A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges

1. I n t r o d u c t i o n R o u g h l y speak ing , a fuzzy set is a class w i t h u n s h a r p b o u n d a r i e s , tha t is, a class in w h i c h the t r ans i t i on f rom m e m b e r s h i p t o n o n m e m b e r s h i p is g r a d u a l r a the r t h a n a b r u p t . In this sense , t h e class of ta l l m e n is a fuzzy set , as are t he classes o f beaut i fu l w o m e n , y o u n g m e n , red f lowers , smal l cars , e t c . Fuzz iness p l ays a n essent ia l role in h u m a n c o g n i t i o n because m o s t of t he classes e n c o u n t e r e d in t h e real wor ld a re fuzzy—some o n l y s l ight ly and some m a r k e d l y so . T h e pervasiveness o f fuzziness in h u m a n t h o u g h t p rocesses suggests t ha t m u c h o f t he logic b e h i n d h u m a n r ea son ing is n o t t h e t r ad i t i ona l two-valued o r even mul t i -va lued logic, b u t a logic w i t h fuzzy t r u t h s , fuzzy connec t ives a n d fuzzy rules o f inference . I n d e e d , it m a y be argued t h a t it is t h e abi l i ty o f t he h u m a n b ra in t o m a n i p u l a t e fuzzy c o n c e p t s t h a t d is t inguishes h u m a n inte l l igence from m a c h i n e in te l l igence . A n d ye t , de sp i t e its funda­ m e n t a l i m p o r t a n c e , fuzziness h a s n o t b e e n a c c o r d e d m u c h a t t e n t i o n in t h e scientif ic l i t e r a t u r e , 1 p a r t l y b e c a u s e it is a n t i t h e t i c t o t h e d e e p l y e n t r e n c h e d t r ad i t i ons of scientif ic t h ink ing based o n Ar i s to t e l i an logic a n d o r i en t ed t o w a r d exac t quan t i t a t i ve analysis , and t Requests for reprints should be sent to Prof. Lotfi A. Zadeh, Dept. of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkeley, California 94720. 1 Discussions of vagueness and related questions from a philosophical point of view may be found in [11]-[ IS]. 4 Copyright ©1973 by Scripts Publishing Company. D ow nl oa de d by [ U ni ve rs ity o f C al if or ni a, B er ke le y] a t 1 6: 43 1 9 A ug us t 2 01 3 FUZZYSETTHEORETIC INTERPRETATION OF LINGUISTIC HEDGES 5 par t ly because fuzziness is suscep t ib le o f confus ion w i t h r a n d o m n e s s . In fact , fuzziness a n d r a n d o m n e s s a re d i s t inc t p h e n o m e n a w h i c h requ i re d i f fe ren t m o d e s o f t r e a t m e n t a n d m a t h e m a t i c a l analys is . T h e t h e o r y o f fuzzy s e t s 2 r ep resen t s a n a t t e m p t a t c o n s t r u c t i n g a concep tua l f r a m e w o r k for a s y s t e m a t i c t r e a t m e n t o f fuzziness in b o t h q u a n t i t a t i v e and qua l i t a t i ve w a y s . A bas ic c o n c e p t in th is t h e o r y is t h a t o f a fuzzy subse t A o f a universe o f discourse U , w i t h A charac ter ized b y a m e m b e r s h i p func t ion ¡±A ( x ) wh ich associa tes w i t h each po in t x in U its "g rade o f m e m b e r s h i p " in A . Usual ly , b u t n o t necessar i ly , \iA (x ) is a s sumed to range in t he interval [ 0 , 1 ] , so t h a t t he g r a d e o f m e m b e r s h i p is a n u m b e r b e t w e e n 0 and 1, w i t h 0 a n d 1 co r r e spond ing t o n o n m e m b e r s h i p and full m e m b e r s h i p , r e spec t ive ly . F o r e x a m p l e , if U is the set o f in tegers f rom 0 t o 100 , t h e n t h e g r a d e of m e m b e r s h i p o f a m a n w h o is 2 3 years old in t h e class o f y o u n g m e n migh t b e specified to . b e 0 . 9 . I n genera l , t he g rades of m e m b e r s h i p a re subjec t ive , in t he sense t h a t the i r spec i f ica t ion is a m a t t e r o f de f in i t ion r a the r t h a n objec t ive e x p e r i m e n t a t i o n o r analys is . I t shou ld b e n o t e d t h a t , a l t h o u g h nA ( x ) m a y b e i n t e r p r e t e d as t h e t ru th -va lue of t h e s t a t e m e n t " x be longs t o A , " it is m o r e na tu r a l t o view it s imp ly as a g rade o f m e m b e r s h i p because t h e s t a t e m e n t " x belongs t o A " is n o t meaningfu l w h e n A is a fuzzy se t . T h e pervasive fuzziness o f t h e seman t i c s a n d t o a lesser e x t e n t — t h e syn tax o f n a t u r a l languages suggests t h a t s o m e aspec t s of l inguis t ic t h e o r y m i g h t b e a m e n a b l e t o ana lys i s b y t echn iques der ived f rom t h e t h e o r y of fuzzy se t s . A few p r e l i m i n a r y s teps in th i s d i r e c t i o n were t a k e n in [ 5 ] , [ 6 ] , and [ 7 ] w i t h t he a im of c o n s t r u c t i n g a f r a m e w o r k for a q u a n t i t a t i v e a p p r o a c h t o fuzzy s e m a n t i c s a n d fuzzy s y n t a x . In w h a t fo l lows, t h e focus of a t t e n t i o n will be o n a m o r e specif ic a p p l i c a t i o n , n a m e l y , t he c o n s t r u c t i o n o f a fuzzy-se t theore t ic i n t e r p r e t a t i o n o f hedges , e.g., " v e r y , " " s o m e w h a t , " " q u i t e , " " m u c h , " " m o r e o r l e s s , " " s o r t of," " e s s e n t i a l l y , " e t c . , as o p e r a t o r s ac t ing o n fuzzy subse t s of t h e universe of d i scourse . O u r m a i n c o n c e r n , h o w e v e r , will b e w i t h t he bas ic aspec t s o f th is i n t e r p r e t a t i o n r a t h e r t h a n wi th de ta i l ed ana lyses o f p a r t i c u l a r hedges . T h e poss ib i l i ty of def in ing hedges as o p e r a t o r s ac t ing o n fuzzy se t s provides a basis for a b e t t e r u n d e r s t a n d i n g o f t he i r ro le in n a t u r a l languages . M o r e i m p o r t a n t , it suggests a w a y of cons t ruc t ing a sys tem o f b o t h na tu r a l a n d artificial hedges w h i c h cou ld b e used t o devise a lgor i thmic languages for t h e desc r ip t ion o f t h e b e h a v i o r of c o m p l e x sys t ems . S u c h languages m i g h t find s ignif icant app l i ca t ions in p s y c h o l o g y , soc io logy , pol i t ical sc ience , phys io logy , e c o n o m i c s , m a n a g e m e n t sc ience , i n f o r m a t i o n re t r ieval a n d o t h e r fields in w h i c h sys tem behav io r is f r e q u e n t l y t o o c o m p l e x o r t o o ill-defined t o a d m i t of analys is in conven t iona l m a t h e m a t i c a l t e r m s . 2 . F u z z y Se t s a n d Languages—Nota t ion a n d T e r m i n o l o g y Let U b e a universe o f d i s cou r se , i.e., a co l lec t ion of ob jec t s d e n o t e d generical ly b y y . F o r ou r p u r p o s e s , i t will b e c o n v e n i e n t t o regard a l anguage , L , as a c o r r e s p o n d e n c e b e t w e e n a set of t e r m s , T , and t h e un iverse of d i scourse , U . 3 T h i s c o r r e s p o n d e n c e will b e a s sumed t o be def ined b y a naming relation, N , w h i c h associa tes w i t h e a c h t e r m x in T and each ob jec t y in U t h e degree , nN(x, y ) , t o wh ich x appl ies t o y . 4 pN(,x, y ) wil l be assumed t o b e a n u m b e r in t h e interval [ 0 , 1 ] , so t h a t N is, in effect , a fuzzy r e l a t i on f rom T t o U . 5 2 Topics in the theory of fuzzy sets which are relevant to the subject of the present paper are discussed in [3]-[10]. 3 A more detailed discussion may be found in [6] and [7]. For simplicity, we asume that T is a non-fuzzy set. 4 It should be noted that J I ^ ( X , y) may not be defined for all xeT and yeU. For example, the degree to which the term jealous applies to an inanimate object such as chair may be asumed to be undefined rather than zero. SA fuzzy relation, R, from a set X toa set Y isa fuzzy subset of the cartesian product X X Y. (X X Y is the colection (Footnote continued) D ow nl oa de d by [ U ni ve rs ity o f C al if or ni a, B er ke le y] a t 1 6: 43 1 9 A ug us t 2 01 3