7_A Computational Theory of Dispositions-1984

8/9/2019 7_A Computational Theory of Dispositions-1984

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A COMPUTATIONAL THEORY OF DISPOSITIONS

Lotfi A. ZadehComp uter Science DivisionUniversity of California,Berkeley, Californ ia 94720, U.S.A.

A B S T R A C TInformally, a d i s p o s i t i o n is a proposition which is prepon-derantl y, but no necessarily always, true. For example, b i rdsc a n f l y is a disposition, as are the propositions S w e d e s a r eb l o n d and S p a n i a r d s a r e d a r k .An idea which underlies the theory described in thispaper is that a disposition may be viewed as a propositionwith implicit fuzzy quantifiers which are approximations to a lland a l w a y s , e.g., a l m o s t a l l, a l m o s t a l w a y s , m o s t , f r e q u e n t l y ,etc. For example, b i r d s c a n f l y may be interpreted as theresult of supressing the fuzzy quantifier m o s t in the proposi-tion m o s t b i r d s c a n f ly . Similarly, y o u n g m e n l i k e y o u n g w o m e nmay be read as m o s t y o u n g m e n l i k e m o s t l y y o u n g w o m e n . Theprocess of transforming a disposition into a proposition is

referred to as e z p l i c i t a t i o n or r e s t o r a t i o n .Explicitation sets the stage for representing the meaningof a proposition through the use of test-score semantics(Zadeh, 1978, 1982). In this appr oach to semantics, the mean-ing of a proposition, p, is represented as a procedure whichtests, scores and aggregates the elastic constraints which areinduced by p.The paper closes with a description of an approach toreasoning with dispositions which is based on the concept of afuzzy syllogism. Syllogistic reasoning with dispositions has animpo rtan t bearing on commonsense reasoning as well as onthe man ageme nt of uncerta inty in expert systems. As a sim-ple application of the techniques described in this paper, weformulate a definition of t y p i c a l i t y - a concept which plays animportant role in human cognition and is of relevance todefault reasoning.

1 . In t roduc t ionInformally, a disposition is a proposition which is prepon-derantly, but not necessarily always, true. Simple examples ofdispositions are: S m o k i n g i s a d d i c t iv e , e x e r c i s e i s g o o d f o r y o u rh e a l t h , l o n g s e n t e n c e s a r e m o r e d i f f ic u l t t o p a r s e t h a n s h o r t s e n -t e n c e s , o v e r e a t i n g c a u s e s o b e s i t y , T r u d i i s a l w a y s r i g h t , etc.Dispositions play a central role in human reasoning, sincemuch of human knowledge and, especially, commousenseknowledge, may be viewed as a collection of dispositions.

The concept of a disposition gives rise to a number ofrelated concepts among which is the concept of a d i s p o s i t i o n a lp r e d i c a t e . Familiar examples of unary predicates of this typeare: H e a l t h y , h o n e s t , o p t i m i s t , s a f e , etc., with binary disposi-tional p redicates exemplified by: t a l l er than in S w e d e s a r e t a l l e rt h a n F r e n c h m e n , l i k e in I t a l i a n s a r e l i k e S p a n i a r d s , l i k e iny o u n 9 m e n l ik e y o u n g w o m e n , and s m o k e s in R o n s m o k e sc igare t t es . Another related concept is that of a d i s p o s i t i o n a lc o m m a n d {or i m p e r a t i v e ) which is exemplified by p r o c e e d w i t hc a u t i o n , a v o i d o v e r e x e r t i o n , k e e p u n d e r r e f r i g e r a t i o n , b e f r a n k ,etc.To Protessor Nancy Cartwright. Research supported in part by NASAGrant NCC2-275 and NSF Grant IST-8320416.

The basic idea underlying the approach described in thispaper is th at a disposition may be viewed as a propositio nwith suppressed, or, more generally, implicit fuzzy quantifierssuch as m o s t ~ a l m o s t a ll , a l m o s t a l w a y s , u s u a l l y , r a r e l y , m u c h o fth e t i m e , etc . To illustrate, the disposition g e s t a t i n g c a u s e so b e s i t y may be viewed as the result of suppression of the fuzzyquantifier m o s t in the proposition m o s t o f t h o s e w h o o v e r e a tare obese. Similarly, the disposition y o u n g m e n l i k e y o u n gw o m e n may be interpreted as m o s t y o u n g m e n l i k e m o s t l yy o u n g w o m e n . It should be stressed, however, that r e s t o r a t i o n(or e z p l i c i t a t i o n ) - - viewed as the inverse of suppression - is aninterpretati on-dependent process in the sense that, in general,a disposition may be interpreted in different ways dependingon the manner in which the fuzzy quantifiers are restored anddefined.

The implicit presence of fuzzy quantifiers stands in theway of representing the meaning of dispositional conceptsthrough the use of conventional methods based on truth-conditional, possible-world or model-theoretic semantics(Cresswell, 1973; McCawley, 1981; Miller and Johnson-Laird,1970),~-tn the c omputa tional a pproach which is described inthis paper, a fuzzy quantifier is manipulated as a fuzzynumber. This idea serves two purposes. First, it provides abasis for representing the meaning of dispositions; and second,it opens a way of reasoning with dispositions through the useof a collection of syllogisms. This aspect of the concept of adisposition is of relevance to default reasoning and non-monotonic logic (McCarthy , 19 80; McDermo tt and Doyle,1980; McDermott, 1982; Reiter, 1983).To illustrate the ma nner in which fuzzy quantifiers may

be manipulated as fuzzy numbers, assume that, after restora-tion, tw o dispo sitions d I and d 2 may be expressed as proposi-tions of the formP l A Q t A t s a r e B I s (1.1)

P 2 A Q 2 B e s a r e C I s , (1.2)in which Ql and Q2 are fuzzy quantifiers, and A, B and C arefuzzy predicates. For example,

P l & - m o s t s t u d e n t s a r e u n d e r g r a d u a t e s (1.3)P 2 ~ m o s t u n d e r g r a d u a t e s a r e y o u n g .

By treating Pl and P2 as the major and minor premises ina syllogism, the following c h a i n i n g syllogism may be esta-blished if B C A (Zadeh, 1983):

1. In the literature of linguistics, ogic and philosophyof languages, fuz-zy quantifiers are usually referred to as ~agne or generalized quantifiers(Barwise and Cooper, 1981; Peterson, 1979). In the approach describedin this paper, a fuszy quantifier s interpreted as a fuzzy number whichprovides an approximate characterization of absolute or relative cardi-nality.

312


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Q 1 A ' s o r e B t s (1.4)Q : B I s a r e C I s> _ ( QI ~ Q 2 ) A # s a r e C ' s

i n w h i c h Q 1 ~ Q 2 r e p r e s e n t s t h e p r o d u c t o f t h e f u z z yn u m b e r s Q I a n d Q 2 ( F i g u r e 1 ).

I I1

- / / / - - - ; O s s o l @[ a = b cP r o p o r t i o n

0 2

F i g u r e 1 . M u l t i p l i c a t i o n o f f u z z y q u a n t i f ie r s

a n d ~ _ ( Q l ~ Q : t ) s h o u l d b e r e a d a s " a t le a s t Q 1 ~ Q 2 . " A ss h o w n i n F i g u r e 1 , Q ~ a n d Q 2 a r e d e f in e d b y t h e i r r e s p e c t i v ep o s s i b i l i ty d i s t r i b u t i o n s , w h i c h m e a n s t h a t i f t h e v a l u e o f Q 1a t t h e p o i n t u i s a , t h e n a r e p r e s e n t s t h e p o s s i b i l it y t h a t t h ep r o p o r t i o n o f A ~s in B ~s is u .I n t h e s p e c i a l c a s e w h e r e P l a n d P 2 a r e e x p r e s s e d b y( 1 .3 ) , t h e c h a i n i n g s y l l og i s m y i e l d s

m o s t s t u d e n t s a r e u n d e r g r a d u a t e sm o s t n n d e r q r a d n a t e s a r e v o u n qm o s t 2 s t u d e n t s a r e y o u n g

w h e r e m o s t ~ r e p r e s e n t s t h e p r o d u c t o f t h e f u z z y n u m b e r m o s tw i t h i t s e l f ( F i g u r e 2 ) .

/zI

/ /m o s t =

m o s t

Propor t i onF i g u r e 2 . R e p r e s e n t a t i o n o f m o s t a n d m o s t 2 .

2 . M e a n i n g R e p r e s e n t a t i o n a n d T e s t - S c o r e S e m a n t i c sT o r e p r e s e n t t h e m e a n i n g o f a d is p o s i t i o n , d , ~ e e m p l o ya t w o - s t a g e p r o c e s s . F i r s t , t h e s u p p r e s s e d f u z z y q u a n t i f i e r s i nd a r e r e s t o r e d , r e s u l t i n g i n a f u z z i l y q u a n t i f i e d p r o p o s i t i o n p .T h e n , t h e m e a n i n g o f p i s r e p r e s e n t e d - - t h r o u g h t h e u s e o ft e s t - sco r e sem an t i c s ( Zad eh , 1 9 78 , 1 98 2 ) - a s a p r o ce d u r ew h i c h a c t s o n a c o l l e c t i o n o f r e l a t io n s i n a n e x p l a n a t o r y d a t a -b a s e a n d r e t u r n s a t e s t s c o r e w h i c h r e p r e s e n t s t h e d e g r e e o f

c o m p a t i b i l i t y o f p w i t h t h e d a t a b a s e . I n ef f ec t , t h i s i m p l ie st h a t p m a y b e v i e w e d a s a c o l l e c t io n o f e l a s t ic c o n s t r a i n t sw h i c h a r e t e s t e d , s c o r e d a n d a g g r e g a t e d b y t h e m e a n i n g -r e p r e s e n t a t i o n p r o c e d u r e . I n t e s t - s c o r e s e m a n t i c s , t h e s e e l a s t i cc o n s t r a i n t s p l a y a r o l e w h i c h i s a n a l o g o u s t o t h a t t r u t h -c o n d i t i o n s i n t r u t h - c o n d i t i o n a l s e m a n t i c s ( C r es s w e l l , 1 9 73 ) .A s a s i m p l e i l l u s t r a t i o n , c o n s i d e r t h e f a m i l i a r e x a m p l e

d A s n o w is w h i t ew h i c h w e i n t e r p r e t a s a d i s p o s i t i o n w h o s e i n t e n d e d m e a n i n g i st h e p r o p o s i t i o n

p A u s u a l l y s n o w i s w h i t e .T o r e p r e s e n t t h e m e a n i n g o f p , w e a s s um e t h a t t h e e z p l a n a -t o r y d a t a b a s e , E D F ( Z a d e h , 1 9 8 2 ) , c o n s i s t s o f t h e f o l l o w i n gr e l a ti o n s w h o s e m e a n i n g i s p r e s u m e d t o b e k n o w n

E D F A W H I T E [S am p le ;p ] + U S U A L L Y [ P r o p o r t i o n ; p ] ,i n w h i c h + s h o u l d b e r e a d a s and . T h e i t h r o w i n W H I T E isa t u p l e ( S i , r i ) , i = 1 , . . . , m , i n w h i c h S i i s t h e i t h s a m p l e o fs n o w , a n d r i i s i s t h e d e g r e e t o w h i c h t h e c o l o r o f S i m a t c h e sw h i t e . T h u s , r i m a y b e i n t e r p r e t e d a s t h e t e s t s c o r e fo r t h ec o n s t r a i n t o n t h e c o l o r o f S i i n d u c e d b y t h e e l a s ti c c o n s t r a i n tW H I T E . S i m i l a r l y , t h e r e l a t i o n U S U A L L Y m a y b e i n t e r -p r e t e d a s a n e l a s t i c c o n s t r a i n t o n t h e v a r i a b l e P r o p o r t i o n , w i t hp r e p r e s e n t i n g t h e t e s t s c o r e a s s o c i a t e d w i t h a n u m e r i c a l v a l u eo f P r o p o r t i o n .

T h e s t e p s i n t h e p r o c e d u r e w h i c h r e p r e s e n t s t h e m e a n i n go f p m a y b e d e s c r i b e d a s f o ll o w s :1 . F i n d t h e p r o p o r t i o n o f s a m p l e s w h o s e c o l o r i s w h i t e :

r l -k -b r mm

i n w h i c h t h e p r o p o r t i o n i s e x p r e s s e d a s t h e a r i t h -m e t i c a v e r a g e o f t h e t e s t s c o r e s .2 . C o m p u t e t h e d e g r e e t o w h i c h s a t i s fi e s t h e c o n -s t r a i n t i n d u c e d b y U S U A L L Y :

r ~ ~ U S U A L L Y [ P r o p o r t i o n ~ p] ,i n wh ic h r i s t h e o v e r a l l t e s t sco r e , i .e ., t h e d eg r ee o fc o m p a t i b i l i t y o f p w i t h E D , a n d t h e n o t a t i o n~ R [ X = a ] m e a n s : S e t t h e v a r i a b l e X i n t h e r e l a-t i o n R e q u a l t o a a n d r e a d t h e v a l u e o f t h e v a r i a b l ep .

M o r e g e n e r a l l y , t o r e p r e s e n t t h e m e a n i n g o f a d i s p o s i t i o ni t i s n e c e s s a r y t o d e f i n e t h e c a r d i n a l i t y o f a f u z z y s e t .Sp ec i f i c a l l y , i f A i s a su b se t o f a f i n i t e u n iv e r se o f d i sco u r seU - -- - { u l , . . . , u , } , t h e n t h e s i g m a - c o u n t o f A i s d e f in ed a s~Count(A ) = I : ~ p A ( U ~ ) , ( 2 . 1 )

i n w h i c h p A ( U i ) , i - - -- l , . . . , n , i s t h e g r a d e o f m e m b e r s h i p o f u /i n A ( Z a d e h , 1 9 8 3 a ) , a n d i t i s u n d e r s t o o d t h a t t h e s u m m a y b er o u n d e d , i f n e e d b e , t o t h e n e a r e s t i n t e g e r . F u r t h e r m o r e , o n em a y s t i p u l a t e t h a t t h e t e r m s w h o s e g r a d e o f m e m b e r s h i p f al lsb e l o w a s p e c i f i e d t h r e s h o l d b e e x c l u d e d f r o m t h e s u m m a t i o n .T h e p u r p o s e o f s u c h a n e x c l u s i o n i s t o a v o i d a s i t u a t i o n i nw h i c h a l a r g e n u m b e r o f t e r m s w i t h l o w g r a de s o f m e m b e r s h i pb e c o m e c o u n t - e q u i v a l e n t t o a s m a l l n u m b e r o f t e r m s w i t h h i g hm e m b e r s h i p .T h e r e l a ti v e s i g m a - c o u n t , d e n o t e d b y ~ C o u n t ( B / A ) , m a y

b e i n t e r p r e t e d a s t h e p r o p o r t i o n o f e l e m e n t s o f B i n A . M o r eex p l i c i t l y ,~ C o u n t ( B / A ) - - ~ ~ C o u n t ( A f l B ) ( 2 . 2 )

E C o u n t ( a ) 'w h e r e B D A , t h e i n t e r s e c t i o n o f B a n d A , is d e f i n e d b y

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i t B n A ( U ) f U S / U ) ^ U S (U ) , U e U ,w h e r e A d e n o t e s t h e s i n o p e r a t o r in i n fi x f o r m . T h u s , i nt e r m s o f t h e m e m b e r s h i p f u n c t i o n s o f B a n d A , t h e r e l a t i v es l g m a - c o u n t o f B a n d A i s g i v e n b y

~,#B(u,) A t i n ( u , )Z C o u n t ( B / A } = ( 2 . 3 }~ , t J a ( u , )A s a n i l l u s t r a t i o n , c o n s i d e r t h e d i s p o s i t i o n

d A o v e r a t i n g c a u s e s o b e s i t y (2 .4)w h i c h a f t e r r e s t o r a t i o n i s a s s u m e d t o r e a d 2

p A m o s t o f t h o s e w h o o v e r e a t a re o b e s e . (2 .5)T o r e p r e s e n t t h e m e a n i n g o f p , w e s h a l l e m p l o y a n e x p l a -n a t o r y d a t a b a s e w h o s e c o n s t i t u e n t r e l a ti o n s a r e:

E D F ~ - P O P U L A T I O N [ N o m e ; O v e re a t; O be se ]+ M O S T ( P r o p o r t i o n ; i t ] .

T h e r e l a t i o n P O P U L A T I O N i s a l i s t o f n am es o f i n d iv id u a l s ,w i t h t h e v a r i a b l e s O v e r e a t a n d O b e s e r e p r e s e n t i n g , r e s p e c -t i v e l y , t h e d e g r e e s t o w h i c h N a m e o v e r e a t s a n d i s o b e s e . I nM O S T , p i s t h e d e g r e e t o w h i c h a n u m e r i c a l v a l u e o f P r o p o r -t i o n f it s t h e i n t e n d e d m e a n i n g o f M O S T .T o t e s t p r o c e d u r e w h i c h r e p r e s e n ts t h e m e a n i n g o f pi n v o l v e s t h e f o l l o w i n g s t e p s .

1 . Le t N a m e ~ , i - - 1 . .. . m , b e t h e n a m e o f i t h i n d i v i -d u a l i n P O P U L A T I O N . F o r e a c h N a m e , f i n d t h ed e g r e e s t o w h i c h N a m e i o v e r e a t s a n d i s o b e s e :a i A P OV ER EA r ( N a m e i ) A 0 . . . . t P O P U L A T / O N ( N a m e = N a m e i ]

# , A I to n E sE ( N a m e i } ~ o 6 , , P O P U L A T l O N [ N a m e ~ N a m e i ] .2 . C o m p u t e t h e r e l a ti v e s i g m a - c o u n t o f O B E S E inO V E R E A T :

= i a i A #ip @ ~ C o u n t ( O B E S E / O V E R E A T ) = E , a i3 . C o m p u t e t h e t e s t s c o r e f o r t h e c o n s t r a i n t i n d u c e db y M O S T :

r - ~ ~ M O S T [ P r o p o r t i o n --~ p ] .T h i s t e s t s c o r e r e p r e s e n t s t h e c o m p a t i b i l i t y o f p w i t h t h e

e x p l a n a t o r y d a t a b a s e .3 . T h e S c o p e o f a F u z z y Q u a n t i f i e r

I n d e a l i n g w i t h t h e c o n v e n t i o n a l q u a n t i f i e r s al l a n d s o m ei n f l i n t - o r d e r l o g i c , t h e sco p e o f a q u an t i f i e r p l ay s an e ssen t i a lr o l e in d e f in in g i ts m ean in g . I n t h e ca se o f a f u zzy q u an t i f i e rw h i c h i s c h a r a c t e r i z e d b y a r e l a ti v e s i g m a - c o u n t , w h a t m a t t e r si s t h e i d e n t i t y o f t h e s e t s w h i c h e n t e r i n t o t h e r e l a t i v e c o u n t .T h u s , i f t h e s i g m a - c o u n t i s o f t h e f o r m E C o u n t ( B / A ), w h i c hs h o u l d b e r e a d a s t h e p r o p o r ti o n o f B I s i n A I s , t h e n B a n dA wi l l b e r e f e r r ed t o a s t h e n - s e t [ w i t h n s t a n d i n g f o r n u m e r a -t o r ) a n d b - s e t ( w i t h b s t a n d i n g f o r b as e ) , r e s p e c t iv e l y . T h eo r d e r e d p a i r { n - s e t, b - s e t } , t h e n , m a y b e v i e w e d a ~ a g e n e r a l i-za t i o n o f t h e co n cep t o f t h e sco p e o f a q u an t i f i e r . No te , h o w-ev e r , t h a t , i n t h i s sen se , t h e sco p e o f a f u zzy q u an t i f i e r i s as e m a n t i c r a t h e r t h a n s y n t a c t i c c o n c e p t.A s a s i m p l e i l l u s t r a t i o n , c o n s i d e r t h e p r o p o s i t i o np A m o s t s tu d e n t s a r e u n d e r g r a d u a t e s . I n t h i s c a se , t h e n -se t o f m o s t i s u n d e r g r a d u a t e s , t h e b - s e t i s s t u d e n t s , a n d t h esco p e o f m o s t i s t h e p a i r { u n d e r g r a d u a t e s , s t u d e n t s } .

2. It should be understood that (2.5) is just one of m any possible in-terpret~.tions of (2.4), with no imp licat;on tha t is constitutes a prescrip-tive interpretation of causa lity. See Suppes (1970}.

A s a n a d d i t i o n a l i l l u s t r a t i o n o f t h e i n t e r a c t i o n b e t w e e ns c o p e a n d m e a n i n g , c o n s i d e r t h e d i s p o s i t i o nd A y o u n g m e n l i k e y o u n g w o m e n . (3 .1)

A m o n g t h e p o s s i b le i n t e r p r e t a t i o n s o f t h i s d i s p o s i t i o n , w es h a l l fo c u s o u r a t t e n t i o n o n t h e f o l l o w i n g ( t h e s y m b o l r dd e n o t e s a r e s t o r a t i o n o f a d i s p o s i t i o n ) :r d I A m o s t y o u n g m e n l i k e m o s t y o u n g w o m e n

r d 2 A m o s t y o u n g m e n l i k e mostlyy o u n g w o m e n .T o p l a c e i n e v i d e n c e t h e d i f f e r e n c e b e t w e e n r d I a n d r d z ,i t is e x p e d i e n t t o e x p r e s s t h e m i n t h e f o r m

r d l - ~ - m o s t y o u n g m e n P Ir d 2 ~ m o s t y o u n g m e n P 2 ,

w h e r e P l a n d P 2 a r e t h e f u z z y p r e d i c a t e sP l A l i ke s m o s t y o u n g w o m e n

a n dP 2 A l ik e s m o s t l y y o u n g w o m e n ,

w i t h t h e u n d e r s t a n d i n g t h a t , f o r g r a m m a t i c a l c o r r e c tn e s s , l i kesi n P I a n d P 2 s h o u l d b e r e p l a c e d b y l lke w h e n P l a n d P 2 a c ta s c o n s t i t u e n t s o f r d I a n d r d 2 . I n m o r e e x p l i c i t t e r m s ,P I a n d P 2 m a y b e e xp r e s s e d a sP I A P , [ N a m e ; p ] ( 3 . 2 )

P 2 ~ - P 2 [ N a m e ; p ] ,i n w h i c h N a m e i s t h e n a m e o f a m a l e p e r s o n a n d # i s t h ed e g r e e t o w h i c h t h e p e r s o n i n q u e s t i o n s a t i s f i e s t h e p r e d i c a t e .[ E q u i v a l e n t l y , p i s t h e g r a d e o f m e m b e r s h i p o f t h e p e r s o n i nt h e f u z z y s e t w h i c h r e p r e s e n t s t h e d e n o t a t i o n o r , e q u i v a l e n t l y ,t h e e x t e n s i o n o f t h e p r e d i c a t e . )

T o r e p r e s e n t t h e m e a n i n g o f P I a n d P 2 t h r o u g h t h e u s eo f t e s t - s c o r e s e m a n t i c s , w e a s s u m e t h a t t h e e x p l a n a t o r y d a t a -b a s e c o n s i s t s o f t h e f o l l o w i n g r e l a t i o n s ( g a d e h , 1 9 8 3 b ):E D F A P O P U L A T I O N ( N a m e ; Ag e ; S ex ] +L l K E [ N a m e l ; N a m e 2 ; p ] + Y O U N G ( A g e ; p ] +M O S T ( P r o p o r t i o n ; It] .

In L I K E , i t i s t h e d e g r e e t o w h i c h N a m e l l i k e s N a m e 9 ;a n d i n Y O U N G , i t i s t h e d e g r e e t o w h i c h a p e r s o n w h o s e a g e i sA g e i s y o u n g .

F i r s t , w e s h a l l r e p r e s e n t t h e m e a n i n g o f P I b y t h e f o ll o w -i n g t e s t p r o c e d u r e .1 . Div ide P O P U L A T I O N i n t o t h e p o p u l a t i o n o f m a l e s ,M . P O P U L A T I O N , a n d t h e p o p u l a t i o n o f f e m a l e s ,F . P O P U L A T IO N :

M . P O P U L A T I O N A N . . . A g, P O P U L A T I O N [ S e z -- - M a l e ]F . P O P U L A T O N A N e,,,,a ge P O P U L A T I O N [ S e z - - -F e m a l e ] ,

w h e r e N ~ m c , A o c P O P U L A T I O N d e n o t e s t h e p r o j e c -t i o n o f P O P U L A T I O N o n t h e a t t r i b u t e s N a m e a n dA g e .2 . Fo r each N a m e : , j ~ 1 . .. . L , in F . P O P U L A T I O N ,f i n d th e ag e o f N a m e i :

A i A Age F . P O P U L A T I O N [ N a m e ~ N a m e i ] .3 . F o r e a c h N a m e i , f i n d t h e d e g r e e t o w h i c h N a m e i isy o u n g :a i A ~ Y O U N G [ A g e = A i ] ,

w h e r e a i m a y b e i n t e r p r e t e d a s t h e g r a d e o f

314


4/7

m e m b e r s h i p o f N a m e i i n t h e fu zzy se t , Y W , o fy o u n g w o m e n .4 . Fo r each N a m e i , i = l , . . . , K , in M . P O P U L A T I O N ,f i n d t h e ag e o f N a m e i :

B i A A ge M . P O P U L A T l O N [ N a m e - - - N a m e i ] .5 . Fo r each N a m e i , f i n d t h e d eg ree t o w h i ch N a m e il ikes N a m e i :~ ii ~ - ~ L I K E [ N a m e l = N a m e l ; N a m e 2 = N a m e i ] ,w i t h t h e u n d e r s t a n d i n g t h a t ~ i / m a y b e i n t e r p r e t e da s t h e g r a d e o f m e m b e r s h i p o f N a m e i i n t h e fu zzyset , W L i , o f w o m e n w h o m N a m e , l ikes .6 . Fo r each N a m e / f i n d t h e d eg ree t o w h i ch N a m e ,l ikes N a m e an d N a m e i i s y o u n g :

"Tii A a i A # i i N o t e : A s i n p rev i o u s ex amp l e s , w e emp l o y t h e ag g re -g a t i o n o p e r a t o r r a i n ( A ) t o r e p r e s e n t t h e m e a n i n go f co n j u n c t i o n . In e f fec t , 7 0 i s t h e g rad e o fm e m b e r s h i p o f N a m e i i n t h e i n t e r sec t i o n o f t h efu zzy se t s W L I a n d Y W .

7 . C o m p u t e t h e r e l a t i v e s i g m a - c o u n t o f w o m e n w h o mN a m e i l ik e s a m o n g y o u n g w o m e n :P i A ~ C o u n t t W L i / Y W ) (3.4)E C o u n t ( W L i N Y W )

~ C o u n t( Y W )_ ~ i 7 6

a i

F. a i8 . Co m p u t e t h e t e s t s co re fo r t h e co n s t ra i n t in d u cedb y M O S T :

r = ~ M O S T [ P r o p o r t i o n ---- i] (3.5)Th i s t e s t - sco re w ay b e i n t e rp re t ed a s t h e d eg ree t ow h i ch N a m e i sat i sfies PI, i .e . ,

r i = p P I [ N a m e = N a m e i ]Th e t e s t p ro ced u re d e sc r i b ed ab o v e rep re sen t s t h emea n i n g o f P , . In e f fec t, i t t e s t s t h e co n s t ra i n tex p re ssed b y t h e p ro p o s i t i o n

E C o u n t ( Y W / W L i ) i s M O S Tan d i mp l i e s t h a t t h e n -se t an d t h e b -se t fo r t h eq u an t i f i e r m o s t i n PI a re g i v en b y :

n - s e t = W L i = N . , . , 2 L I K E [ N a m e 1 -- ~ Na m e i ]f l F . P O P U L A T I O Nan d

b - s et = Y W = Y O U N G f l F . P O P U L A T I O N .By co n t ra s t , i n t h e ca se o f P2 , t h e i d en t i t i e s o f t h en -se t an d t h e b -se t a re i n t e rch an g ed , i . e . ,

n - s e t = Y Wan d

b - s e t = W L i ,w h i ch i mp l i e s t h a t t h e co n s t ra i n t w h i ch d e f i n e s P2 isex p re ssed b y

E C o u n t ( Y W [ W L i ) is M O S T .

9 .

10 .

11 .

T h u s , w h e r e a s t h e s c o p e o f t h e q u a n t i fi e r m o s t i n PIis { W L i , Y W } , t h e s c o p e o f m o s t l y in P2 i s{ YW, WL~}.H a v i n g r e p r e s e n te d t h e m e a n i n g o f P 1 a n d P ~ , i tb e c o m e s a s i m p l e m a t t e r t o r e p r e s e n t t h e m e a n i n go f r d , an d rd ~. Tak i n g rd D fo r ex amp l e , w e h av e t oad d t h e fo l l o w i n g s t ep s t o t h e t e s t p ro ced u re w h i chd e f i n e s PrF o r e a c h N a m e i , f i n d t h e d eg ree t o w h i ch N a m e i isy o u n g :

6 i A u Y O U N G [ A g e = B i] ,w h ere / f m ay b e i n t e rp re t ed a s t h e g rad e o fm e m b e r s h i p o f N a m e i i n t h e fu zzy se t , Y M , o fy o u n g m e n .C o m p u t e t h e r e l a t i v e s i g m a - c o u n t o f m e n w h o h a v ep r o p e r t y P * a m o n g y o u n g m e n :

6 &-- ~ C o u n t ( P l / Y M )~ C o u n t ( P i f l Y M )

C o u n t ( Y M )~i r i A $ i

~ i~ iT e s t t h e c o n s t r a i n t i n d u c e d b y M O S T :r = ~ M O S T [ P r o p o r t i o n = - - p ] .Th e t e s t s co re ex p re ssed b y (3 .6 ) rep re sen t s t h eo v e ra l l t e s t s co re fo r t h e d i sp o s i t i o n

d A y o u n g m e n li k e y o u n g w o m e ni f d i s i n t e rp re t ed a s r d 1 . I f d i s i n t e rp re t ed a s rd2 ,w h i ch i s a mo re l i k e l y i n t e rp re t a t i o n , t h e n t h e p ro -c e d u r e i s u n c h a n g e d e x c e p t t h a t r i in (3 .5) should herep l aced b y

r = ~ M O S T [ P r o p o r t i o n -~ - 6i]where

6, A ~ C o u n t ( Y W / W L , )

4 . R e p r e s e n t a t i o n o f D h s p o s| t l o na l C o m m a n d s a n dC o n c e p t sTh e ap p ro a ch d esc r i b ed i n t h e p reced i n g sec t io n s can b ea p p l ie d n o t o n l y t o t h e r e p r e s e n t a t io n o f t h e m e a n i n g o f di s p o -s i t i o n s an d d i sp o s i t i o n a l p red i ca t e s , b u t , mo re g en e ra l l y , t ov a r i o u s t y p es o f seman t i c en t i t i e s a s w e l l a s d i sp o s i t i o n a l co n -cep t s .A s an i l l u s t ra t i o n o f i t s ap p l i ca t i o n t o t h e rep re sen t a t i o no f t h e m e a n i n g o f d i s p o s i ti o n a l c o m m a n d s , c o n s i d er

d c A s t a y a w a y f r o m b a l d m e n , (4.1)w h o se ex p l i c i t r ep re sen t a t i o n w i l l b e a s su med t o b e t h e co m-m an d

c A s t a y a w a y f r o m m o s t b a l d m e n . (4.2)Th e mea n i n g o f c i s d e f i n ed b y i t s co mp l i an ce c r i t e r i o n (g ad eh ,1 9 8 2 ) o r , eq u i v a l en t l y , i t s p ro p o s i t i o n a l co n t en t (Sea r l e , 1 9 7 9 ) ,w h i ch may b e ex p re ssed a se e A s t a y i n g a w a y f r o m m o s t b a l d m e n .

T o r e p r e s e n t t h e m e a n i n g o f c e t h r o u g h t h e u s e o f t e s t -s c o r e s e m a n ti c s , w e s h a l l e m p l o y t h e e x p l a n a t o r y d a t a b a s e

3 1 5


5/7

E D F A R E C O R D [ N a m e ; p B a l d; A c ti o n ]+ M O S T [ P r o p o s i t i o n ; # ] .

T h e r e l a t i o n R E C O R D m a y b e i n t e r p r e t e d a s a d i a r y - -k e p t d u r i n g t h e p e r i o d o f i n t e r e s t -- i n w h i c h N a m e i s t h en a m e o f a m a n ; p B a l d i s t h e d e g r e e t o w h i c h h e i s b a l d ; a n dA c t i on d e s c r i b e s w h e t h e r t h e m a n i n q u e s t io n w a s s t a y e d a w a yf r om ( A c t i o n ~ l ) o r n o t ( A c t i o n = 0 ) .

T h e t e s t p r o c e d u r e w h i c h d e f i n e s t h e m e a n i n g o f dc m a yb e d e s c r i b e d a s f o l lo w s :

1 . F o r e a c h N am e i , i ~ I . .. .. n , f i n d (a ) t h e d e g r e e t ow h i c h N a m e l i s b a l d ; a n d ( b ) t h e a c t i o n t a k e n :# B a l di A , B ~ I d R E C O R D [ N a m e - - . Namei]

A c t i on i A a~ t i onR E C O R O [N am e - -. N ame i ] .2 . C o m p u t e t h e r e l a t iv e s i g m a - c o u n t o f c o m p l i a n c e :

1 [~ i pB al d l A A c t i n i } " ( 4 .3 )= - - #3 . T e s t t h e c o n s t r a i n t i n d u c e d b y M O S T :

r = ~ M O S T [ P r o p o M t i o n = p] (4 .4 )T h e c o m p u t e d t e s t s c o r e e x p r e s s e d b y ( 4 .4 )r e p r e s e n t s t h e d e g r e e o f c o m p l i a n c e w i t h c , w h i l e t h ep r o c e d u r e w h i c h l e a d s t o r r e p r e s e n t s t h e m e a n i n g o fde .

T h e c o n c e p t o f d i s p o s it i o n a l it y a p p l i e s n o t o n l y t o s e m a n -t ic e n t i t i e s s u c h a s p r o p o s i t i o n s , p r e d i c a t e s , c o m m a n d s , e t c . ,b u t , m o r e g e n e r a l l y , t o c o n c e p t s a n d t h e i r d e f i n i t io n s . A s a ni l l u s t r a t i o n , w e s h a l l c o n s i d e r t h e c o n c e p t o f t y p i c a l i t y - - ac o n c e p t w h i c h p l a y s a b a s ic r o l e i n h u m a n r e a s o n i n g , e s p e c i a l l yi n d e f a u l t r e a s o n i n g ' ( R e i t e r , 1 9 83 ) , c o n c e p t f o r m a t i o n ( S m i t ha n d M e d i a , 1 9 8 1) , a n d p a t t e r n r e c o g n i t i o n ( Z a d e h , 1 97 7} .

L e t U b e a u n i v e r s e o f d i s c o u r s e a n d l e t A b e a fu z z y s e ti n A ( e .g . , U A c ar s a n d A ~ s t a t ion w agons ) . T h ed e f i n i t i o n o f a typical e l e m e n t o f A m a y b e e x p r e s s e d in v e r b a lt e r m s a s f o l l o w s :

t is a t ypical e l e m e n t o f A i f a n d o n l y i f ( 4 . 5 )( a ) t h a s a h i g h g r a d e o f m e m b e r s h i p i n A , a n d( b ) m o s t d e m e n t s o f ,4 a r e s i m i la r t o t .

i t s h o u ld b e r e m a r k e d t h a t t h i s d e f i n it i o n s h o u l d b e v i e w e d a sa disposition al definition, t h a t i s , a s a d e f i n i t i o n w h i c h m a yf a i l, i n s o m e c a s e s , t o r e f l e c t o u r i n t u i t i v e p e r c e p t i o n o f t h em e a n i n g o f t y p i c a l it y .

T o p u t t h e v e r b a l d e f i n i t i o n e x p r e s s e d b y ( 4 .5 ) in t o am o r e p r e c i s e f o rm , w e c a n e m p l o y t e s t - s c o r e s e m a n t i c s t or e p r e s e n t t h e m e a n i n g o f ( a ) a n d ( h ). S p e c i f ic a l l y , l e t S b e as i m i l a r i t y r e l a t i o n d e f i n e d o n U w h i c h a s s o c i a t e s w i~ h e a c h e l e -m e n t u i n U t h e d e g r e e t o w h i c h u i s s i m i l a r t o t ~ . F u r t h e r -m o r e , l e t S ( t ) b e t h e Mmilari ty c las~ o f t , i . e . , t h e f u z z y s e t o fe l e m e n t s o f U w h i c h a r e s im i l a r t o t . ~ V h a t t h i s m e a n s i s t h a tt h e g r a d e o f m e m b e r s h i p o f u i n S ( t ) i s eq u a l t o # s ( t , u ) , t h ed e g r e e t o w h i c h u i s s i m i l a r t o t ( Z a d e h , 1 9 7 1) .

L e t H I G H d e n o t e t h e f u z z y s u b s e t o f t h e u n i t i n t e rv a lw h i c h i s t h e e x t e n s i o n o f t h e f u z z y p r e d i c a t e h i g h . T h e n , t h ev e r b a l d e f i n i t i o n ( 4 . 5 ) m a y b e e x p r e s s e d m o r e p r e c i s e l y i n t h ef o r m :

t is a t ypical e l e m e n t o f A i f a n d o n l y i f ( 4 . 6 )3. For consistency with the defini tion of A, S must be such that i f uand u I have a high degree of similari ty, then their grades of member-ship in A should be close in magnitude.

( a ) P a ( t ) i s H I G H( b ) E C o u n t ( S ( t ) / A ) is M O S T .

T h e f u z z y p r e d i c a t e high m a y b e c h a r a c t e r i z e d b y i t sm e m b e r s h i p f u n c t io n PHtCH o r , e q u i v a l e n t l y , a s t h e f u z z y r e i n -t o n I I IGf I [Grade ; P L i n w h i c h Grade i s a n u m b e r i n t h e i n t e r -val [0 ,1 ] and p . i s t h e d e g r e e t o w h i c h t h e v a l u e o f Grade f i t st h e i n t e n d e d m e a n i n g o f high.

A n i m p o r t a n t i m p l i c a t io n o f t h i s d e f in i t io n i s t h a t t y p i -c a l i t y i s a m a t t e r o f d e g r e e . T h u s , i t f o l lo w s a t o n c e f r o m ( 4 . 6 )t h a t t h e d e g r e e , r , to w h i c h t i s t y p i c a l o r , e q u i v a l e n t l y , t h eg r a d e o f m e m b e r s h i p o f t i n th e f u z z y s e t o f t y p i c a l e l e m e n t so f A , i s g i v e n b y

r = t H I G H [ G r a d e = t ] A ( 4 .7 )a M O S T [ P r o p o r t i o n = ~ , C o u n t ( S ( t ) /A ] .

I n te r m s o f t h e m e m b e ~ h i p f u n c t io n s o f H I G H , M O S T , Sa n d A , ( 4. 7} m a y b e w r i t t e n a s[ ~ , P s t t , u ) A P A ( u ) Ir A V.-LF. J ' (4 .8 )w h e r e tHIGH, PMOSr, PS a n d PA a r e t h e m e m b e r s h i p f u n c t i o n so f H I G H , M O S T , S a n d A , r e s p e c t iv e l y , a n d t h e s u m m a t i o nZ u e x t e n d s o v e r t h e e l e m e n t s o f U .

I t i s o f i n t e r e s t t o o b s e r v e t h a t i f p a ( t ) ----- 1 a n d. s ( t ,n ) = ~ a ( u ) , ( 4. 9 )

t h a t i s, t h e g r a d e o f m e m b e r s h i p o f u i n A i s e q u a l t o t h ed e g r e e o f s im i l a r i ty o f u t o t , t h e n t h e d e g r e e o f ty p i c a l i t y o f ti s u n i t y . T h i s i s r e m i n i s c e n t o f d e f i n i t i o n s o f p r o t o t y p i c a l i t y( R o s c h , 1 9 7 8) i n w h i c h t h e g r a d e o f m e m b e r s h i p o f a n o b j e c ti n a c a t e g o r y i s a s s u m e d t o b e i n v e r s e l y r e l a t e d t o i t s " d i s -t a n c e " f r o m t h e p r o t o t y p e .I n a d e f i n it i o n o f p r o t o t y p i c a l i t y w h i c h w e g a v e i n g a d e h( 1 9 8 2) , a p r o t o t y p e i s i n t e r p r e t e d a s a s o - c a l l e d a - s u m m a r y .I n r e l a t i o n t o t h e d e f i n i t i o n o f t y p i c a l i t y e x p r e s s e d b y ( 4 . 5) , w em a y s a y t h a t a p r o t o t y p e i s a a - s u m m a r y o f ty p i c a l e l e m e n t so f A . I n t h is s e n se , a p r o t o t y p e i s not , i n g e n e r a l , a n e l e m e n to f U w h e r e a s a t y p i c a l e l e m e n t o f A i s , b y d e f i n i t i o n , a n c l e -m e n t o f U . A s a s i m p l e i l l u s t r a t i o n o f t h i s d if f e r e n c e , a s s u m et h a t U i s a c o l l e c t i o n o f m o v i e s , a n d A i s t h e f u z z y s e t o fW e s t e r n m o v i e s . A p r o t o t y p e o f A i s a s u m m a r y o f t h e s u m -

m a r i e s { i .e . , p l o t s ) o f W e s t e r n m o v i e s , a n d t h u s i s n o t a m o v i e .A t y p i c a l W e s t e r n m o v i e , o n t h e o t h e r h a n d , i s a m o v i e a n dt h u s i s a n e l e m e n t o f U .5 . F u z z y S y l l o g is m s

A c o n c e p t w h i c h p l a y s a n e s s e n t i a l r o l e i n r e a s o n i n g w i t hd i s p o s i t i o n s i s t h a t o f a fuzzy sy l logism ( Z a d e h , 1 9 8 3 c ) . A s ag e n e r a l i n f e r e n c e s c h e m a , a f u z z y s y l lo g i sm m a y b e e x p r e s s e di n t h e f o r mQIA'a are B in (5 . 1 )Q 2 C I 8 a re D I sf Q s E ' a a re F ~ a

w h e r e Q l a n d Q 2 a r e g i v e n f u z z y q u a n t i f i e r s , Q 3 i s f u z z yq u a n t i f i e r w h i c h i s t o b e d e t e r m i n e d , a n d A , /3 , C , D , E a n d Fa r e i n t e r r e l a t e d f u z z y p r e d i c a t e s .

I n w h a t f o l l o w s , w e s h a l l p r e s e n t a b r i e f d i s c u s s i o n o f t w ob a s i c t y p e s o f f u z z y s y l l o g is m s . A m o r e d e t a i l e d d e s c r i p t i o n o ft h e s e a n d o t h e r f u z z y s y l lo g i s m s m a y b e f o u n d i n Z a d e h( 1 9 8 3 c , 1 9 8 4 ) .T h e intersec t ion~product sy l logism m a y b e v ie w e d a s a ni n s t a n c e o f ( 5 .1 ) i n w h i c h

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6/7

6 ' ~ A and BEA AF A B andD ,

an d Q a = Q 1 ~ Q 2 , i .e - , Q a i s t h e p ro d u c t o f Q I an d Q 2 i nf u z z y a ri t h m e t i c . T h u s , w e h a v e a s th e s t a t e m e n t o f t h e s y ll o -g i sm:Q 1 A ' s a r e B ' s ( 5 . 2 )QT(A and B) ' s arc CI s(Q1 (~ Q2) A Is are ( B a n d C) ls

In part ic u lar , i f B i s conta ine d in A , i .e ., PB --< PA, whe re PAa n d P 8 a r e t h e m e m b e r s h i p f u n c t i o n s o f A and B, re sp ec -t i v e l y , t h en A and B = B , and (5 .2) becomesQ1 A' s are Be s (5.3)Q~ B' s arc C I s( Q I ~ Q2) A's are (B andC )'s .

Si n ce B and C i mp l i e s C , i t fo l l o w s a t o n ce f ro m (5 .3 )t h a tQ1 A I s arc BI s (5.4)Q2 BI s are C ' s>( QI ~ Q2) A' s arc C' s,

which i s the chaining syllogism ex p re ssed b y (1 .4 ). Fu r t h e r -mo re , i f t h e q u an t i f i e r s Q ] an d Q 2 a re mo n o t o n i c , i . e . ,>- QI -- Q1 and _> Q2 = Q2, then (5 .4) becom es the productsyllogismQI A e s are B ' s (5.5)Q~ BI s are CI s(QI ~ Q2) A' s ore C 'st h e ca se o f t h e consequent conju nction syllogism, w enh a v eC ~ _ AE ~ _ AF = B and D .

In t h i s e a se , t h e s t a t emen t o f sy l l o g i sm i s :Q I A ' s a r e B ' s ( 5 . 0 )Q:A fs are CIsQa A e s are ( B and C) Is

w h ere Q i s a fu zzy n u mb e r (o r i n t e rv a l ) d e f in ed b y t h e i n e -q u a l i t i e s0 ~ ( Q 1 Q 2 0 1 ) _ ~ Q _~ Q I ~ ) Q 2 , ( 5.7 )

w h e re (~ , ~ ~ an d @ a re t h e o p e r a t i o n s o f ad d i t i o n , su b t rac -t i o n , ra i n an d max i n fu zzy a r i t h me t i c .A s a s i mp l e i l l u s t ra t i o n , co n s i d e r t h e d i sp o s i t i o n s

dl A students are youngd 2 ~-- students are sin gl e.

U p o n re s t o ra t i o n , t h e se d i sp o s i t i o n s b eco me t h e p ro p o s i t i o n sPl A most students are youngP2 A most students are single

Th en , ap p l y i n g t h e co n seq u en t co n j u n c t i o n sy l l o g i sm t o P l an dP2, we can i n fe r t h a t

Q students are single an d youngw h e r e

2 m o s t 0 1 _ ( o a t ( n o t q ) ) A ' o o r e n o t Bt ,w h i ch red u ces t o

n o t ( q A ' s a re B ' * ) = ( 0 .8 )( a n t ( n o t q ) ) A ' , are not B' *

i f Q i s mo n o t o n i c (e .g ., Q A most).A s an i l l u s t ra t i o n , i f d A birds can fl y a n d Q A most,t h en (0 .8 ) y i e l d s

n o t ( b i r d s c a n f l y ) ( a n t ( n o t m o s t ) ) b ir ds c a n n o t f l y . ( o. g)I t s h o u l d b e o b s e r v e d t h a t i f Q i s a n a p p r o x i m a t i o n t oall, t h e n ant(not Q) i s a n a p p r o x i m a t i o n t o some. F o r t h er i g h t -h an d me mb er o f (0 .9 ) t o b e a d i sp o s i t i o n , most m u s t b e

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a n a p p r o x i m a t i o n t o at least a half. In this case ant [not most]wil l be an approx im at ion to most, and consequen t ly the r igh t -hand m em ber o f (0 .9 ) m ay be expressed - - upon the suppres -sion of most -- as the d ispos i t ion birds cannot fly.R E F E R E N C E S A N D R E L A T E D P U B L I C A T IO N S

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7_A Computational Theory of Dispositions-1984