Adams, et al., Uncertainties, Premises & Conclusions in Deductive Inferences [32 pgs]

7/28/2019 Adams, et al., Uncertainties, Premises & Conclusions in Deductive Inferences [32 pgs]

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E R N E S T W . A D A M S A N D H O W A R D P . L E V I N E

O N T H E U N C E R T A I N T I E S T R A N S M I T T E D F R O MP R E M I S E S T O C O N C L U S I O N S I N D E D U C T I V E

I N F E R E N C E S

1 . THE PROBLEMI t i s e v i d e n t t h a t w h e r e t h e c o n c l u s i o n o f a d e d u c t i v e i n f e r e n c e d e p e n d s o ni t s p r e m i s e s , t h e c o n c l u s i o n w i l l n o t b e a l o g i c a l c e r t a i n t y u n l e s s t h e p r e m i -se s t h e m s e l v e s a r e l o g i c a ll y c e r t a i n , w h i c h i s ra r e , a n d u n c e r t a i n t i e s i nt h e p r e m i s e s w i l l b e i n s o m e m e a s u r e p a s s e d o n , o r t r a n s m i t t e d , t o t h ec o n c l u s io n . M o r e o v e r , t h e L o t t e r y P a r a d o x 1 s h o w s t h a t t h e r e a r e d e d u c t -i v e ly s o u n d i n f e r en c e s w h o s e p r e m i s e s a r e i n d i v i d u a l l y v e r y h i g h l y p r o b -a b le , w h i l e t h e i r c o n c l u si o n s a r e t o t a l l y im p r o b a b l e . T h e a i m o f th i s p a p e ri s to i n i t i a t e s y s t e m a t i c in q u i r y i n t o t h e c i r c u m s t a n c e s i n w h i c h t h i s s o r t o fp h e n o m e n o n c a n o c c u r , a n d m o r e g e n e r a l ly i n t o th e q u e s t i o n as t o h o wh igh a p r o b a b i l i t y is g u a r a n t e e d i n t h e c o n c l u s i o n o f a d e d u c t i v e ly s o u n di n f e r e n c e , g i v e n p l a u s i b l e b o u n d s o n t h e u n c e r t a i n t i e s o f t h e p r e m i s e s.T h i s q u e s t i o n i s i m p o r t a n t t o a p p l i e d l o g i c a l t h e o r y , b e c a u s e i t i s n o tn o r m a l l y t h e c a s e t h a t p e r s o n s m a k i n g i n f e r e n c e s a r e s a t i s f i e d m e r e l y t ok n o w t h a t t h e i r c o n c l u si o n s a r e e n t a il e d b y p r e m i s es w h i c h t h e y a c ce p t ,b u t t h e y a l s o w a n t s o m e a s s u r a n c e t h a t t h e i r c o n c l u s i o n s a r e p r o b a b l e .T h e f o l l o w i n g s ec t io n w i ll p re s e n t r e s ul ts b e a r i n g o n m a x i m u m c o n c l u s i o nu n c e r t a i n t i e s c o m p a t i b l e w i t h g i v e n p r e m i s e u n c e r t a i n t i e s i n c e r t a i n k i n d so f d e d u c t i v e l y s o u n d i n fe r en c e s, a n d t h e f in a l se c ti o n s m a k e s o m e i n -f o r m a l r e m a r k s o n t h e m e t h o d o l o g i c a l s ig n i fi ca n c e o f p r o b a b i l it y c o n s id e -r a t i o n s i n d e d u c t i v e l o g i c.B e f o r e s ta r t i n g w e s ta t e t w o e l e m e n t a r y t h e o r e m s o f p r o b a b i l i ty t h e o r yw h i c h t h r o w s o m e l i g h t o n o u r p r o b l e m , a n d w h i c h s u g g e s t t h e d i r e c ti o no f t h e i n q u i r y t o f o l l o w . D e f i n e t h e u n c e r t a i n t y o f a p r o p o s i t i o n t o b e t h ep r o b a b i l i t y t h a t i t is f al s e (t h i s u n c e r t a i n t y i s n o t t o b e c o n f u s e d w i t h t h ee n t r o p i c u n c e r t a i n t y m e a s u r e o f I n f o r m a t i o n T h e o r y ) . T h e f ir s t t h e o r e ms t at e s t h a t t h e u n c e r t a i n t y o f t h e c o n c l u s i o n o f a d e d u c t i v e l y s o u n d i n fe r -r e n c e c a n n o t e x c e e d t h e s u m o f t h e u n c e r t a i n t i e s o f t h e p r e m i s e s . 2 H e n c e ,t h e d e p e n d e n c e o f a c o n c l u s io n o n m a n y p r e m i s e s , a s i n t h e L o t t e r yP a r a d o x e x a m p l e , is e s se n t i al i f h i g h p r o b a b i l i t ie s o f t h e i n d i v i d u a l p r e -Synthese 30 (1975) 429--460.A ll Rights ReservedCopyright 1975 by D. Reidel Publishing Company, Dordrecht-Holland


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430 ERNEST W. ADAMS AND HOWARD P. LEVINEr a is e s a r e t o b e c o m p a t i b l e w i t h z e r o p r o b a b i l i t y o f c o n c l u s i o n s . C o n v e r -s e ly , p e r s o n s r e a s o n i n g f r o m o n l y t w o , t h r e e o r f o u r p r e m i s e s , w h i c h a r et y p i c a l i n t e x t b o o k a p p l i c a t i o n s , w i l l n o t b e l i k e l y t o a r r i v e a t h i g h l y i m -p r o b a b l e c o n c l u s i o n s - t h o u g h t h e p r e m i s e u n c e r t a i n t y s u m m i g h t b e' u n a c c e p t a b l y h i g h ' .

T h e s e c o n d t h e o r e m , a p a r t i a l c o n v e r s e o f t h e f i rs t , s a y s t h a t i f a l l o f t h ep r e m i s e s o f a d e d u c t i v e i n f e re n c e a r e essential t o t h e i n f e r e n c e , t h e n f o rg i v e n p r e m i s e u n c e r t a i n t i e s i t i s l o g ic a l l y p o s s i b l e e i t h e r t h a t t h e c o n c l u -s i o n u n c e r t a i n t y s h o u l d equal t h e s u m o f t h e p r e m i s e u n c e r t a i n t ie s , o r , i ft h a t s u m is g r e at e r t h a n 1 , t h a t t h e p r o b a b i l i t y o f t h e c o n c l u s i o n s h o u l d b ez e r o . A c o n s e q u e n c e o f t h i s t h e o r e m i s t h a t i f p e r s o n s a r e t o g u a r a n t e eb e t t e r t h a n t o t a l i m p r o b a b i l i t y i n th e c o n c l u s i o n s t h e y d r a w f r o m p r e m i -se s o f a priori u n c e r t a i n t y e , t h e y m u s t e i t h e r r e s t r i c t t h e m s e l v e s t oi n f e r e n c e s w i t h 1 / ~ o r f e w e r p r e m i s e s o r e l s e i n t r o d u c e s o m e redundancyi n t o t h e i r r e a s o n i n g , s o t h a t t h e i r c o n c l u s i o n s d o n o t d e p e n d o n a l l o ft h e i r p r e m i s e s . I n w h a t f o l l o w s w e w i l l b e l a r g e l y c o n c e r n e d w i t h t h e l a t t e rp o s s ib i li ty , s in c e it a p p e a r s t h a t a c e r t a i n a m o u n t o f r e d u n d a n c y i s c h a r ac -t e ri s ti c o f r e a l l if e re a s o n i n g f r o m m a n y p r e m i s e s , w h e r e u n c e r t a in t i e sc a n n o t b e n e g l ec t ed .C o n s i d e r t h e f o l l o w i n g e x a m p l e . A t e l e p h o n e s u r v e y is m a d e o f t h ep o l i t i c a l p a r t y a f f il i a ti o n s o f 1 ,0 00 p e o p l e , w h i c h r e s u l t s i n t h e c o l l e c t i o no f ' d a t a ' o f t h e f o r m ' p e r s o n 1 is a D e m o c r a t ' , ' p e r s o n 2 is a R e p u b l i c a n ' ,a n d s o o n u p t o p e r s o n 1 ,0 00 . 62 9 o f t h e p e r s o n s i n t e r v i e w e d r e p o r t t h e m -s e lv e s t o b e D e m o c r a t s : i .e . , e x a c t l y 6 29 ' d a t a i t e m s ' a r e o f t h e f o r m ' p e r -s o n i i s a D e m o c r a t ' . I t i s i n t u i t i v e l y e v i d e n t , h o w e v e r , t h a t g i v e n th e u n -c e r t a i n t i e s t y p i c a l o f d a t a c o l l e c t e d i n t h i s w a y , i t w o u l d b e u n s a f e t oc o n c l u d e ' e x a c tl y 6 2 9 o f t h e p e r s o n s i n te r v ie w e d a r e D e m o c r a t s ' , e v e nt h o u g h t h i s w o u l d b e a d e d u c t i v e c o n s e q u e n c e o f t h e d a t a c o ll e ct e d, e a c hi t e m o f w h i c h w o u l d b e s u f f ic i e n tl y p r o b a b l e t o b e a c c e p t e d b y i t s e lf . O nt h e o t h e r h a n d , i t w o u l d s e e m primafac ie r e a s o n a b l e t o c o n c l u d e ' a t l e a s t6 0 0 o f th e p e r s o n s i n te r v i ew e d a r e D e m o c r a t s ' . T h i s c o n c l u s i o n w o u l d n o td e p e n d d e d u c t i v e ly o n a ll o f t h e p r em i s e s , o r e v e n o n a n y o n e o f t h e m , a n db e c a u se o f th i s 'r e d u n d a n c y ' i n th e d a t a , o u r s e c o n d t h e o r e m d o e s n o ta p p l y . W e w i ll s e e i n t h e n e x t s e c ti o n t h a t i n f a c t theprimafacie r e a s o n a b l e -n e s s o f t h is a n d s i m i la r c o n c l u si o n s f r o m r e d u n d a n t p r e m i s e s i s p a rt i a ll yj u s t i f i e d t h e o r e t i c a l ly .

T h e p r o b l e m o f c a l c u l a t i n g t h e p r o b a b i l i t y t h a t a t l e a s t 6 0 0 o u t o f


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ON UNCERTA INTIES IN DEDUCTI VE INFERENCES 4311 ,0 00 p eo p l e a r e D em o cr a t s , w i t h ' d a t a ' l ik e t h a t j u s t d e s c r i b ed w i t h g i v enind iv idua l d a tum ' e r ro r p rob ab i l i t ies ' , l oo ks a t f i r s t s i gh t l ike a v ery e le -m en t a r y o n e o f p r o b ab i l it y t h eo r y . H o w ev e r , o u r a p p r o a ch t o i t w ill d if fe rf rom the usua l approach in s t a t i s t i cs , i n t ha t we wi l l no t assume a pr io r it h a t e r r o r s a r e i n d ep en d en t ( e.g ., t h a t t h e ch an ce t h a t b o t h o f t w o r ep o r t sare i ncor rec t i s equa l t o t he p ro du ct o f the i r i nd iv idua l uncer t a in ti es ) .I n d ep en d en ce a s s u m p t i o n s a r e p lau s i b l y r eg a r d ed a s inductive i n charac t e ran d w e w an t t o a s ce r t a i n h o w h i g h a co n c l u s i o n u n ce r t a in t y i s logicallycom pat ib l e wi th g iven p rem ise uncer t a in t i es (we wi l l assum e in f ac t t ha tan y p r o b ab i l i t y fu n c t i o n s a ti sf y in g th e K o l m o g o r o v A x i o ms i s a l o g i cal lyp o s s i b l e o n e ). C o m p ar i n g t h e logical b o u n d s o n co n c l u s io n u n ce r t a in t ie sw i t h t h o s e w h i ch f o l lo w w h en i n d ep en d en ce a s s u mp t io n s a r e i n v o k ed w i lli n f ac t a f f o rd u s s o m e i n d ica t i o n o f t h e d eg r ee t o w h i ch ce r t a in k i n d s o fdeduct ive ly sound in ferences i n r ea l i t y r es t on i nduct ive assumpt ions fo rthe i r j u s t i f i ca t i on . We wi l l even f i nd some which a re deduct ive ly sound ,are t o t a l l y un jus ti f ied w hen log i ca l l y poss ib l e conc lus ion uncer t a in t i es a reco n s i d e red , b u t w h i ch a r e i n d u c ti v e ly ju s t if ied b y i n d ep en d en ce a s s u m p -t ions ! Su ch in ferences a re p l aus ib ly t e rme d deceptively deductive.

O u r f o r mu l a t i o n o f t h e p r o b l em o f d e t e r mi n in g l o g i ca l ly m ax i mu m co n -c lus ion uncer t a in t i es co m pat ib l e wi th g iven p rem ise uncer t a in t ies r edu cesi t t o on e o f ma x imiz ing a l i near func t ion r ep resen t ing t he con c lus ion un-cer t a in ty , sub j ec t t o l i near cons t r a in t s r ep resen t ing a p r i o r i b o u n d s o n t h epremise uncer t a in t i es . Th i s i s s imply a p rob lem of l inear programming.M os t o f the r esu l t s we sha ll s t a t e a re i n f ac t f a i r ly s t r a igh t fo rw ard app l i ca-t io n s o f b a s i c th eo r em s o f t h a t t h e o r y ( p r in c i p a ll y t h e s o - ca l led dualitytheorems), and fo r t h i s r eason we sha l l s t a t e t hem ra ther i n fo rmal ly , andrefer t o t he r e l evan t l it e r a tu re fo r t he p roof s . O ur p r im ary c once rn wi ll bewi th t he usefu lness and s ign i fi cance o f t hese r esu lt s i n app l i ca t i on , and no tw i t h th e d ev e l o p men t o f y e t an o t h e r u n w a n t ed ' s y s tem ' o f n o n - s t an d a r dlogic.

A s ign if ican t limi t a t i on o n the p resen t s t u dy mus t be no t ed . T h i s is tha ti t i s r es t r i c t ed t o i n ferences i nvo lv ing on ly what migh t be ca l l ed ' f ac tua l 'p r o p o s i t i o n s , t o w h i ch p r o b ab i l i t i e s s a t i s f y i n g t h e K o l mo g o r o v A x i o msproper ly app ly . In par t i cu l a r , we sha l l no t cons ider i n ferences i nvo lv ingconditional p r o p o s i t i o n s , w h o s e p r o b ab i l i t i e s a r e p l au s i b l y meas u r ed a scond i t i ona l p robab i l i ti es . 3 I t t u rns ou t t ha t w hen cond i t iona l s a re i n t ro -d u ced i t c an h ap p en t h a t co n c l u s i o n p r o b ab il it ie s a r e b o u n d e d b y p r o d u c t s


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432 E R N E S T W . A D A M S A N D H O W A R D P . L E V I N E

o f premise probabilities r a t h er t h a n b y sums of premise uncertainties, a n dt h e r e f o r e c o n c l u s i o n u n c e rt a i n ti e s c a n s o m e t im e s b e g r e a t ly r e d u c e d i n t h ec o n d i t i o n a l c a se . H o w e v e r , o u r i n v e s t i g a t i o n o f t h i s c a s e i s s t il l a t a ne a r l y s ta g e , a n d w e h o p e t o p r e s e n t r e su l t s o n i t i n a l a t e r p a p e r .

2 . U N C E R T A I N T Y M A X I M AA n inference wil l be a sys tem I = ( (~bi , . . . , ~b ,> , i f> , w he re q~i, . . . , ~bn an d ~Oare s en tences o f an unsp ec i f i ed f i r s t -o rd e r l ang uag e ; q51 . . . , q~, a re thepremises o f / , {~b l, . . ., ~bn} is i ts total premise set ( s u b se t s o f w h i c h a r esimply premise sets), ~ is its conclusion, a n d {qS1,.. ., ~b,, -~O} is i ts refu-tation set. Oc cas ion a l ly we wi l l u se the a l t e rna t iv e n o t a t i on (~ l . . . . , ~bn) (~b)f o r L T h e u s u a l c o n c e p t s o f lo g i ca l c o n s e q u e n c e , c o n t r a d i c t i o n , a n d s o o na r e p r e s u p p o s e d ( w h e n e x p l ic i tl y s p e c if i ed t h e l a n g u a g e m a y h a v e c o n -s i s t e n t a x i o m s w h i c h a r e t r e a t e d a s l o g i c a l t r u t h s ) , a s w e l l a s t h a t o f ap r o b a b i l i t y f u n c t i o n f o r t h e l a n g u a g e . I t i s n o t p r e s u p p o s e d t h a t t h e i n fe r -e n c e s w e d e a l w i t h a re d e d u c t i v e l y s o u n d , o r t h a t t h e i r t o t a l p r e m i s e s e t sa r e c o n s i s t e n t - t h o u g h i n c o n s i s t e n t p r e m i s e s e ts i n t r o d u c e s o m e s u rp r i s -i n g u n c e r t a i n t y p h e n o m e n a w h o s e i n t e r p r e ta t i o n s i n v o lv e p r o b l e m s , a n dw h i c h w i l l n o t b e e n t e r e d i n t o i n d e t a i l.

A r b i t r a r y f i r s t- o r d e r i n f e r e n c e s a r e t r i v i a ll y r e d u c i b l e t o s e n t e n t i a l in f e r -e n c es f o r th e p u r p o s e o f u n c e r t a i n t y m a x i m u m d e t e r m i n a t i o n . T w o i n fe r -ences (~bl , . . . , q~ , ) I ( ~) an d ( f f~ , . . . . q~) I (~0 ') a re equivalent w ith respectto uncertainty maximization i f a n y g i v e n s u b s e t o f t h e f i r s t r e f u t a t i o n s e t{ ~ b l, .. ., ~bn, - ~ } is c o n s i s t e n t i f a n d o n l y i f t h e c o r r e s p o n d i n g s u b s e t o f{~b~, . . . , ~b ', -~k ' } i s cons i s t en t . I t i s e a sy to show th a t two in fe rencesw h i c h a r e e q u i v a l e n t i n t h i s s e n s e h a v e t h e s a m e c o n c l u s i o n u n c e r t a i n t ym a x i m a . O b v i o u s l y a n y f i r s t - o r d e r i n f e r e n c e i s e q u i v a l e n t t o a s e n t e n t i a li n f e r e n c e in t h e d e f i n e d se n se . W e w i l l s h o w n e x t t h a t t h e f o r e g o i n g r e d u c -t i o n c a n b e c a r r ie d c o n s i d e r a b ly f a r th e r .

F i x i n g a t t e n t i o n o n t h e i n f e r e n ce ( (~1, .. ., ~ ) 1(~) , t w o k i n d s o f p r e m i ses e ts w i l l p r o v e i m p o r t a n t w h e r e t h e t o t a l p r e m i s e s e t P = { ~b l, . . , ~b ,} i sc o n s i s t e n t , a n d a t h i r d k i n d m u s t b e c o n s i d e r e d w h e n i t i s i n c o n s i s t e n t .A p r e m i s e s et P ' _ P is sufficient fo r I i f ~ i s a l o g i c a l c o n s e q u e n c e o f P ' ,a n d is essential or I i f ~ i s n o t a l o g i c a l c o n s e q u e n c e of P. .~P' . Suf f i c i en tand e s sen t i a l p remise s e t s a re dua l in ce r t a in re spec t s . Eve ry su f f i c i en tp r e m i s e s e t i n t e r se c t s ( h a s a n o n - e m p t y i n t e r s e c t i o n w i t h ) e v e r y e ss e n t i a l


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O N U N C E R T A I N T I E S I N D E D U C T I V E I N F E R E N C E S 433

o n e , a n d t h e s e t s o f e i th e r k i n d a r e d e f i n a b l e a s j u s t t h o s e w h i c h i n t e r s e c ta l l s e ts o f t h e o t h e r . T h e m i n i m a l s e t s o f e a c h t y p e ( i. e. , t h e s e t s o f t h a tt y p e w h i c h h a v e n o p r o p e r s u b s e ts o f t h a t t y p e ) w i ll p r o v e e s p e ci a ll y i m -p o r t a n t . T h e m i n i m a l s u f fi c ie n t p r e m i s e s e t s f o r I ( m . s . s . f o r I ) w i l l b e s u p -p o s e d t o b e t h e s e ts $ 1 . . . . , S r ( o n o c c a s i o n r m a y b e 0 ), a n d t h e m i n i m a le s s e n t ia l p r e m i s e s e t s (m . e . s. f o r I ) w i ll b e s u p p o s e d t o b e E l , . . . , E s ( s m a yb e 0 ). T h e s iz e o f t h e s m a l l e s t m . e . t o w h i c h a g i v e n p r e m i s e b e l o n g s w i l lb e s i g n i f i c a n t i n t h a t i t g i v es a r o u g h i n d e x o f t h e ' w e i g h t ' o f t h a t p r e m i s es o f a r a s it s u n c e r t a i n t y c o n t r i b u te s t o t h e m a x i m u m u n c e r t a i n t y o f t h ec o n c l u s i o n . To tally inessential, o f irrelevant p r e m i s e s a r e o n e s w h i c hb e l o n g t o n o m . e . s ., a n d t h e i r u n c e r t a in t i e s m a k e n o c o n t r i b u t i o n t o t h ec o n c l u s i o n ' s u n c e r t a i n t y .

C on t in u in g w i th the fo reg o in g in fe rence , (~b l . . . . ~b,) I (~ ,) , it s a ssoc ia t edm i n i m a l s u ff ic ie n t a n d m i n i m a l essential orm s a r e t h e s e n t e n c e s m s ( I ) a n dm e ( / ) d e f i n ed as f o ll o w s . L e t t i n g A S~ b e t h e c o n j u n c t i o n o f t h e p r e m i s e s i nS ~, o r a t a u t o l o g y T i f S j i s e m p t y , m s ( I ) i s t h e d i s j u n c t i o n A S 1 v . .. v A S ,o r e l s e i s a n a r b i t r a r y c o n t r a d i c t i o n F i r t h e r e a r e n o m . s . s , f o r t h e i n fe r e n c e .L e t t i n g V E ~ b e t h e d i s j u n c t i o n o f t h e p r e m i s e s o f E j , o r F i f E i i s e m p t y ,m e ( / ) i s t h e c o n j u n c t i o n V E 1 & . ." & V E ~ , o r is T i f t h e r e a r e n o m . e . s .T h e t w o s e n te n c es m s ( / ) a n d m e ( / ) a r e e a si ly s e e n t o b e l o g i c a ll y e q u i v a -l e n t, b u t m o r e i m p o r t a n t l y f o r o u r p u r p o s e s t h e o r i g i n a l c o n c l u s i o n , ~ ,,c a n b e r e p l a c e d b y e it h e r m s ( / ) o r m e ( / ) , a n d t h e r e s u lt i n g i n f er e n c e w i llb e e q u i v a l e n t t o t h e o r i g i n a l w i t h r e s p e c t t o u n c e r t a i n t y m a x i m i z a t i o n .N o t e t h a t w e h a v e n o w r e d u c ed t h e u n c e r t a in t y m a x i m i z a t io n p r o b l e mf o r a n i n f er e n c e w i t h a n a r b i t r a r y c o n c l u s i o n 0 t o t h a t o f m a x i m i z i n g t h eu n c e r t a i n t y o f a n o t h e r c o n c l u s i o n m s ( / ) o r m e ( / ) w h i c h w il l l o g ic a l lyi m p l y ~k b u t w i ll n o t i n g e n e r a l b e l o g i c a l l y i m p l i e d b y ~ , a n d w h i c h i s o ft h e f o r m o f a d i s j u n c t io n o f c o n j u n c t i o n s o f p re m i s e s o r o f a c o n j u n c t i o no f d i s j u n c t i o n s o f p r e m i s e s . T h e r e d u c t i o n c a n b e c a r r i e d o n e s t e p f a r t h e r .I f t h e p r em i s e s a r e c o n s i s t e n t t h e n e a c h p r e m i s e c a n b e r e p l ac e d b y ad i s t in c t a t o m i c l e t te r w i t h t h e s a m e r e p l a c e m e n t s b e i n g m a d e i n m s ( I ) o rm e ( I ) , a n d t h e n e w i n f e r e n c e w i l l b e e q u i v a l e n t t o t h e o r i g i n a l w i t h r e s p e c tt o u n c e r t a i n t y m a x i m i z a t i o n . T h e s a m e r e d u c t i o n c a n a l s o b e c a r ri e d o u tw h e n t h e p r e m i s e s a r e i n c o n s i s t e n t , e x c e p t t h a t i n t h i s c a s e i t is n e c e s s a r yt o a d d n o n - l o g i c a l a x i o m s t o t h e l a n g u a g e w h i c h s p e c i f y i n e f fe c t t h a t s e tso f a t o m i c f o r m u l a s w h i c h c o r r e s p o n d t o i n c o n s i s t e n t p r e m i s e s e t s a r ei n c o n s i s t e n t . I t is s i g n i fi c a n t t h a t w h e n t h i s r e d u c t i o n is c a r r i e d o u t n e g a -


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43 4 ERNEST W. ADAMS AND HOW ARD P. LEVINEt i o n d i a p p e a r s e n t ir e ly , si n ce p r em i s e s a r e a t o m i c f o r m u l a s a n d c o n c lu -s i o n s a re c o n j u n c t i o n s o f d i s j u n c t i o n s o r d i s j u n c t io n s o f c o n j u n c t i o n s o fp r e m i s e s .

S o m e s p e c i a l c a se s s h o u l d b e n o t e d . I f ~k i s a l o g i c a l c o n s e q u e n c e a tm o s t o f i n c o n s i s te n t p r e m i s e se ts t h e n m s ( / ) a n d m e ( / ) a r e b o t h e q u i v a -l e n t t o a c o n t r a d i c t i o n , F , a n d i n t h i s c as e a n y p r e m i s e p r o b a b i l i t ie s a r ec o m p a t i b l e w i t h O ' s h a v i n g p r o b a b i l i t y 0 . I f t h e c o n c l u s i o n is a l o g i c alt r u t h ( i t i s i n d e p e n d e n t o f th e p r e m i s e s ) , t h e r e a r e n o m . e .s . , t h e o n l y m . s .i s t h e e m p t y s et , a n d b o t h m s ( / ) a n d m e ( / ) a r e e q u i v a le n t t o a l o g ic a lt r u t h . I n t h e c a s e i n w h i c h t h e t o t a l p r e m i s e s e t is s u ff i ci e n t b u t n o p r o p e rs u b s e t o f i t i s , e v e r y i n d i v i d u a l p r e m i s e i s e s s en t i a l ( i. e. , th e s i n g l e t o n s e tc o n t a i n i n g i t is es s en t ia l) , a n d m s ( / ) a n d m e ( / ) a r e b o t h e q u i v a l e n t t o t h ec o n j u n c t i o n o f a l l o f t h e p r e m i s es . T h i s i s t h e s o u n d , c o n s i s te n t , irredun-dan t i n fe r e n ce . I t f o ll o w s e as i ly f r o m t h e f o r e g o i n g t h a t a n y t w o s o u n d ,c o n s i s te n t , i r r e d u n d a n t i n f e re n c e s w i t h t h e s a m e n u m b e r o f p re m i s e s a r ee q u i v a l e n t w i t h t h e r e s p e c t t o u n c e r t a i n t y m a x i m i z a t i o n , n o m a t t e r h o w' s t r o n g ' o r ' w e a k ' t h e i r c o n c l u s i o n s a r e . T h e f o r e g o i n g g e n er a li z e s a b i t :i f a l l o f th e p r e m i s e s o f a n i n f e r e n c e a r e e i t h e r e s s e n t ia l o r i r r e l e v a n t t h e nt h e c o n c l u s i o n c a n b e r e p l a ce d b y t h e c o n j u n c t i o n o f t h e e s s e n ti a l p r e m i s esa n d t h e r e s u l t i n g i n f e r e n c e w i l l b e e q u i v a l e n t t o t h e o r i g i n a l w i t h r e s p e c tt o u n c e r t a i n t y m a x i m i z a t io n .

N o w s u p p o s e t h e t o t a l p r e m i s e s e t P o f a n i n f e r e n c e I i s i n c o n s i s t e n t .A s u b s e t P ' _ P w i ll b e c a ll e d nega tive ly suf fi cien t fo r I i f P N p ' i s cons i s -t e n t a n d s u f f ic i e n t f o r L T h e min ima l neg a t ive ly su f f i c i en t p rem ise s se t sf o r 1 ( m . n . s . s . f o r 1 ) w i l l b e w r i t t e n NS1, . .. , NS t . In the spec ia l c a se inw h i c h t h e c o n c l u s i o n o f I is e n t a i le d a t m o s t b y i n c o n s i s t e n t p r e m i s e s e t st h e r e a r e n o m . n . s . s , f o r I ( t = 0 ) , a n d i n t h e c a se i n w h i c h th e t o t a l p r e m i s es e t is s u f f ic i e n t f o r I a n d c o n s i s t e n t , t h e o n l y m . n . s , f o r ! i s t h e e m p t yp r e m i s e s e t.

Th e m.e . s , an d m .n . s . s , fo r an in fe rence (q51 . . . . qSn) I ( 0 ) a re c lose lyr e l a t e d t o t h e m i n i m a l s u b s e t s o f th e r e f u t a t i o n s e t R = { ~ b l, .. ., qSn, - 0 }w h i c h c a n b e f a l s if i e d i n a n y s t a t e o f a ff a i rs . T h e s e m i n i m a l f a l s if i a b les u b s e ts o f R a r e t h e c o m p l e m e n t s w i t h r e sp e ct s t o R o f t h e m a x i m a l c o n -s i s t e n t s u b s e ts o f R . I f R ' i s s u c h a s u b s e t , i t w i l l b e a n m . e . f o r 1 i f i t d o e sn o t c o n t a i n - ~ , a n d i f R ' d o e s c o n t a i n - ~ t h e n R ' ~ { - ~ } i s a n m . n .s .f o r L I n a n y c a s e t h e m . e .s , a n d m . n . s . s , f o r I r e p r e s e n t ' m i n i m a l f al s if ic a -t i o n s t a t e s ' re l a t iv e t o R , a n d i t p r o v e s u s e f u l t o r e p r e s e n t t h e s e s t a te s i n a


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minimal alsif ication m atrix a s f o ll o w s . T h e r o w s i n t h e m a t r i x c o r r e s p o n dt o t h e m . e .s , a n d t h e m . n . s . s . ( w i t h t h e m . e .s , c o m i n g fi r st ) , a n d t h e c o l -u m n s c o r r e s p o n d t o t h e p r em i s es o f / , w i t h a f in a l c o l u m n f o r th e c o n -c l u s i o n . ' l ' s a n d ' O 's a r e n o w w r i t t e n i n t o t h e c e l ls o f t h e m a t r i x a c c o r d i n gt o t h e r u l e s : ( 1 ) w h e r e t h e c e l l i s t h e i n t e r s e c t i o n o f a p r e m i s e s e t r o w a n dp r e m i s e c o l u m n , a ' 1 ' is w r i t t e n i n i f t h e p r e m i s e b e l o n g s t o t h e s e t , a n d a' 0 ' i s w r i t t e n i n o t h e r w i s e , ( 2 ) ' l ' s a r e w r i t t e n in t o t h e c o n c l u s i o n c o l u m ni n t h e m . e . r o w s , ( 3 ) 'O 's a r e w r i t t e n i n t o t h e c o n c l u s i o n c o l u m n i n m . n . s .r o w s . A s i m p le i l l u s tr a t i o n is t h e i n f er e n c e o f t h e c o n c l u s i o n ' A ~ - B 'f r o m t h e t h r e e p r e m i s e s ' A ' , ' B ' , a n d ' - ( A & B ) ' ( t h e p r e m i s e s a re b o t hr e d u n d a n t a n d i n c o ns i s te n t ). I n t h i s e x a m p l e t h e r e a r e t w o m i n i m a l s u f fi -c i e n t p r e m i s e s e t s , $ 1 - - { A , - ( A & B ) } a n d $ 2 = { B , - (A & B ) } , t w o m i n i-m a l e s s en t ia l p r e m i s e s et s E l = { A , B } a n d E z - - { - ( A & B ) } , a n d t w onega t ive ly su f f i c i en t p remise se t s , NSx = {A}, and N S z = {B}. The re fo ret h e m i n i m a l f a l s i fi c a ti o n m a t r i x i s :

p r e m i s e s c o n c l u s i o nA B -(A&B) ( A~ -B )

E ~ = { A , B } 1 1 0 1E z = { - ( A & B ) } 0 0 1 1NS I= {A} 1 0 0 0N S 2= { B } 0 1 0 0

I t i s al s o c o n v e n i e n t t o r e g a r d t h e e n t r ie s i n t h e r o w s o f t h e m i n i m a lf a l s i f ic a t i o n m a t r i x a s t h e v a l u e s o f minim al falsification functions cor re -s p o n d i n g t o t h e s e r o w s , t h e f u n c t i o n c o r r e s p o n d i n g t o t h e r o w o f a nm . e . , E j , b e i n g s y m b o l i z e d ej , a n d t h e f u n c t i o n c o r r e s p o n d i n g t o t h e r o wo f t h e m . n .s . NS2 b e i n g w r i t te n nsj. T h e s e f u n c t io n s , m a p p i n g t h e p r e m -i se s o f t h e i n f e r e n c e p l u s i ts c o n c l u s i o n i n t o { 0,1 }, a r e i m p o r t a n t b e -c a u s e t h e y a r e i n f a c t re s t ri c ti o n s o f ' e x t r e m e ' u n c e r t a i n t y f u n c t i o n s j u s tt o t h i s s e t o f s e n te n c es . U n c e r t a i n t y f u n c t i o n s a r e d e f i n e d s u c h t h a tu ( q ) = 1 - p ( q ) f o r s o m e p r o b a b i li ty f u n c t i o n p a n d f o r a ll se n te n ce s e o ft h e l a n g u a g e . I t i s t r i v i a l t h a t t h e o n l y u n c e r t a i n t y f u n c t i o n s w h i c h w en e e d t o c o n s i d e r f o r o u r s e n t e n t i a l l a n g u a g e s a r e g e n e r a t e d a s c o n v e xc o m b i n a t i o n s o f falsity functions f o r t h e l a n g u a g e , w h i c h a r e f u n c t i o n s fs u c h t h a t f o r s o m e m o d e l , f q ) = 1 f o r a l l s en t e n ce s ~ / w h ic h ar e f a ls e in t h em o d e l , a n d f ( q ) = 0 f o r a ll se n te n ce s w h i c h ar e t ru e i n th e m o d e l . I n m a x i -


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436 ERNEST W. ADAMS AND HOWARD P . LEVINEm i z i n g c o n c l u s i o n u n c e r t a i n t i e s i t i s s u ff i ci e n t t o c o n s i d e r c o n v e x c o m b i -n a t i o n s o f f u n c t io n s w h i c h f a ls i f y m i n i m u m s u b s et s o f t h e r e f u t a t i o n s e tf o r t h e i n f e r e n c e , t h e r e s t r i ct i o n s o f w h i c h t o t h e p r e m i s e s a n d c o n c l u s i o no f t h e i n f e r e n c e a r e j u s t t h e m i n i m a l f a l s i f i c a t io n f u n c t i o n s e l . . . . e , a n dn s l , . . . , n s t . A c c o r d i n g l y w e c a n r e s tr i ct a t t e n t i o n t o u n c e r t a i n t y f u n c t i o n sw h i c h c a n b e e x p r e s s ed i n t h e f o r m :

t(1 ) u ( r / ) = ~ p j e j ( t l ) + ~ q j n s j ( t l ) ,.i=1 i=1

w h e r e P l , . . . , P , a n d q l , . . ., q t a r e n o n - n e g a t i v e re a ls s u m m i n g t o 1, a n dr / i s e i th e r a p r e m i s e o r t h e c o n c l u s i o n o f t h e i n f e r e n c e i n v o l v e d .

T h e ' w e i g h t s ' P l , . . . , P s a n d q l , . . . , q t i n E q u a t i o n (1 ) a r e t h o s e a t t a c h i n gt o t h e c o r r e s p o n d i n g m i n i m a l fa l s if i ca t io n f u n c t i o n s e l , . . . , e s a n dn s l . . . . , n s t , a n d i n d i r e ct l y t o t h e s e ts E l , . . . , E ~ a n d N S 1 . . . . . N S t . T h u s , t h ew e i g h t s P l . . . . , p , w i l l b e c a l l e d t h e m . e . w e i g h t s w h i c h t o g e t h e r w i t h t h em . n . s , w e i g h t s q l . . . . , q t , g e n e r a t e t h e f u n c t i o n u d e f i n e d b y E q u a t i o n (1 ).N o t e t h a t t h e u n c e r t a i n t y o f th e c o n c l u s i o n ~k g i v en b y t h e u n c e r t a i n t yf u n c t i o n d e f i n e d in t h i s w a y i s j u s t t h e s u m o f t h e m . e . w e i g h t s , P l , . - ., P , ,a n d t h e u n c e r t a i n t y o f a n y p r e m i s e is e q u a l to t h e s u m o f t h e w e ig h t s o ft h e m . e .s , a n d m . n . s . s , t o w h i c h i t b e l o n g s .

B o u n d s o n t h e u n c e r t ai n t ie s o f t h e p r e m i se s i n t h e i n f e re n c e 1 == < < ~ b l , . . . , ~ b,) , ~ ) w i l l b e r e p r e s e n t e d b y n o n - n e g a t i v e r e a l v e c t o r s~; =


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O N U N C E R T A I N T I E S I N D E D U C T I V E I N F E R E N C E S 437t o b e e q u a l t o t h a t m a x i m u m ( a n d w e d e f i n e I t ( l ; ~ ) t o b e t h e s a m e a sI t ( l : ~ ) f o r p .u . b .v . s , w i t h u n i f o r m c o m p o n e n t s e). F o r f i x e d / , # ( I ; e ) a n dI t ( l ; 5 ) a r e c l e a r l y r e a l v a l u e d f u n c t i o n s , w h i c h w i ll b e c a l l e d t h e conclu-s io n u n c e r ta i n t y m a x i m i z a t i o n f u n c t i o n ( c . u . m . f . ) a n d t h e uniform conclu-s i o n u n c e r t a i n ty m a x i m i z a t i o n f u n c t i o n (u . c .u .m. f . ) , r e spec t ive ly , fo r / ,w h i c h a r e d e f i n e d f o r a l l c o n s i s t e n t p . u . b . v .s . T h e s e a r e t h e f u n c t i o n sw h o s e p r o p e r t i e s w i l l c o n c e r n u s i n w h a t f o l lo w s . O b v i o u s l y b o t h f u n c -t i o n s a r e m o n o t o n e i n c re a s in g ( t h o u g h n o t s t ri c tl y m o n o t o n e i n c r e a s-i n g ) i n a l l o f t h e i r a r g u m e n t s o v e r t h e i r d o m a i n s o f d e f i n it io n , a n d i nt h e c a s e in w h i c h t h e p r e m i s e s a r e c o n s i s t e n t a n d e n t a i l a n o n - l o g i c a l l yt r u e c o n c l u s i o n , p ( I ; 0 ) = 0 a n d I t ( I ; 1 ) = 1 ( l o g i c a ll y i m p l i e d c o n s e q u e n c e so f p e r f e c t l y c e r t a i n p r e m i s e s a r e p e r f e c t l y c e r ta i n a n d c o n s e q u e n c e s de-p e n d i n g o n ' p e r f e c t l y u n c e r t a i n ' p r e m i s e s a r e p e r f e c t l y u n c e r t a i n ) . T o i n -v e s t i g a te t h e s e f u n c t i o n s i n d e t a i l, i t p r o v e s h e l p f u l t o u t i li z e r e s u l t sf r o m t h e t h e o r y o f L i n e a r P r o g r a m m i n g ( se e es p ec ia ll y G o l d m a n a n dTucke r , 1956) .

R e s t r i c ti n g o u r s el v es t o u n c e r t a i n t y f u n c t i o n s o f th e f o r m (1 ), o u rp r o b l e m b e c o m e s t h a t o f m a x i m i z i n g th e c o n c l u si o n u n c e r t ai n t y

( 2 ) u ( O ) = + ' " + p ,( i f s = 0 , u ( ~ ) m u s t e q u a l 0 ) s u b j e c t to t h e ' p r i m a r y c o n s t r a i n t s ' :

t(3 ) u (~ , ) = ~ pj ej (?p,) + E qj ns j ((~,)


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438 E R N E S T W . A D A M S A N D H O W A R D P , L E V I N EObserve tha t the 'weights', wl ,.. ., w, in the dual weighting system(w, v>=((wl, ..., w,>, v> entering into (4), (5.1), and (5.2) above corre-

spond to the individual premises ~bl, .., q~,, and for this reason we willcall each w~ the ith premise weight of the dual weighting system. Forreasons which will become apparent later, it is appropriate to call theparameter v the consistency index of the dual weighting system. If thesystem (w, v) satisfies the dual constraints (5.1) and (5.2) we will say thatit is consistent with them. These inequalities can be restated in words asfollows: the necessary and sufficient condition for (w, v> to be consistentwith the dual constraints is that the sum o f the weights o f the prem ises ineach minim al essential set plus th e consistency index m ust be at least 1, andthe sum o f the weights o f the prem ises in each m inimal negat ively suff ic ientset plus the consistency index m ust be at least O. Whether or not the dualweighting system is consistent with the dual constraints, it generates alinear functional w" e+v according to (4) over the space o f p.u.b.v.s, e,which will be called the conclusion uncertainty bound fun ctio n (c.u.b.f.)g en er a ted b y (w , v ) .

The essential facts about the maximization and dual minimizationproblems and their interconnections are as follows. The dual system ofconstraints (5) is always consistent, since the dual weighting system(( 0, .. ., 0>, 1> is always consistent with it, but for a fixed p.u,b.v, e therewill be a min imu m of form (4) if and only if the primary constraints (3)are consistent. I f the uncertainty function u is consistent with the primaryconstraints, and the dual weighting system (w, v> is consistent with thedual constraints, then

(6) u()


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O N U N C E R T A I N T I E S I N D E D U C T I V E I N F E R E N C E S 43 9

T o c o n c l u d e o u r s u m m a r y o f t h e r e le v a n t li n e ar p r o g r a m m i n g r es u lt s,i t is t o b e n o t e d t h a t t h e c l a ss o f a l l d u a l w e i g h t in g s y s t e m s ( w , v ) c o n -s i s te n t w i t h t h e d u a l c o n s t r a i n t s ( 5 ) f o r m s a c o n v e x s e t in n + 1 d i m e n -s i o n a l v e c t o r s p a ce , a n d t h is c o n v e x s e t h a s o n l y a f in i te n u m b e r o f e x -t r e m e p o i n t s ( c o n s i s t e n t d u a l w e i g h t i n g s y s t e m s w h i c h a r e n o t c o n v e xc o m b i n a t i o n s o f o t h e r c o n s i s te n t d u a l w e i g h t i n g sy s te m s ) , w h i c h w e m a yd e s i g n a t e ( w 1 , v l ) , . . . , ( w m , v m ). T h e s e w i ll b e c a l l e d t h e charac t e r i s t i cd u a l w e i g h t i n g s y st e m s f o r t h e i n f e r e n c e / , s in c e i f t h e r e e x is ts a m i n i m u mv a l u e o f w " ~ + v f o r a r b i t r a r y ( w , v ) c o n s i s t e n t w i t h t h e d u a l c o n s t r a in t s ,t hi s v a lu e m u s t b e a s s u m e d a t o n e o f th e e x t r e m e p o i n ts , a n d t h e r e f o r e w ec a n w r i te :

( 7 ) / ~ (I ; ~ ) = m i n ( w 1 . e + v1 . . . , w~ " e + v '~) .T h e v a l u e s wk'F. -~-Vk f o r k = l , . . . , m , t h e r e f o r e d e t e r m i n e p ( 1 ; ~ ) i n t h er a n g e o f i ts d e f i n i ti o n . I n c e r t a i n c a s e s t h e s e v a l u e s a ls o g i v e i n f o r m a t i o na b o u t t h i s r a n g e , s in c e th e f a c t t h a t w k . e + vk i s a c t ua l l y n e g a t i v e m e a n st h a t e is n o t a c o n s i s t e n t p . u . b . v , f o r L H o w e v e r , t h e f a c t t h a t a ll o f t h ev a l u e s w k . e + v k, k = 1 .. . , m , a r e n o n - n e g a t i v e i s n o t a l w a y s a s u f f i c i e n tc o n d i t i o n f o r e t o b e c o n s i s t e n t f o r L

M o r e t e r m i n o l o g y . I f ( w k, v k ) , k = 1 , . . ., m , a r e t h e c h a r a c t e r is t i c d u a lw e i g h ti n g s y st em s f o r a n i n f e r e n c e / , t h e n t h e f u n c t io n s w k ' e + vk w h i c ha r e d e f i n e d o v e r t h e p . u . b . v , s p a c e f o r t h e i n f e re n c e w il l b e c a l l e d t h ec h a r a c t er i st ic f u n c t i o n s f o r t h e i n f e re n c e . F o r a p a r t i c u l a r e , t h e c h a r a c t e r -i st ic f u n c t i o n o r f u n c t i o n s w h i c h m i n i m i z e wk. e + vk w i l l be ca l l ed t h eapp l i cab le f u n c t i o n s f o r e , a n d t h e a s s o c i a te d w e i g h t i n g s y s t e m o r s y s te m s(w k , vk ) w i ll b e t h e a p p l i c a b l e w e i g h t i n g s y s t e m o f s y s t e m s f o r ~ . T h ec la s s o f al l e f o r w h i c h a p a r t i c u l a r c h a r a c t e r i s t i c f u n c t i o n i s a p p l i c a b l e i sa l w a y s a c o n v e x ( p o s s i b l y e m p t y ) s e t o f p . u . b . v . s , w h i c h w i l l b e c a l l e d t h ea p p l ic a b l e d o m a i n o f t h e f u n c t i o n ( a n d o f it s a s s o c ia t e d w e i g h t in g s y s t e m ) ,a n d t h e r e s t r i c ti o n o f t h e f u n c t i o n t o i ts a p p l i c a b l e d o m a i n w i ll b e c a ll e dt h e a p p l i c a b l e p a r t o f t h e f u n c t i o n .

S o m e g e n e r a l p r o p e r t i e s o f # ( I ; ~ ) f o l l o w i m m e d i a t e l y . T h i s f u n c t i o nm u s t b e c o n t i n u o u s a n d p i e ce - w i se l i n e a r o v e r it s d o m a i n o f d e fi n it io n .P i e c e- w i s e l i n e a ri t y f o l lo w s f r o m t h e f a c t t h a t # ( I ; e ) is t h e f in i te u n i o n o ft h e a p p l i c a b le p a r t s o f th e c h a r a c t e r i s ti c f u n c t i o n s f o r / , r e s t r ic t e d to t h es e t o f c o n s i s t e n t p . u . b . v . s . , w h e r e e a c h c h a r a c t e r i s t i c f u n c t i o n i s it s e l f


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440 ERNEST W. ADAMS AND HOWARD P . LEVINEl in ea r. N o w co n s i d e r a f i xed p .u .b .v . , e = ( e 1 . . . . e , ) an d a p a r t i cu l a rprem ise q~i. Th e a p p l i c a b l e we igh t o r weigh t s o f qSi fo r ~ wil l be j us t t hei t h c o m p o n e n t s wk o f t he app l i cab l e d ua l we igh t ing sys t em o r sys t emsfo r e . These weigh t s hav e t he fo l l owing s ign if icance . I f the unce r t a in tybo un d e~ i s i n c r e a s e d b y a s ma l l amo u n t 6 , a l l o t h e r u n ce r t a i n t y b o u n d sr ema i n i n g t h e s ame , t h en t h e co n c l u s io n u n ce r t a i n t y ma x i m u m / ~ ( I ; 5 )

k is th e s m a l l e s t o f t h e ap p l icab l eill i n cr ea s e b y t h e am o u n t w ~ 6 , w h er e w ii t h p rem ise weigh t s fo r e . I f the un cer t a in ty bo un d e i is d e c r e a s e d b y as ma l l am o u n t 6 t h en , p r o v i d ed t h e r e s u lt in g u n ce r t a i n t y b o u n d s a r e co n -s is ten t , th e m ax i m u m co n c l u s i o n u n ce r ta i n t y w i ll d ec r ea s e b y t h e am o u n tw ~ 6 , where now w~ i s t he l a r g e s t of t he app l i cab l e i t h p remise w eigh ts .Thus , t he app l i cab l e i t h p remise weigh t s a f fo rd a ' l oca l l y app l i cab l eindex ' o f the imp or t anc e o f qS~ so f a r as i ts unc er t a in ty i n f luences t hem ax i m u m u n ce r t a i n t y o f t h e co n c l u s io n . Th a t / ~ ( I ; ~ ) i n cr ea se s i n p r o -p o r t i o n t o s ma l l e st ap p l icab l e p remi s e w e i g h ts b u t d ec r ea s e s i n p r o p o r -t i on t o l a rges t app l i cab l e p remise weigh t s means , rough ly , t ha t t he moreunce r t a in a p rem ise i s, t he l ess wi l l chang es i n i ts unce r t a in ty a f f ec t t hem ax i m u m u n ce r t a i n t y o f t h e co n c l u s io n . M o r e r o u g h l y s till, t h e m o r ep r o b a b l e p r emi s e s w i ll b e t h e o n es w h o s e u n ce r t a i n ti e s m o s t i mp o r t an t lya f fec t t h e m ax i m u m co n c l u s i o n u n ce rt a in t y .

A l l o f t h e f o r eg o i n g ap p l i e s m u t a t i s m u t a n d i s t o u n i f o r m p .u .b .v . s .an d t o t h e v a l u es o f t h e u n i f o r m c .u .m . f . /~ ( I ; 5 ). H e r e a l l th a t ma t t e rs a r es u m s of p rem ise weigh t s an d cons i s t ency ind i ces in charac t e r i s t ic dua lweigh t ing sys t ems , s i nce i t fo l l ows t r i v i a l l y f rom (7 ) t ha t i n t he un i fo rmb o u n d c a s e,

(8) # ( I ; 5) = m in(s (w 1) ~ + v l , . . . , s ( w m ) ~ + v m )w h e r e s ( w k ) i s t he s um of t he p rem ise weigh t s in t he charac t e r i s t ic sy s t em( w k , v k ) . Th o s e s y s t ems ( w k , v k ) wh ich min imize (8 ) fo r par t i cu l a r e ma yb e t e r m e d a p p B e a b l e fo r e , and so on . T r iv ia l ly , p ( I ; 5 ) wi l l be c on t inuo us ,p i ece-wi se l i near, a nd increase wi th sm al l es t app l icab l e p rem ise-weigh ts u ms an d d ec r ea s e w i t h l a r g es t ap p l i cab l e p r emi s e w e i g h t s u ms . I n t h ecase i n wh ich t he p rem ises a re cons i s t en t an d en t a i l t he con c lus ion i t wi l lbe seen tha t t he co ns i s t ency index i s e i t her 1 o r 0, and

( 9 ) t 1 (I ; 5 ) = m i n ( s ( w k ) 5 , 1)w h e r e ( w k , v k ) i s t he charac t e r i s t i c sys t em wi th cons i s t ency index 0 and


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O N U N C E R T A I N T I E S I N D E D U C T I V E I N F E R E N C E S 4 4 1

s m a l l e st p r em i s e w e i g h t s um . N o t e a l s o t h a t e x c e p t i n t h e c a s e in w h i c ht h e c o n c l u s i o n i s a lo g i c a l t r u t h , a l l p r e m i s e w e i g h t s u m s m u s t b e g r e a t e rt h a n o r e q u a l t o 1 , a n d t h e r e f o r e p ( I ; e ) c a n n o t b e l es s t h a n e u n l e ss t h ec o n c l u s i o n i s a l o g i c a l t r u t h .

M o s t o f th e r e s ul ts t o f o l l o w c o n c e r n t h e c o n s i s te n t a n d s u ff ic ie n tp r e m i s e c a se , a n d t h e s e a r e e a s y g e n e r a l i z a t i o n s f r o m t w o s i m p l e e x a m p l e s .T h e f ir s t i s t h e i n f e re n c e w i t h p r e m i se s ' A ' , ' g ' , a n d ' C ' , a n d c o n c l u s i o n' A & ( B v C ) ' . T h i s c o n c l u s i o n i s a l r e a d y i n m i n i m a l e s s e n ti a l f o r m , t h et w o m i n i m a l e s s e n t i a l p r e m i s e s e t s b e i n g E 1 = { A }, a n d E 2 = {B, C}. As ina l l c o n s i s t e n t s u f f i c ie n t p r e m i s e c a se s , t h e o n l y m i n i m a l n e g a t i v e l y s u f fi -c i e n t p r e m i s e s e t i s t h e e m p t y s e t : i .e . , NS1 = A. T h e m i n i m a l f a l s if i c at i o nm a t r i x t h e r e f o r e i s:

p r e m i s e s c o n c l u s i o nA B C A& (B vC )

E1 = (A } 1 o o 1E2 =(B,C} 0 1 1 1NS1 = A 0 0 0 0

T h e p r i m a r y u n c e r t a in t y m a x i m i z a t i o n p r o b le m is t h a t o f m a x i m i z i n g t h eu n c e r t a i n t y u(A & ( B y C))=p~+p2 s u b j e c t t o t h e p r i m a r y c o n s t r a i n t su(A)=pl


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442 E R N E S T W , A D A M S A N D H O W A R D P . L E V I N El it i es , wh ere th e apriori inequa l i t ies w >f 0 , w2 >i 0 a nd w3 >i 0 a re includ ed) ,a r e :

( w ~ , v j ) = ( ( 0 , O , 0 ) , 1 )(w 2, v2 ) = ( ( 1 , 1 , 0 ) , O )( w s , v~> = ( ( 1 , 0 , 1 > , 0 ) .

In th i s case , therefo re , Eq ua t ion (7) r educes to :# (I ; 81, s2, e3) = ra in (l, el + e2, sl + ~3).

A m o n g th e t h in g s w h ich g en e r a li z e ea si ly f ro m th e ex am p le , t h e f o l l o w -i n g m a y b e n o t e d . T h e total uncertainty weight ing system, ( ( 0 . . . . ,0 ) , 1) ,with a l l p remise weigh ts 0 and cons is tency index 1 , i s a lways among thecharac te r i s t ic weigh t ing sys tems , and the assoc ia ted charac te r i s t ic func-t ion has the c ons tan t v a lue 1 . Fo r eac h min im al su f f ic ien t p rem ise se t, S j ,there i s a co r respond ing unitary charac te r i s t ic sys tem ( w k, 0) , w h e r e t h ekcom pon en t s w~ are O 's o r l ' s and a re equa l to 1 fo r those i such tha t q~ib e lo n g s t o S j . I n t h e ex am p le t h e r e w e r e tw o m in im a l s u ff ic ien t p r em is esets , S~ = (A, B} an d $2 = (A , C} , an d th e co r r e s p o n d in g u n i t a r y ch a r ac -te r i s tic sys tems were (w 2 v2 = ( ( 1 , 1 , 0 ) , 0 ) a nd ( w 3 , v3 ) = ( ( 1 , 0 , 1 , 0 ) .Th e ch a r ac t e ri s t ic f u n c t io n s w h ich co r r e s p o n d to t h e s e m in im a l s u ff ic i en tp r em is e s e ts a r e s im p ly t h e s u m s o f t h e u n ce r t a in ty b o u n d s o f th e p r em is e sin t h e s e t s t o w h ich t h ey co r r e s p o n d . I n t h e p r e s en t c a s e t h e o n ly ch a r ac -t e ri s ti c f u n c t io n s w e r e t h e t o t a l u n ce r t a in ty f u n c t io n an d th e u n i t a r y f u n c -t ions co r resp ond in g to m in imal essen t ia l p remise se ts . In cases o f th i sk in d t h e m ax im u m co n c lu s io n u n ce r t a in ty e it h e r eq u a ls 1 ( t o t a l u n ce r -t a in ty ) o r e l se eq u a l s t h e s u m o f t h e p r em is e u n ce r t a in ty b o u n d s i n t h e' leas t unc er ta in ' ( leas t p remise unc er ta in ty b ou nd sum) o f i t s su f fic ien tp r em is e s e t s . S u ch in f e r en ces ac t a s t h o u g h th e i r co n c lu s io n s d ep en dso le ly on the i r l eas t unce r ta in su f fic ien t p remise set s.

In the consis tent , suff icient , i r redundant case, the only suff icient premis es e t is th e t o t a l p r em is e s e t, an d i n t h is c a s e i t is ev id en t t h a t t h e m ax im u mco n c lu s io n u n ce r t a in ty i s e it h e r 1 o r e l se eq u a l s t h e s u m o f a l l o f t h ep r em is e u n ce r t a in ty b o u n d s , w h ich ev e r i s l e a s t . Th i s co m b in es t h e tw o' t h eo r em s ' o f e l em en ta r y p r o b ab i l i t y s t a t ed i n th e i n t r o d u c t io n .

Th e on ly m in imal nega t ive ly su ff ic ien t se t in the cons is ten t a nd su f fic ien tp rem ise case be ing the e m pty p remise se t , i t fo l lows t r iv ia l ly tha t the co n-


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ON UNCERTAI NTIES IN DEDUCTIV E INFERENCES 443sistency index must always be non-negative, and in fact will equal 0 in allbut the total uncerta inty weighting system. In effect, then, we can elimi-nate the consistency index from our considerations so long as the prem-ises are consistent, and concentrate on the 'reduced' dual constraints whichresult from the original constraints when v is set equal to O. The extremesolutions to the reduced constraints then give the premise weights in allcharacteristic systems but the total uncertainty system, and we get thecomplete set of characteristic weighting systems by simply adding thetotal uncertainty system to the extreme solutions to the reduced constraints.When we come to inconsistent premise sets it will be seen that consistencyindices play a more important role.

The final generalization illustrated in our first example has to do withthe fact that the conclusion of the inference was a conjunction, each ofwhose conjuncts could be looked on as the conclusion of a "sub-inference'.Tha t is, our original inference had the form (A, B, C) I(A & (B v C)), andthe two sub-inferences can be represented as (A, B, C) I(A) and (A, B, C)1(By C). In this particular case, the subinferences are separate in thefollowing sense: each premise is relevant to at most one o f the subinfer-ences, and is irrelevant (totally inessential) to the other. Where an infer-ence with consistent premises and a conjunctive conclusion can be separ-ated in this way, the non-total uncertainty characteristic functions of thecompound inference will always be sums of the non-total uncertaintycharacteristic functions of the sub-inferences. It is trivial tha t the onlynon-total uncertainty characteristic function for I(A) is el (the maximumuncerta inty of its conclusion, 'A', is equal to the that of its first premise),and the only non-total uncertainty characteristic functions for I(B v C)are e2 and %. In virtue of the fact that the sub-inferences are separable, itfollows that the nontotal uncertainty characteristic functions of 1(4 &(B v C)) must be e1 + e2 and e1 + ~a-4

So far we have only encountered premise weights of O and 1, and this isessential because we have arrived at characteristic functions which areconstructible from 'unitary' functions by just two operations: minimiza-tion, which corresponds roughly to disjunction, and addition, whichcorresponds roughly to conjunction. We will now see that there is anotherpossibility: redundant but not irrelevant premises can lend 'statisticalsupport' in limiting the maximum uncertainty of conclusions. As ourexample we will take the inference with premises 'A', 'B', and 'C' , as be-


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444 E R N E S T W . A D A M S A N D H O W A R D P . L E V I N Ef o r e , b u t w i t h t h e c o n c l u s i o n $ = ' ( A & B ) v (A & C ) v ( B & C ) '. T h ec o n c l u s i o n e x p r es s e s t h e f a c t t h a t a t l e a s t t w o o f t h e p r e m i s e s a r e t r u e , a n dt h e r e is o b v i o u s r e d u n d a n c y a m o n g t h e p r e m i se s - i n f a c t n o s i n gl e p r e m -i se i s e s se n t ia l . H e r e t h e r e a r e t h r e e m i n i m a l e s s e n t i a l p r e m i s e s e ts ,E~ = {A, B} , E2 = {A, C} ,m a t r i x i s:

a n d E 3 = {B, C) . T h e m i n i m a l f a l s i f i c a t i o np r e m i s e s c o n c l u s i o nA B C ~O

E, = { A , B } 1 1 0 1E 2 = { A , C } 1 0 1 1E a = { B , C } 0 1 1 1NS1 = A 0 0 0 O.

T h e ' r e d u c e d ' s y s t e m o f d u a l c o n s t r a i n t s ( a r r i v e d a t b y s e t t i n g v = 0 ) i s:w l + w 2 " 1 + w 3 " 0 > t 1w l ' l + w 2 " l + w 3 " 0 t> 1w1 - 0 + w 2 " l + w 3 . 1 > / 1 .

T h e e x t r e m e s o l u t i o n s t o t h e s e i n e q u a l i ti e s a r e :w 1 = , a r e o f a n e w t y p e w h i c h w ew i s h t o c o n s i d e r in d e t a il . O b s e r v e t h a t t h e d o m a i n o f a p p l i c a ti o n o f t h isfu nc t ion i s the s e t o f p .u .b .v . s . < sx , ~2, %> such t ha t e1+82+8a .- .


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s u c h t h a t n o s in g l e w e i g h t e x c ee d s t h e s u m o f t h e o t h e r t w o . T h i s i n c lu d e st h e u n i f o r m b o u n d e a s a s p e c ia l c a s e, a n d i n t h i s c a se th e n e w c h a r a c t e r -i s ti c f u n c t i o n g i v es a m a x i m u m u n c e r t a i n t y b o u n d o f 1 .5 e, w h i c h iss m a l l e r t h a n t h e u n c e r t a i n t y b o u n d s o f 2e w h i c h ar e g i v e n b y t h e u n i t a r yc h a r a c t e r i s t i c f u n c t i o n s . T h e f a c t t h a t t h e m a x i m u m c o n c l u s i o n u n c e r -t a i n t y h e r e d o e s n o t d e p e n d o n t h e p r e m i s e u n c e r t a i n t i e s i n a n y u n i q u em i n i m a l s u f f ic i e n t s e t, b u t r a t h e r i s c o n t r i b u t e d t o b y a l l, s u g g e s ts t h ea p p r o p r i a t e n e s s o f c a ll i n g t h e s e c h a r a c t e r is t i c f u n c t i o n s a n d w e i g h t i n gs y s t e m s s t a t i s t i c a l .

T h e s t a ti s t ic a l p r e m i s e w e i g h t s o f i n t h e e x a m p l e a r e s i g n i f ic a n t i n t h a tt h e y a r e r e c i p r o c a ls o f t h e s iz es o f th e m . e . s, i n t h e e x a m p l e . T h i s g e n e r a l i -ze s a s fo l lows . L e t c~ be the s i ze o f the sm a l l e s t m .e . to w h ich a g ivenp r e m i s e q ~ b e l o n g s , i f i t b e l o n g s t o a n y , a n d l e t w ~ = 1/e~ i f e i is d e f i n e d , a n do t h e r w i s e l e t w ~ = 0 . T h e n t h e w e i g h t s ( w l . . . . , e ,> a r e e a s il y se e n t o s a t i s f yt h e r e d u c e d d u a l c o n s t r a i n t s f o r t h e i n f e r e n c e i n q u e s t i o n , h e n c e b y ( 7 ) .

(10) / t ( I ; e l , 8 , ) < ~1 e ,. . , - - + . . . + - - ,1 Cnw h e r e t h e s u m o n t h e r i g h t is t a k e n o v e r t h o s e t i / c i fo r w hic h ~b~ is re l eva nthen ce c~ i s de f ined . T he re i s a l so a p a r t i a l conve rse o f (10 ) wh ich app l i e si n t h e u n i f o r m b o u n d c a s e. L e t D t , D 2 , . . . b e a r b i t r a r y e s s e n t ia l ( n o t n e c -e s s a r il y m i n i m a l e ss e n t ia l ) p r e m i s e s e ts h a v i n g t h e p r o p e r t y t h a t e v e r yp r e m i s e b e l o n g s t o t h e s a m e n u m b e r o f t h e s e e s s e n ti a l s e ts a s e v e r y o t h e r .L e t t i n g e b e t h e s i z e o f t h e l a r g e s t o f t h e se s e t s, it is n o t h a r d t o s h o w b yc o n s i d e r in g t h e p r i m a r y c o n s t r a i n t s t h a t :

n8(10 cB o t h (1 0 ) a n d (1 1) t h e r e fo r e r e l at e m a x i m u m c o n c l u s i o n u n c e r t ai n t ie sto re c ip roca l s o f s i ze s o f e s s en t i a l p rem ise s e t s .

As a f i r s t app l i ca t ion o f (10 ) and (11 ) con s ide r an in fe rence (~b1 . . . ,q~,) I (~O) in w hic h ~ / is n o t e n t a i l e d b y a n y p r e m i s e s e t w i t h a o r f e w e rm e m b e r s , a n d i s e n t a il e d b y e v e r y p r e m i s e s e t w i t h m o r e t h a n b m e m b e r s .T h e n n o e s s e n t i a l p r e m i s e s e t c a n h a v e l e s s t h a n n - b m e m b e r s , a n d i tf o l l o w s f r o m ( 1 0) t h a t

~ i + "'" + en( l z ) . . . . .n - b


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446 ERNEST W. ADAMS AND HOWARD P. LEVINEE v e r y n - a m e m b e r p r e m i s e s e t i s e s s en t ia l, a n d s i n ce e a c h p r e m i s e be -l o n g s t o t h e s a m e n u m b e r o f n - a m e m b e r p re m i s e s et s, (1 1) a p pl ie s a n dw e g e t :

n8( 1 3 ) - -n - - a

T h e s p e c ia l s t a ti s t ic a l ca s e is t h a t i n w h i c h t h e c o n c l u s i o n i s n o t e n t a i l e db y a n y p r e m i s e s e t w i t h a m e m b e r s , a n d i s e n t a i l e d b y a n y p r e m i s e s e tw i t h m o r e t h a n a m e m b e r s . I n t h i s c as e th e c o n c l u s i o n is e q u i v a le n t to t h ep r o p o s i t i o n ' m o r e t h a n a p r e m i s e s a r e tr u e ' , a n d (1 2) a n d (1 3) c o m b i n e t oi m p l y :

8( 1 4 ) # ( I ; e ) = a "n

T h e c o n c l u s i o n u n c e r t a i n t y m a x i m u m w i ll n e v e r b e le ss t h a n t h e u n i f o r mp r e m i s e b o u n d , b u t t h a t i t m a y s t il l b e ' a c c e p t a b l y s m a l l' i f ~ i s s m a l l a n d1 - a / n i s n o t t o o c l os e t o 0 . F o r e x a m p l e , i f t h e p r e m i s e s a r e ' s u r v e y d a t a 'i t e m s ' p e r s o n 1 i s a D e m o c r a t ' , 'p e r s o n 2 is a D e m o c r a t ' a n d s o o n u p t o1 ,0 00 , e a c h i t e m o f w h i c h h a s a n a p r i o r i u n c e r t a i n t y o f 0 .0 2 , a n d t h e c o n -c l u s i o n is t h a t m o r e t h a n 9 0 0 o f t h e p e r s o n s s u r v e y e d ar e D e m o c r a t s , t h e nt h e m a x i m u m c o n c l u s i o n u n c e r t a in t y w i ll b e 0 . 0 2 / ( 1 - 0 . 9 ) = 0 . 2 . T h i s' s m a l l is h ' b o u n d is o f c o u r s e m u c h h i g h e r t h a n w o u l d b e e x p e c te d in -t u it i v el y , a n d m u c h h i g h e r t h a n t h e b o u n d w h i c h f o l lo w s i f e r r o rs a r ea s s u m e d i n d e p e n d e n t . N o n e t h e l e s s i t s h o w s t h a t n o t a l l o f t h e c o n f i d e n c er e p o s e d i n s u c h s t a ti s ti c a l c o n c l u s io n s d e p e n d s o n t a c i t i n d u c t iv e a s s u m p -t i o n s .

A s l i g h t g e n e r a l i z a t i o n o f th e s i m p l e s t a t i s ti c a l c a s e is t h a t i n w h i c h t h ep remise s ~b1 . . . . q~, c a n b e p u t i n t o t h e f o r m q S,= A ~ f o r i = 1 , . , . , k , a n d~ b , = - A t f o r i - - - k + 1 , . . . , n , a n d t h e c o n c l u s i o n is ' t h e n u m b e r o f A~ 'sw h i c h a r e t r u e l ie s b e t w e e n k l a n d k 2 ( e x c lu s i v e) '. T h i s i s a c a s e i n w h i c ht h e c o n c l u s i o n c a n b e e x p r es s e d a s a c o n j u n c t i o n , ' m o r e t h a n k 1 A f ' s a ret r u e ' a n d ' le s s t h a n k 2 A { s a r e t r u e ' w h e r e t h i s c o n j u n c t i o n s e p a r a t e s t h ep r e m i s e s s o t h a t t h e p o s i t i v e p r e m i s e s A I . . . . , A k a r e r e l e v a n t o n l y t o t h ef i r s t c o n j u n c t w h i l e t h e n e g a t i v e p r e m i s e s - - A k + l . . . . , - - A , a r e r e l e v a n to n l y t o t h e s e c o n d . L e t t i n g / 1 a n d I 2 b e th e s u b - in f e re n c e s w h o s e c o n c l u -s i o n s a r e t h e f i r s t a n d s e c o n d c o n j u n c t s , r e s p e c ti v e ly , 11 a n d I 2 a r e s e p a r a t e -


19/32


l y o f t h e ' s i m p l e ' s t a ti s t ic a l f o r m p r e v i o u s l y d i s c u s s e d ( a f t e r t h e i r ir r el e -v a n t p r e m i s e s h a v e b e e n d e l e t e d ) , a n d E q u a t i o n ( 14 ) i m p l i e s :

a n d

# ( Ix ; ~ ) - _ _ k l1 ~ - - k

n ~ k 2 1 n - kI n v i r t u e o f th e s e p a r a t i o n o f t h e s u b -i n fe r e nc e s , t h e m a x i m u m u n c e r t a i n t yo f t h e c o m p o u n d i n fe r e nc e , # ( I ; e ), m u s t e q u a l t h e s u m o f t h e t w o c o m -p o n e n t c o n c l u si o n u n c e r ta i n t y m a x i m a a b o v e.

A f i n a l s t a t is t i c a l e x a m p l e l e a d s t o a p o s s i b l y s u r p r i s i n g r e s u l t. I n t h i sc a s e t h e p r e m i s e s o f t h e i n f e r e n c e a r e i d e n t i ti e s o f t h e f o r m a ~ = a j f o r a l ll < i < j ~ n ( t h e r e a r e n ( n - 1 ) s u c h p r e m i s e s ) a n d t h e c o n c l u s i o n i sa l = a 2 . . . . . a , ( as . . . . . a , a r e a l l e q u a l ). T h e i n f e r e n c e i s s y m m e t r i c i n t h ep r e m i s e s , a n d i t is s u f f ic i e n t t o c o n s i d e r t h e e s s e n t i a l s e ts t o w h i c h t h ep r e m i s e a l = a 2 b e l o ng s . O n e s u c h is th e n - l m e m b e r s e t D a = {ax = a 2 ,a l = a 3 . . . . , a l = a , } , w h i c h i s e a s i ly s e e n to b e n o t o n l y e s s e n t ia l , b u t t o b et h e s m a l l e s t e s s e n t i al s e t t o w h i c h a a = a2 b e l o n g s . S e t t i n g e a c h c~ = n - 1a n d e a c h e k = e i n (10 ), an d rec a l l ing the r e a re n ( n - 1) p rem ise s in a l l, wet h e n g e t :

/ 2 ( / ; ~)


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448 ERNEST W. ADAMS AND HOWARD P . LEVINEA v a r i a n t o n t h e a b o v e e x a m p l e l e a d s t o s i m i la r s o m e w h a t s u r p r is i n gr e su l ts . H e r e w e m a y t a k e o u r n ( n - 1 ) i te m s o f d a t a t o b e inequalit ies o f

t h e f o r m a~


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O N U N C E R T A I N T I E S I N D E D U C T I V E I N F E R E N C E S 449A m o n g t h e t h i n g s t o n o t e i n t h e e x a m p l e a r e th e f o ll o w i n g . T h e f o u r t hc h a r a c t e r i s t i c f u n c t i o n a n d w e i g h t i n g s y s t e m h a v e n e g a t i v e c o n s i s t e n c y

inde x v~ = - 1 . Th i s is a sys tem in wh ich a l l p rem ise s a re e s s en t i a l O.e., a llh a v e p o s i ti v e w e i gh t ), b u t t h e t o t a l i t y a r e i n c o n s is t e n t. T h e d o m a i n o fa p p l i c ab i l it y o f t h is s y s t e m i s t h e s e t o f p r e m i s e b o u n d s s u c h t h a t b o t h82 + 83 and 82 + 83 do n o t exceed 1 . Th i s inc lud es the un i fo rm bo un d ca sei n w h i c h 8 is b e t w e e n a n d . N o t e t h a t t h e c o n c l u s i o n u n c e r t a i n t y b o u n di n t h is c a s e i s 4 8 - 1 , a n d t h a t t h i s m a y b e s i g n i f ic a n t l y l es s t h a n t h a t g i v e nb y t h e t w o ' c o n s i st e n t ' c h a r ac t e ri s ti c f u n c t i o n s , b o t h o f w h o s e v a l u es a r e2 8. T h e t h i r d p r e m i s e ' - ( A & B ) ' h a s a p p l i c a b l e w e i g h t 2 i n t h e i n c o n s is -t e n t c h a r a c t e r i s t i c f u n c t i o n , a n d t h i s m e a n s t h a t w h e r e t h a t f u n c t i o na p p l i e s , t h e m a x i m u m c o n c l u s i o n u n c e r t a i n t y i s ' d o u b l y d e p e n d e n t ' o nt h e u n c e r t a i n t y o f t h e fi n a l p r e m i s e . I n o n e e x t r e m e c a se , t h a t i n w h i c h t h i sp r e m i s e i s cer ta in , i t i s p o s s i b l e f o r t h e t w o r e m a i n i n g p r e m i s e s t o b eh ig h ly u nc e r t a in wh i l e th e con c lus io n i s c e r t a in : i f 81 = e2 = wh i le 8a = 0 ,t h e n p ( I ) = 0 . T h i s a g a i n is st r an g e , t o s a y t h e le a st , a n d w e s h a ll c o m m e n to n i ts s i g n i f ic a n c e i n t h e f o l l o w i n g s e c t i o n .

A s a l a s t e x a m p l e , c o n s i d e r t h e i n f e r e n c e w h o s e p r e m i s e s a r e n a t o m i cf o r m u l a s A1 . . . . . A , a n d w h o s e c o n c l u s i o n is ' m o r e t h a n a o f t h e p r e m i se sa r e t r u e ' , w h e r e o u r l a n g u a g e w i l l n o w b e a s s u m e d t o c o n t a i n a n o n -l o g ic a l a x i o m e q u i v a l e n t t o ' n o t m o r e t h a n b o f t h e p re m i s e s a re t r u e ' f o rs o m e b > a . I f b i s l es s t h a n n t h e t o t a l p r e m i s e s e t i s i n c o n s i s t e n t . T h i s i n f e r -e n c e is e a s i l y a n a l y z e d a l o n g t h e l i n e s a l r e a d y i n d i c a t e d , a n d w e w i l l o n l ys t a te t h e r e s u lt s c o n c e r n i n g t h e u n i f o r m u n c e r t a i n t y b o u n d c as e. H e r e t h ed o m a i n o f d e f i n i t i o n o f /~ ( I ; 8 ) i s t h e c l a s s o f 8 > /1 - b / n O f b = n a ll b o u n d sa r e c o n s i s te n t ) , t h e p r e m i s e w e i g h t s a re a l l e q u a l t o 1 ~ ( b - a ) , t he cons i s -t e n c y i n d e x i s - ( n - b ) / b - a ), a n d f o r a l l c o n s i s t e n t 5 ,

( b - n ( 1 - e ) ~p ( I ; 8 ) = m i n 1 , b - - a - J "O b s e r v e t h a t t h e c o n s i st e n c y i n d e x , w h i c h d e p e n d s o n a , b , a n d n , c a n h a v ef r a c t i o n a l a n d a r b i t r a r i l y l a rg e n e g a t i v e v a l u e s. F i n a l l y , n o t e a s o m e w h a tp a r a d o x i c a l r e s u lt a n a l o g o u s t o o n e e n c o u n t e r e d w i t h t h e f i rs t i n f er e n c ef r o m i n c o n s i st e n t p r e m i se s . T h i s is t h a t a s t h e ' a m o u n t o f i n c o n s i s te n c y 'i n c re a s e s f r o m ' n o i n c o n s i s te n c y ' ( b = n ) t o ' m a x i m a l in c o n s i s t e n c y '( b = n ( 1 - 8 ) ) , t he u n c e rt a in t y b o u n d p ( I ; 8 ) a c t u a l l y d e c r e a s e s f r o m


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45 0 ERNEST W. ADAMS AND HOW ARD P. LEVINE

ne/(n-a) t o 0 . O n c e a g a i n w e h a v e m a t h e m a t i c a l r e s u lt s i n s e a r c h o fi n t e r p r e t a ti o n , a n d i t is t im e t o t u r n t o q u e s t i o n s o f s ig n if ic a n ce a n d o t h e ri ss u es o f m e t h o d o l o g y .

3 . LOG ICAL SIGNIFI CANCE OF UNCERTAINTY BOUNDS

I n t h is a n d t h e s u c c e ed i n g t w o s e c ti o n s w e m a k e s o m e m e t h o d o l o g i c a lo b s e r v a t i o n s m o r e w i t h t h e a i m o f e x p la i n i n g t h e s i g n if ic a n ce o f o u r f o r m a lr e s u l t s t h a n t o r e s o l v e a n y o f t h e d i f fi c u l t i s su e s in v o l v e d . F i r s t w e w a n tt o a s k o f w h a t i m p o r t i t is to t h e l o g i ci an to k n o w t h a t t h e m a x i m u mun ce r t a in ty w h ich the con c lus ion o f the in fe rence s chem e (q51, . . . , q~ ,) I (~k )c a n h a v e c o m p a t i b l e w i t h s o m e g i v e n p r e m i s e u n c e r t a i n t y b o u n d s i se q u a l t o t h e v a l u e / ~ ( I ; e~ . . . , e ,) . L e t u s t a k e t h e f o l l o w i n g a s p l a u s i b l e :t h e f o r e g o i n g m e a n s t h a t t h e r e a r e a c t u a l p r o p o s i t i o n s , P l . . . . . p , a n d q o fthe fo r m s o f q51 . . . , q~, an d 0 , r e spec t ive ly , ( i .e . , P l . . . . , p , an d q w ou ld bep ro pe r ly s ym bo l i zed a s q51 . . . , ~b, an d ~ , r epsec t ive ly ) , and som e occ as io no n w h i c h i t w o u l d b e r a t i o n a l t o e s t i m a t e e a c h p l a s h a v i n g u n c e r t a i n t y n og r e a t e r t h a n e~ f o r i = 1 , . . . , n , w h i l e t h e u n c e r t a i n t y o f q w o u l d b e r a t i o -n a l l y e s t i m a t e d a s e q u a l t o / z ( I ; e l , . - . , e , ). N o w c o n s i d e r a l o g i c i a n s i t t in gi n e x p e r t j u d g m e n t w h e n s o m e o n e a s k s h i m c o n c e rn i n g p r o p o s it i o n sp~ , . . . , p ', an d q ' w h ich a re a l so o f the fo r m s o f q~ l , .. ., ~b, an d ~k, r e spec -t i v e l y : i s i t r a t i o n a l f o r m e t o c o n c l u d e q ' o n t h e b a s i s o f p ~ , . . . , p ' , ? T h el o g i c i a n c a n a n s w e r t h a t w i t h o u t f u r t h e r i n f o r m a t i o n h e c a n n o t te ll h o wc e r t a i n h i s i n t e r l o c u t o r i s o f h i s p r e m i s e s , b u t g i v e n b o u n d s e l , . . ., e , w h i c ha r e p l a u s i b l e f o r p r e m i s e s o f t h e s o r t s i n v o l v e d , t h e c o n c l u s i o n ' s u n c e r -t a i n t y c a n n o t e x c e e d # ( I ; e l . . . . , ~ ,) , a n d f u r t h e r m o r e t h e r e a r e p r e m i s e sa n d c o n c l u s io n s o f th e s a m e f o r m i n w h i c h th e c o n c l u s i o n ' s u n c e r t a i n t yw o u l d a c t u a l l y e q u a l t h e v a l u e / ~ (I ; 81 . . . , ~ ,) . T h u s , g r a n t e d o n l y t h ep l a u s i b l e a s s u m p t i o n t h a t t h e p r e m i s e u n c e r t a i n t i e s d o n o t e x c e e d~ 1 , . - . , e , , a l l t h a t c a n b e a s s u r e d c o n c e r n i n g t h e c o n c l u s i o n ' s u n c e r t a i n t yin virtue of the inferenee's being of the from o f (q51, .. ., ~bt) I ( 0 ) i s t ha t i tc a n n o t e x c e e d # ( I ; ~ 1 ,. .. , ~ ,) . O f c o u r s e a d e d u c t i v e l o g i c i a n q u e r r i e da b o u d t h e r a t i o n a l i t y o f i n f e rr i n g q ' f r o m p ] , . . . , p ~ m a y r e p l y t h a t d e d u c -t iv e l o gi c is c o n c e r n e d o n l y w i t h p o s s ib l e t r u t h v a l u e s a n d t h a t q u e s t i o n sa b o u t d e g re e s o f c e r t a i n ty o f c o n c l u s io n s a r e p r o p e r l y a d d r e s s e d t o i n -d u c t i v e l o g ic i an s . W h e t h e r i n f a c t w e h a v e e n t e r e d t h e d o m a i n o f i n d u c -t i v e l o g i c, o r h a v e a t a n y r a t e b l u r r e d t h e d i s t i n c ti o n b e t w e e n d e d u c t i v e


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ON UNCERTAI NTIES IN DEDUCTIVE INFERENCES 451and induct ive l og i c i n i n t roducing p robab i l i t i es and uncer t a in t i es , i s am at t e r w h ich wil l be r e tu rned to i n Sec t ion 5.

N o w co n s i d e r m o r e c l o s e ly t h e n a t u r e o f t h e ' ex t r eme ' u n ce r t a i n tyfunc t ions , u , wh ich max im ize t he c onc lus ion u ncer t a in ty u (~k) sub j ec t t op rem ise un cer t a in ty bou nds , u (q~)~


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452 ERNEST W. ADAMS AND HOWARD P. LEVINEa f f a i r s '. R e g i o n 6 is o f i m p o r t a n c e s in c e i t i s t h e o n e i n w h i c h a l l p r e m i s e sa r e t r u e , a n d i f t h e p r e m i s es a r e c o n s i s t e n t t h e n a n y p r o b a b i l i t y f r o m 0 t o 1c a n b e p u t i n t o t h i s r e g i o n . R e g i o n s 2 , 4 , a n d 8 a r e i m p o r t a n t b e c a u s et h e s e a r e t h e o n e s i n w h i c h a s m a n y p r e m i s e s a s p o s s ib l e a r e t r u e w h i l e t h ec o n c l u s i o n is fa l se . I n a s e ns e , t h e s e c o r r e s p o n d t o t h e m i n i m a l e s s e n ti a lf a l s i fi c a t i o n f u n c t i o n s f o r o u r i n f e re n c e .

N o w s u p p o s e t h a t t h e p r e m i s e s a r e c o n s i s t e n t a n d w e w a n t e a c h o ft h e m t o h a v e a p r o b a b i l i t y a t le a s t 1 - 8 ( h e n ce a n u n c e r t a i n t y n o g r e a t e rt h a n 8 ) w h i l e k e e p i n g t h e c o n c l u s i o n ' s p r o b a b i l i t y a s l o w a s p o s s ib l e . O n ew a y o f a s su r i n g p r o b a b i l i t y a t l e a s t 1 - 8 f o r a l l p r e m i s e s a t o n c e i s to p u t1 - 8 p r o b a b i l i t y i n t o r e g i o n 6 , th e i r r e g i o n o f in t e r s ec t i o n w h e r e a l l o ft h e m a r e t r u e . I f t h e r e m a i n i n g p r o b a b i l i t y 8 i s n o w d i s t r ib u t e d i n t or e g i o n s 1 , 2 , 4 , a n d 8 in w h i c h t h e c o n c l u s i o n i s f a ls e , t h e n t h e c o n c l u s i o nw i ll h a v e a p r o b a b i l i t y n o g r e a t e r t h a n 1 - 8 . H e n c e w e k n o w t h a t w e c a nm a k e a l l p r e m i se s h a v e p r o b a b i l i ty a t l e a st 1 - 8 w h i l e t h e c o n c l u s i o n ' sp r o b a b i l i t y i s n o g r e a t e r t h a n t h i s v a l u e . T h e c o n c l u s i o n c a n b e m a d es ti ll m o r e i m p r o b a b l e w h i l e r e t a i n i n g t h e p r e a s s i g n e d p r e m i s e p r o b a b i l i t yb o u n d s i f a l l p r o b a b i l i t i e s o u t s i d e o f th e p r e m i s e i n t e r s e c t i o n a r e d is t r ib -u t e d i n t o r e g i o n s 2 , 4 , a n d 8 w h e r e m a x i m a l n u m b e r s o f p r e m i s e s a r et r u e w h i l e t h e c o n c l u s i o n i s fa l se . T h i s i n t u r n a l l o w s t a k i n g s o m e p r o b a b -i l it y o u t o f t h e p r e m i s e i n t e r s e c t io n , a n d i n f a c t i t i s e a s i ly s e e n t h a t t h ew a y t o g e t m a x i m u m c o n c l u s i o n u n c e r t a i n t y c o m p a t i b l e w i t h p r e m i s ep r o b a b i l i t i e s a t l e a s t 1 - 8 i s t o p u t p r o b a b i l i t ie s o f 8 i n t o r e g i o n s 2 , 4 , a n d

38 a n d a p r o b a b i l i t y o f 1 - 7 8 i n t o r e g i o n 6 .O f c o u r s e t h e f o r e g o i n g c o n c l u s i o n u n c e r t a i n t y m a x i m i z a t i o n p r o c e -

d u r e w o n ' t w o r k i f th e p r e m i s e s a r e i n c o n s i s t e n t , s in c e in t h a t c a s e w e c a no n l y p u t 0 p r o b a b i l i t y i n t o t h e i r r e g i o n o f i n te r s ec t io n . H e r e i n s t e a d o fp u t t i n g p r o b a b i l i t y i n t o r e g i o n 6 w e t r y to p u t s u f fi c ie n t p r o b a b i l i t y i n t om a x i m a l c o n s i s t e n t se ts o f p r e m i s e s , c o r r e s p o n d i n g t o r e g i o n s 3 , 5 , a n d 7( w h o s e f a l s i ty f u n c t i o n s a r e t h e n e g a t i v e s u f f i c ie n t f a l s it y f u n c t i o n s f o ro u r i n f e r e n c e ) . I f i t i s p o s s i b l e t o g e t e n o u g h p r o b a b i l i t y i n t o 3 , 5 , a n d 7 t og i v e e a c h p r e m i s e a p r o b a b i l i t y o f a t l e a s t 1 - 8 , t h e n th e r e m a i n i n g p r o b a b -i l it y i s d i s t r i b u t e d i n t o r e g i o n s f a ls i f y i n g t h e c o n c l u s i o n , a s b e f o r e . B u te v i d e n t l y i t w i l l n o t a l w a y s b e p o s s i b l e t o g e t e n o u g h p r o b a b i l i t y i n t o 3 , 5 ,a n d 7 t o m a k e e a c h p r em i s e h a v e p r o b a b i l i t y a t l ea s t 1 - 8 a n d s ti ll h a v ee n o u g h l e f t o v e r t o k e e p t h e c o n c l u s i o n 's p r o b a b i l i ty b e l o w 1 - 8 ( a n d i tm a y n o t b e p o ss ib l e t o g e t e n o u g h i n t o 3 , 5, a n d 7 t o m a k e t h e p r e m i se s


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ON UNCERT AINTI ES IN DEDUC TIVE INFERENCES 453h a v e p r o b a b i l i t y a t l e a s t 1 - 8 , n o m a t t e r w h a t t h e c o n c l u s i o n ' s p r o b a b i l i t yi s) , a n d t h i s w i l l b e t h e s i t u a t i o n w h e n / z ( I ; 5 ) i s s m a l l e r t h a n e .

A v e r y i m p o r t a n t t h i n g t o n o t e a b o u t t h e ' e x t r e m e ' p r o b ab i l it ie s a n d u n -c e r t a in t i e s m a x i m i z i n g u (~ k) i s t h a t t h e s e a r e a l s o o n e s i n v o l v i n g e x t re m e so f non- independence. T h i s i s m o s t e v i d e n t i n t h e c o n s i s t e n t p r e m i s e c a s ei n t h e p r e s e n t e x a m p l e , f r o m t h e f a c t t h a t t h e conditional p r o b a b i l i t y o fa n y p r e m i s e g i v e n t h e f a l s i t y o f a n y o t h e r p r e m i s e i s e q u a l t o z e r o i n th ee x t r e m e p r o b a b i l i s t i c s t a t e o f a f f ai r s . T h i s i s n o t t h e s t a t e o f a f f a ir s w h i c hw e n o r m a l l y e n v is a g e w h e n s o m e o n e a s s e rt s t h a t h e a c c ep t s ' p re m i s e s ' o fs u c h a n d s u c h f o r m s - s a y o f f o r m s q51, . . ., qS, - w h e r e w e a r e a p t t o i m a g i n ei n s t e a d t h a t t h e p r e m i s e s r e p re s e n t i t e m s o f i n d e p e n d e n t l y a c q u i r ed i n f o r -m a t i o n a n d w h e r e t h e f a ls i t y o f a n y o n e i t e m w o u l d n o t n e c e s sa r il y c a lla n y o t h e r i t e m i n t o q u e s t i o n . P e r h a p s i t w o u l d e v e n b e s o m e w h a t m i s -l e a d i n g f o r a p e r s o n t o d e s c ri b e t h e s o r t s o f h i g h l y in t e r d e p e n d e n ts y s te m s o f p r o p o s i t i o n s w h i c h w e n e e d t o c o n s i d e r i n a r r iv i n g a t c o n c l u -s i o n u n c e r t a i n t y m a x i m a a s ' p r e m i s e s '. B e t h a t a s i t m a y , t h e f a c t t h a te x t r e m e p r o b a b i l i s t i c s t a t e s o f a f f a i r s i n v o l v e e x t r e m e i n t e r d e p e n d e n c es u g g es t s t h a t w e m i g h t e x p e c t l o g ic a l ly p o s s i bl e c o n c l u s i o n u n c e r t a i n t yb o u n d s t o d i ff e r g r e a t ly f r o m t h e c o n c l u s i o n u n c e r ta i n t ie s w h i c h f o l l o wi f ' n o r m a l ' i n d e p e n d e n c e a s s u m p t i o n s a r e m a d e . T h i s is n o t t h e p l a ce t oe n t e r i n d e t a i l i n t o t h e e r r o r p r o b a b i li ti e s w h i c h f o l lo w f r o m i n d e p e n d e n c ea s s u m p t i o n s ( t h i s b e i n g a s t a n d a r d a s p e c t s t at is t ic s ) , b u t a c o u p l e o f r e -m a r k s a r e i n o r d e r b y w a y o f c o m p a r i n g l o g i ca ll y m a x i m u m c o n c l u si o nu n c e r t a i n t i e s w i t h t h o s e f o l l o w i n g u n d e r i n d e p e n d e n c e a s s u m p t i o n s .

T h e e a s ie s t u n c e r t a i n t y b o u n d c o m p a r i s o n c a n b e m a d e w h e r e t h e p r e m -i se s o f th e i n f e r e n c e a r e n i n d e p e n d e n t p r o p o s i t i o n s s y m b o l i z a b l e b ya t o m i c f o r m u l a s ' A I ' , . . . , ' A : a n d t h e c o n c l u s i o n i s e q u i v a l e n t t o t h ea s s e r ti o n t h a t m o r e t h a n a o f th e p r o p o s i t i o n s a r e t r u e . A b b r e v i a t i n g t h ec o n c l u s i o n a s ' M ( a , n )' ( m o r e t h a n a o u t o f t h e n p re m i s e s a r e tr u e ) w eh a v e t h e i n f e r e n c e I ( M ( a , n ) ) = ( A 1 . . . , A , ) I ( M ( a , n )) , w h e r e w e h a v ea l r e a d y s e en t h a t t h e u n i f o r m c o n c lu s i o n u n c e r t a i n t y b o u n d f u n c t i o n isg i v e n b y :

/~ ( I ( M ( a , n ) ) ; s ) - - a"n

T h e i n f e r e n c e j u s t c o n s i d e r e d w a s t h e s p e c i al c a s e i n w h i c h n = 3 a n d a = 1


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454 ERNEST W. ADAMS AND HOWARD P. LEVINEw h e r e w e h a v e

# ( I ( M O , 3)) ; 5) = -~ .I f 5 = 0 .1 , f o r i n s ta n c e , t h e l o g ic a l ly m a x i m u m c o n c l u s i o n u n c e r t a i n t y w i llb e 0 . 15 . H o w e v e r , w h e n t h e p r e m i s e s a r e a s s u m e d t o b e s t a t i s t ic a l l y i n d e -p e n d e n t t h e p r o b a b i l i t y o f a n y s u b s e t o f t h e m b e i n g fa l se is e q u a l t o t h ep r o d u c t o f t h e i r i n d i v i d u a l p ro b a b i li t ie s o f f al s it y , w h i c h i s h e r e a s s u m e dt o b e e . W r i t i n g t h e i n d e p e n d e n t u n c e r t a i n t y f u n c t i o n w h i c h a s s i g n sp r o b a b i l i t y e a s t h e p r o b a b i l i t y o f fa l s it y f o r e a c h i n d i v i d u a l p r e m i s e a su , , w e h a v e

'

j= Ow here (~) i s t he b ino m ia l coe t t i c i en t n t] j t ( n - j ) ! W h e r e n = 3 a n d a = 1 w eh a v e

u~ (M (1, 3)) = 53 + 352(1 - 5).I f n = 0 . 1 , f o r i n s t a n c e , u ~ (M ( 1 , 3 )) m u s t b e 0 .0 2 8 w h i c h i s m u c h s m a l l e rt h a n t h e m a x i m u m l o g ic a l ly p o s s ib l e c o n c l u s i o n u n c e r t a i n t y o f 0 .1 5 .T h e d i f f e r e n c e b e t w e e n t w o v a l u e s g i v e s a m e a s u r e o f t h e d e g r e e t ow h i c h t h e c o n c l u s i o n d e p e n d s o n u n e x p r e s s e d i n d e p e n d e n c e a s s u m p t i o n s ,w h i c h m u s t b e r e g a r d e d a s e m p i r i c a l i n c h a r a c t e r s i n c e t h e s a m e k i n d so f a s s u m p t i o n s c a n b e u s e d t o j u s t i f y t h e o b v i o u s l y in d u c t iv e i n f e re n c e o ft h e c o n c l u s i o n M ( a , n ) f r o m t h e s i n g l e p r e m i s e M ( a - 1 , n ) w h e n a a n d na r e l a rg e e n o u g h ( t h is i s t h e i n f e re n c e o f ' m o r e t h a n a o f th e p r e m i s e s a r et r u e ' f r o m ' a t l e a s t a o f t h e p r e m i s e s a r e t r u e ' ) .

A m o r e s t r ik i n g r e s u lt e m e r g e s w h e n w e r e c o n s i d e r t h e i n f e re n c e o f th ec o n c l u s i o n a t = a 2 . . . . . a , f r o m t h e n ( n - 1 ) [ 2 p r e m i s e s a i = a j f o r1


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ON UNCER TAIN TIES IN DEDUCT IVE INFERENCES 455

t h a t t h e p r o b ab i l it i e s o f e r r o r i n th e r ep o r t s a r e i n d ep en d en t . P r o ceed i n gin t h is w ay , we migh t assum e tha t a l l poss ib l e equa t iona l s t a t es o f a f f a ir sam o n g , s ay , t h e t en i tems a l , . . . , a x o were e qual ly l i ke ly a p r i o r i ( each s u cheq u a t i o n a l s t a te o f a f fa i rs w o u l d co r r e s p o n d t o a p a r t i t io n o f t h e t eni tems i n to m u t u a l l y ex c lu s iv e su b s e ts ) , an d t h a t t h e p r o b ab i l i t y o f an yi n d i v i d u a l r ep o r t ' s b e i n g co r r ec t w as s o me v a l u e , s ay n . U n d e r t h i sa s s u m p t i o n , ev en i f 0 w e r e o n l y a s la r g e a s ~ , th e p o s t e r i o r u n ce r t a i n t y o fthe conc lus ion a l = a 2 . . . . . a l o f r o m th e g iv e n ' d u b i o u s d a t a ' c o u ld b eno l a rger t han 0 .02 . Th i s shows tha t ou r i n tu i t i on abou t t he co r rec tnesso f th i s con c lus ion is j u s ti f ied i f i n d ep en d en ce a s s u m p t i o n s o f an e s s en -t i a ll y empi r i ca l cha rac t e r a re ju s t i fi ed . Th i s i s c lear ly an i n ference w hichi t is approp r i a t e t o ca l l ' decep t ive ly deduct ive ' .

4. THE POSSIBLE SIGN IFI CANCE OF PROBAB ILITY CHANGE

I

Documents

Adams, et al., Uncertainties, Premises & Conclusions in Deductive Inferences [32 pgs]