sense that the conjunction of the premisesin question may be expressed as(usually)((X is A) A ((X, Y) is R))or, equivalently, as

(usually)(X, Y) is A A R) (3.42)where the conjunction of A and R is defined by

VAAR(U>V) = ^A(U) / * *( , I>). (3.43)Then , on applying the dispositional projection rule to(3.41), we have

( usually)(( XisA)A((X,Y)isR))(usually)(Y is projK (A A R) (3.44)

and sinceA R = p r o j K ( ^ R)

we can assert that(usually)(X is A)(usualfy)((X, Y) is R)(usually)(Y is A R),

(3.45)

which expresses a dispositional version of the composit ional rule of inference. It should be noted that the validityof this version depends on the assumption that the twopremises (3.40) and (3.41) may be combined conjunctivelyas in (3.42). Without this assumption, the fuzzy quantifierin the conclusion would be less specific than usually, e.g., itmight be usually2.An impor tan t way in which usuality enters into syllogistic reason ing is the following.

Cons ider the chaining inference schemaQXA 's are B 'sQ2B's are C'sQA 's are C 's

(3.46)

in which no restrictions are placed on A, B, C. Then, aswas pointed out earlier (see (3.9)), Q is the vacuous quantifier none-to-all. However, under the assumption that B c A,


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762 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. SMC-15, NO. 6, NOVEMBER/DECEMBER 1985(3.46) leads to the multiplicative chaining syllogism (see(3.1))

QXA 's are B 's 2 ' s a r e C ' sB ( i 2 M ' s a r e C " s .

Now assume that the categorical assumption B


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ZADEH: SYLLOGISTIC REASONING IN FUZZY LOGIC

[41] P. Szolovits and S. G. Pauker, "Cate gorica l and probabilistic reasoning in medical diagnosis," Artificial Intelligence, vol. 11, pp.115-144, 1978.[42] R. R. Yager, "Quantifie d pro positions in a linguistic logic," inProc. 2nd Int. Seminar Fuzzy Set Theory, E. P. Klem ent, Ed. Linz,Austria: Johannes Kepler University, 1980.[43] L. A. Zad eh, "Similarity relations and fuzzy ordering s," Info. Sci.,vol. 3, pp. 177-200, 1976.[44] L. A. Zad eh, "Fuzz y logic and approxima te reasoning (in mem oryof Grigore Moisil)," Synthese, vol. 30, pp. 407-428, 1975.[45] L. A. Zad eh, "A theory of approxim ate reasoning," tech. Mem o. no.M 7 7 / 5 8 , University of California, Berkeley, 1977. See also MachineIntelligence, (vol. 9), J. E. Hayes, D. Michie, and L. I. Kulich, Eds.New York: John Wiley, 1979, pp. 149-194.[46] L. A. Zadeh, "Test-sc ore semantics for natural languages and

76 3meaning-representation via PR F," tech. memo. no. 247, AI Center,SRI International, Menlo Park, CA, 1981. See also EmpiricalSemantics, B. B. Rieger, Ed. Bochum : Brockmeyer, 1981, pp.281-349 .[47] L. A. Zad eh, "A comp utationa l approach to fuzzy quantifiers innatural languages," Computers and Mathematics, vol. 9, pp. 149-1 84,1983.[48] L. A. Zad eh, "T he role of fuzzy logic in the manage ment ofuncertainty in expert systems," Fuzzy Sets and Systems, vol. 11, pp.199-227, 1983.

[49] L. A. Zad eh, "A theory of commonsense knowle dge," in Issues ofVagueness, H. J. Skala, S. Termin i, and E. Trillas, Eds. Dord rech t:Reidel, 1984, pp. 257-296.[50] L. A. Za deh , "A com putation al theory of dispositio ns," in Proc.1984 Int. Conf. Computational Linguistics, 1984, pp. 312-318.

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