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Confirmation theory & confirmation logic Lin, Chao-tien 1987

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CONFIRMATION THEORY & CONFIRMATION LOGIC By LIN B.A.,  CHAO-TIEN  N a t i o n a l Taiwan U n i v e r s i t y , 1962  M . A . , N a t i o n a l Taiwan U n i v e r s i t y , I966 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF PHILOSOPHY)  We accept t h i s t h e s i s as conforming to the r e q u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA October 1987 ©  L i n C h a o - t i e n , 1987  In presenting degree  this thesis  in partial fulfilment of the requirements  at the University of British Columbia, I agree that the Library shall make it  freely available for reference and study. copying  for an advanced  of this thesis for scholarly  department  or  by  his or  her  I further agree that permission for extensive  purposes  may be granted by the head of my  representatives.  It  is  understood  that  publication of this thesis for financial gain shall not be allowed without permission.  Department of  Philosophy  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6G/81)  March 23,  1988  copying  or  my written  ABSTRACT The t i t l e firmation of  o f my d i s s e r t a t i o n  logic",  and i t  the d i s s e r t a t i o n  ory that by C a r l In  could G.  Hempel.  Part  One I  of  view,  Hempel's  to  for  to  more r a d i c a l  The  confirmation  of  qualitative  (i)  of  confirmation  introduce  confirmation  discussions,  confirmation,  the-  discovered  things,  (ii)  (iii)  review  present  t h a n o t h e r known v i e w s ,  the paradoxes  motivation  confirmation"  do t h e t h r e e  theory  of  Parts.  an a d e q u a t e of  theory & con-  a  the  a new  and a r g u e  may r e q u i r e  as  that  change  logic. In  gics.  P a r t Two I I  logics  would  solve  logics  construct  employ t h e s e  lying  of  for  that  there  of  of  some  and  and  Unfortunately, inadequate.  This  theories  confirmation.  theory,  relationship  or,  (controversial)  the I  consider as  the  lo-  underhad h o p e d ,  three-valued underlying  because  I  even,  one-to-one  a  believe  three truth-values falsity"  three confirmation-statuses  of  and,  respectively,  of  "confirmation",  "neutrality".  these  theories  became c l e a r  had b e e n d e v e l o p e d .  I  as  which,  logics  " n e i t h e r t r u t h nor  (non-controversial)  three-valued  confirmation logics"  confirmation  intimate  "falsity"  "disconfirmation"  promising  any o t h e r m a n y - v a l u e d  c o r r e s p o n d e n c e between the "truth",  of  some new c o n f i r m a t i o n  any p r o m i s i n g is  a number  "quasi  the paradoxes  instead  logic  the  five  subsequent  the paradoxes  which i s  a solution of  of  of  "the paradoxes  t h e common b a c k g r o u n d  "confirmation  to c o n s t r u c t  t r y mainly  the fundamentals  major views  consists  was  solve  is  were f o u n d t o be  after  a complete  semantically  semantics  for  them  iii Thus,  one n e g a t i v e r e s u l t  of P a r t Two i s t h a t  our  syntactical  approach to c o n f i r m a t i o n t h e o r y i s wrong from the v e r y b e g i n n i n g . However,  from t h i s  negative r e s u l t  we l e a r n a p o s i t i v e  lesson: a  s e m a n t i c a l approach i s more fundamental and d e c i s i v e than a s y n t a c t i c a l one, at l e a s t t h i s  is  so f o r c o n s t r u c t i n g an adequate t h e -  ory of c o n f i r m a t i o n . It  i s rewarding to note t h a t the t h r e e - v a l u e d  out i n P a r t Two i s fact,  s i m p l e , complete and the f i r s t  the new t h r e e - v a l u e d  though the l i n e  semantics i s  of thought  In Part Three  I  shift  of i t s  i n the s p i r i t  i s much n e g l e c t e d the search f o r  semantics worked  logics  a systematic  I come,  at l a s t ,  and  to a semanTwo.  search through s e v e r a l p r o m i s i n g  three-valued  to a p l a u s i b l e c o n f i r m a t i o n l o g i c  and to a  c o n f i r m a t i o n theory that c o u l d s o l v e a l l known paradoxes of firmation.  al-  himself).  a confirmation logic  t i c a l approach because of the l e s s o n l e a r n e d i n P a r t  In  of Frege,  (even by Frege  an adequate t h e o r y of c o n f i r m a t i o n from a s y n t a c t i c a l  After  kind.  con-  The p r o m i s i n g t h r e e - v a l u e d c o n f i r m a t i o n theory i s  called  "the i n t e r n a l c o n f i r m a t i o n t h e o r y " . In Part Four I review and a p p r a i s e the adequacy c o n d i t i o n s down by Hempel as the n e c e s s a r y c o n d i t i o n s f o r mation t h e o r y . cially,  confir-  Under the c r i t i c i s m s of Carnap, Goodman and, e s p e -  with the h e l p of Hanen's thorough s t u d i e s ,  most an i d e n t i c a l c o n c l u s i o n to H a n e n ' s J ori  any adequate  laid  i n a t h e o r y of q u a l i t a t i v e  I come to a l -  we should not impose a p r i -  c o n f i r m a t i o n any adequacy c o n d i t i o n s  l a i d down by Hempel except perhaps the E n t a i l m e n t C o n d i t i o n ,  al-  though the i n t e r n a l c o n f i r m a t i o n theory a l s o adopts the E q u i v a l e n c e Condition for  some i n t r i n s i c  In the l a s t P a r t F i v e  I  reasons. try  to a p p r a i s e the three most impor-  tant  confirmation  dissertation.  After radoxes of  theory  of  at,  confirmation  perhaps  live  logical  constructed  in  confirmation,  confirmation  criticisms  this  Goodman's  and t h e  internal  situation  with)  are not  illusions  internal confirmation  when t h e r e  the b e s t all  thing  more r e a s o n a b l e is  no o b v i o u s  all  right  and t o  Hempel  says.  to  paradoxes  say  think  that  is  theory  theory,  way t o  of  to  to  I un-  overcome dissolve  confirmation,  the paradoxes  otherwise  pa-  conclusiont  t h a t we c a n do i s  new and o l d  genuine as  a r e made and some new  are u n e x p e c t e d l y d e r i v e d i n both the  reluctantly,this  f o r Hempel may be a f t e r firmation  theory of  selective  c o n f i r m a t i o n and t h e  t h e new d i f f i c u l t i e s to  and/or  theory.  der the p r e s e n t  (i.e.  of  some more v i g o r o u s  selective  arrive  discussed  They a r e H e m p e l ' s  and S c h e f f l e r ' s confirmation  theories  have  of  con-  psycho-  V  "One of the most f r u i t f u l kinds of contributions to philosophical discussion i s the discovery of a new conceptual problem . . . . One of the best examples . . . i s the paradoxes of confirmation . . . to which Hempel c a l l e d the attention of the p h i losophical community . . . " (Hintikka [19&9) p.2*0  TABLE OF CONTENTS Page INTRODUCTION PART I .  1  HEMPEL * S THEORY OF CONFIRMATION, DIFFERENT VIEWS OF THE  PARADOXES OF CONFIRMATION, AND THE CLASSICAL LOGIC  4  1. Why Have a Theory of C o n f i r m a t i o n  5  2. Nicod's C r i t e r i o n  6  3. Hempel's C r i t i c i s m s of Nicod's C r i t e r i o n  8  4. The E q u i v a l e n c e C o n d i t i o n 5.  6.  10  Fundamentals o f Hempel*s Theory o f Q u a l i t a t i v e Confirmation  12  The Paradoxes o f C o n f i r m a t i o n  18  7. Max B l a c k ' s D i a g n o s i s of the Paradoxes o f C o n f i r m a t i o n  20  8. Von Wright's Treatment of the Paradoxes o f C o n f i r m a t i o n  23  9* The B a y e s i a n Treatment of the Paradoxes of C o n f i r m a t i o n  27  10.  Quine's S o l u t i o n of the Paradoxes o f C o n f i r m a t i o n  30  11. Goodman's and S c h e f f l e r ' s Concept o f S e l e c t i v e Confirmat i o n and the Paradoxes of C o n f i r m a t i o n , and Grandy's Related Discussion 12. Armstrong's View o f the Paradoxes o f C o n f i r m a t i o n  33 40  13. Some Other Proposed S o l u t i o n s o f the Paradoxes o f C o n f i r mation  ^2  14. Hempel*s View o f the Paradoxes o f C o n f i r m a t i o n 15.  A New View of the Paradoxes o f C o n f i r m a t i o n  16. Challenges  to the C l a s s i c a l L o g i c  ^ 48 5  2  17. The P l a u s i b l e Inadequacy o f the C l a s s i c a l L o g i c  56  18. An Axiomatic  59  Review o f the C l a s s i c a l S e n t e n t i a l L o g i c  vii PART I I .  A SYNTACTIC AND FORMALLY SEMANTIC APPROACH TO THREE-  VALUED CONFIRMATION THEORIES, AND A THREE-VALUED SEMANTICS  63  1.  A Brief Introduction  64  2.  A D e f i n i t i o n of "Confirmation Logic"  65  3.  An Example o f Minimal  67  Confirmation Logic  4. The A x i o m a t i z a t i o n o f MC3  71  5.  I n Search o f more Minimal  72  6.  N e o - c l a s s i c a l Minimal  7.  Quasi C o n f i r m a t i o n L o g i c s and Quasi C o n f i r m a t i o n T h e o r i e s  Confirmation Logics  77  Confirmation Logics  79  8. Some P r o p e r t i e s of the Quasi C o n f i r m a t i o n L o g i c s  81  9. Some more P r o p e r t i e s of the Quasi C o n f i r m a t i o n L o g i c s  84  10. The A x i o m a t i z a t i o n of a T r u t h - f u n c t i o n a l l y Complete  Quasi  (Sentential) Confirmation Logic 11.  C o n f i r m a t i o n a l V a l u a t i o n and  a  Three-valued  86  Complete Semantics f o r any  Q u a n t i f i c a t i o n a l L o g i c with I d e n t i t y  88  12. The F a i l u r e of the 20 Q u a s i - c o n f i r m a t i o n L o g i c s to S a t i s 94  f y a Monotonicity Condition PART I I I .  A CONFIRMATION LOGIC AND A PLAUSIBLE SOLUTION OF THE  PARADOXES OF CONFIRMATION  98  1.  A Brief Introduction  99  2.  A Three-valued  Q u a s i - c l a s s i c a l Logic  100  2 3.  An A x i o m a t i z a t i o n o f L^  4. Some P r o p e r t i e s o f the Three-valued  102 S e n t e n t i a l L o g i c L^  104  5.  A Quasi-Hempelean E x t e r n a l C o n f i r m a t i o n Theory  6.  Paradoxes o f C o n f i r m a t i o n Regained  108  7.  An A n a l y s i s o f the F a i l u r e  110  8. C o n d i t i o n a l Law v s . I m p l i c a t i o n a l Law  107  112  viii 9. I n Search o f an I m p l i c a t i o n I 10.  I n Search o f an I m p l i c a t i o n I I  114 118  11. J u s t i f i c a t i o n o f the T r u t h Rule o f t h e I m p l i c a t i o n I  121  12. J u s t i f i c a t i o n o f the Truth Rule o f the I m p l i c a t i o n I I  123  13. A Long Way t o Reach the T r u t h Rule o f the I m p l i c a t i o n  126  14. A B r i e f View o f C3 and QC3  129  15. Elementary I n t e r n a l C o n f i r m a t i o n Theory  132  16. The General I n t e r n a l C o n f i r m a t i o n Theory  135  17. A Q u a s i - s o l u t i o n o f the Paradoxes o f C o n f i r m a t i o n  136  18. Some C r i t i c i s m s o f the Q u a s i - s o l u t i o n o f the Paradoxes of C o n f i r m a t i o n  138  19. A P l a u s i b l e S o l u t i o n o f the Paradoxes o f C o n f i r m a t i o n PART IV.  140  ADEQUACY CONDITIONS FOR CONFIRMATION AND THE GOODMAN  PARADOX  145  1. Adequacy C o n d i t i o n s f o r C o n f i r m a t i o n L a i d down by Hempel  146  2. The Converse Consequence C o n d i t i o n and i t s R e j e c t i o n  148  3.  150  Carnap and the Consequence  Condition  4. E x p l i c a t i o n o f the Concept o f C o n f i r m a t i o n  152  5. Carnap and the C o n j u n c t i o n C o n d i t i o n  155  6. Goodman^s Paradox and the C o n j u n c t i o n C o n d i t i o n  157  7. C a r n a p s S o l u t i o n o f the Goodman Paradox  160  8. Salmon's S o l u t i o n o f the Goodman Paradox  l6l  9 . An Attempted S o l u t i o n o f the Goodman Paradox  163  0  10. The Goodman Paradox Regained  166  11. A T e n t a t i v e C o n c l u s i o n ^ o f the V a l i d i t y o f the Adequacy C o n d i t i o n s f o r C o n f i r m a t i o n i n L i g h t o f Marsha Hanen's Study  I69  ix PART V.  TENTATIVE CONCLUSIONS  183  1.  Introduction  184  2.  A Summary of Hempel's Theory of C o n f i r m a t i o n  185  3. A Summary of Goodman's and S c h e f f l e r ' s Theory of  Selec-  t i v e Confirmation 4.  A T e n t a t i v e C o n c l u s i o n f o r the Theory  BIBLIOGRAPHY  188 Internal  Confirmation 192. 2  0  0  Acknowledgement Thanks are due to P r o f .  John P. Stewart who as my r e s e a r c h  s u p e r v i s o r has v e r y thoroughly  criticized  the whole content  in  many l o n g and short d i s c u s s i o n s i n the past few years and to P r o f . Howard Jackson and P r o f .  Thomas P a t t o n who have v e r y c a r e -  f u l l y read some v e r s i o n s of my d r a f t s and g i v e n many v a l u a b l e c r i t i c i s m s and s u g g e s t i o n s , semantical aspects.  e s p e c i a l l y about i t s  Thanks are a l s o due to P r o f .  l o g i c a l and Richard E.  Robinson who d i s c o v e r e d a s e r i o u s mistake made i n the s e m i - f i n a l draft  of t h i s  dissertation.  T h i s d i s s e r t a t i o n i s d e d i c a t e d to my parents and my b r o t h e r s and s i s t e r s .  Without t h e i r g r e a t support I d o n ' t know I  have enough p a t i e n c e to f i n i s h  it.  would  1 INTRODUCTION The m o t i v a t i o n  of  this  d i s s e r t a t i o n was  quate c o n f i r m a t i o n t h e o r y t h a t firmation" In  d i s c o v e r e d b y C a r l G.  Part  One I  the fundamentals as  t r y mainly of Hempel's  t h e common b a c k g r o u n d  the major views new v i e w , that  In  of  which i s  a solution  change  of  to  I  of  con-  theory of q u a l i t a t i v e subsequent  (i)  introduce  confirmation  discussions,  (ii)  review  of confirmation, ( i i i ) present t h a n o t h e r known v i e w s ,  the paradoxes  of  and  a  argue  o f c o n f i r m a t i o n may r e q u i r e  solve  "quasi  instead  a  inadequate.  I  any o t h e r many-valued  the  or,  I  had  turn  to  logics  (controversial)  even , a  I  that believe  one-to-one  and,  of  respec-  three confirmation-statuses and  as  three truth-values  and " n e i t h e r t r u t h n o r f a l s i t y "  these  un-  confirmation theory  relationship  "disconfirmation"  Unfortunately,  as  of c o n f i r m a t i o n , mainly becuase  (non-controversial)  "confirmation",  three-valued  of confirmation.  f o r any p r o m i s i n g  c o r r e s p o n d e n c e between the  the  of  some i n t i m a t e  "falsity"  promising  confirmation logics"  the paradoxes  the paradoxes  there i s  a number o f  some new c o n f i r m a t i o n t h e o r i e s w h i c h ,  the u n d e r l y i n g l o g i c  tively,  solve "the paradoxes  t o do t h e t h r e e t h i n g s *  more r a d i c a l  three-valued logics  "truth",  ade-  Hempel.  the paradoxes  employ t h e s e  hoped, would  can solve  an  logic.  derlying logics  that  for  P a r t Two I c o n s t r u c t  logics.  could  to c o n s t r u c t  of  "neutrality".  t h e o r i e s were f o u n d t o be  T h i s became c l e a r a f t e r a c o m p l e t e  semantically  semantics  for  them had b e e n d e v e l o p e d . Thus,one  negative  result  of  Part  approach to c o n f i r m a t i o n theory i s  Two i s  that  our  syntactical  wrong f r o m t h e v e r y  beginning.  2 However,  from t h i s  semantical tactical  approach i s  approach,  quate t h e o r y of It out  negative  is  at  in this  Part  In  Three I  more f u n d a m e n t a l least  is  from a s y n t a c t i c a l i n Part  After logic  I  simple, shift  to  a systematic last,  The p r o m i s i n g  Part  Four I  down b y Hempel as mation theory. cially,  search f o r  can solve  could  solve  ler's  of  worked  kind. and  confirmation  of  the  promising  t h e known p a r a d o x e s  lesson  three-valued and t o of  the n e c e s s a r y  qualitative  a  confircalled  theory".  r e v i e w and a p p r a i s e  the adequacy c o n d i t i o n s  conditions of  f o r any a d e q u a t e  Carnap,  thorough  confir-  Goodman a n d ,  studies,  we s h o u l d n o t  I  come t o  impose  a  laid  espealmost  priori  c o n f i r m a t i o n any a d e q u a c y c o n d i t i o n s the Entailment adopts  Condition,  the E q u i v a l e n c e  laid  although  the  Condition  reasons.  Part  Five I  They are Hempel*s t h e o r y of  ade-  three-valued confirmation theory is  t r y to  confirmation theories discussed tion.  the paradoxes  confirmation logic  confirmation theory also  the l a s t  a  syn-  a confirmation logic  to a p l a u s i b l e  Under the c r i t i c i s m s  some i n t r i n s i c In  its  several  down by Hempel e x c e p t p e r h a p s  for  of  approach because  an i d e n t i c a l c o n c l u s i o n t o H a n e n ' s :  internal  an  search through  w i t h the help of Hanen's  i n a theory of  so f o r c o n s t r u c t i n g  c o m p l e t e and t h e f i r s t  the  internal confirmation In  than a  the t h r e e - v a l u e d semantics  a semantical  confirmation theory that  "the  is  and d e c i s i v e  lesson:  Two.  come, a t  mation.  this  to note t h a t  a confirmation theory that  learned  we l e a r n a p o s i t i v e  confirmation.  rewarding  Part  result  appraise  and/or  theory of  the  t h r e e most  constructed i n  confirmation,  this  Goodman's  s e l e c t i v e c o n f i r m a t i o n and t h e i n t e r n a l  important dissertaand  Scheff-  confirmation  3  theory. After  some more v i g o r o u s  doxes o f c o n f i r m a t i o n of  selective  present  confirmation  and t h e i n t e r n a l  to  live  the best  with) a l l  Hempel may be a t f e r are  para-  i n both the theory  confirmation  more r e a s o n a b l e  theory,  conclusion:  I under  i s no o b v i o u s way t o overcome t h e  t h i n g t h a t we c a n do i s t o d i s s o l v e ( i . e .  new and o l d a l l right  n o t g e n u i n e and t o t h i n k  illusions.  this  s i t u a t i o n when t h e r e  new d i f f i c u l t i e s  a r e made and some new  are unexpectedly derived  come t o , p e r h a p s r e l u c t a n t l y , the  critiicsms  that  paradoxes o f c o n f i r m a t i o n , f o r the paradoxes of c o n f i r m a t i o n  otherwise  i s t o have p s y c h o l o g i c a l  4 PART I.  HEMPEL S THEORY OF CONFIRMATION, DIFFERENT VIEWS OF 1  THE PARADOXES OF CONFIRMATION, AND THE CLASSICAL LOGIC  5  WHY HAVE A THEORY OF CONFIRMATION  1.  We n e e d a t h e o r y o f c o n f i r m a t i o n f o r (A)  The p r o b l e m o f q u a l i t a t i v e  confirmation.  s c i e n c e we o f t e n r e f e r t o t h e e p i s t e m i c (or  of  a theory).  hypothesis  (or  t i o n must,  first  We s a y  all,  t i o n s « of c o n f i r m a t i o n , (B)  pect to this  cal  their relative is  How t o  would f o l l o w However, quate  as  1.  at  from an  least as  empirical hypothesis  of  and  (or  Sometimes  theories)  measure  is  with  How t o  in  res-  obtain  confirmation.  confirmation. does  defini-  neutrality.  confirmation.  comparative  a  confirma-  qualitative  confirmation.  (adequate)  the main c o n c e r n o f  idea of  the problems  confirmation; (or  three  In  empiri-  evidence confirm a the problem of  "warmth")  of just  fundamental  i n philosophy  theory of q u a l i t a t i v e  tackle  of  of  t o what d e g r e e  well  must have a c l e a r  ture"  strength  obtain this  there are  science,  tive  a  1  hypo-  quantita-  confirmation. So,  cal  disconfirmation  The p r o b l e m o f q u a n t i t a t i v e  thesis?  of  theory  three precise  the problem of  s c i e n c e we a s k i  tive  give  an adequate  s c i e n c e we compare h y p o t h e s e s  ordering (C)  So,  The p r o b l e m o f c o m p a r a t i v e  empirical  status  In  reasonsi  t h a t we c a n c o n f i r m o r d i s c o n f i r m  a theory). of  the f o l l o w i n g  theory of this  qualitative  science,  in  whose  for  will  solution  be a n  the reason  that  c o n f i r m a t i o n b e f o r e we  c o n f i r m a t i o n and o f  we must have a c l e a r  b e f o r e we c a n t a l k  empiri-  confirmation.  dissertation  confirmation,  quantitative as  of  problems  about  i d e a of  "warmer"  adewe can  compara"tempera-  and  "degree  temperature". Cf.Carl t i o n 1,  G.Hempelt " S t u d i e s i n the l o g i c of c o n f i r m a t i o n " , i n h i s Aspects of S c i e n t i f i c E x p l a n a t i o n . p . 3 f f .  sec-  6 2.  NICOD'S  CRITERION  This dissertation w i l l  be m a i n l y c o n c e r n e d w i t h t h e i d e a  c o n f i r m a t i o n as developed by Hempel. of c o n f i r m a t i o n i n the case plays  an important  role  Nicod's sentences  idea  is  point  of universal  start  b e f o r e we t u r n t o  theory  idea of  con-  Hempel*s.  universal  i n the f o l l o w i n g  idea  sentences  o f Hempel*s  a review of Nicod's  of c o n f i r m a t i o n about  expressed  Jean Nicod*s  conditional  i n the development  o f c o n f i r m a t i o n , we w i l l f i r m a t i o n on t h i s  Since  of  conditional  wayt^"  C o n s i d e r t h e f o r m u l a o r t h e l a w * A e n t a i l s B. How c a n a p a r t i c u l a r p r o p o s i t i o n , o r more b r i e f l y , a f a c t , a f f e c t its probability? If t h i s f a c t c o n s i s t s of the presence o f B i n c a s e o f A, i t i s f a v o r a b l e t o t h e law 'A e n t a i l s B_*; o n t h e c o n t r a r y , i f i t c o n s i s t s o f t h e a b s e n c e o f B i n a c a s e o f A, i t i s u n f a v o r a b l e t o t h i s l a w . It i s conc e i v a b l e t h a t we h a v e h e r e t h e o n l y two d i r e c t modes i n which a f a c t can i n f l u e n c e the p r o b a b i l i t y o f a l a w . . . . Thus, t h e e n t i r e i n f l u e n c e o f p a r t i c u l a r t r u t h s o r f a c t s on t h e p r o b a b i l i t y o f u n i v e r s a l p r o p o s i t i o n s o r l a w s w o u l d o p e r a t e b y means o f t h e s e two e l e m e n t a r y r e l a t i o n s w h i c h we s h a l l c a l l c o n f i r m a t i o n a n d i n v a l i d a t i o n . L e t us w r i t e form as  a universal  conditional  sentence  in its  simplest  follows» ( x ) (Px =>Qx).  Then,  Nicod's  sentences  i d e a of c o n f i r m a t i o n about  e x p r e s s e d above  is  universal  conditional 2 summed u p b y H e m p e l a s b e l o w t  1. " A n o b j e c t c o n f i r m s a u n i v e r s a l c o n d i t i o n a l h y p o t h e s i s i f and o n l y i f i t s a t i s f i e s b o t h t h e a n t e c e d e n t ( h e r e « * P x * ) and t h e c o n s e q u e n t ( h e r e j ' Q x * ) o f t h e c o n d i t i o n a l } " 2. " i t d i s c o n f i r m s t h e h y p o t h e s i s i f and o n l y i f i t satisfies the antecedent, but not the consequent of the c o n d i t i o n a l j " 1.  See J e a n N i c o d i F o u n d a t i o n s o f G e o m e t r y a n d I n d u c t i o n (trans, by P . P . W i e n e r ) , 1930, p . 2 1 9 ; a l s o q u o t e d b y Hempel i n h i s Aspects of S c i e n t i f i c E x p l a n a t i o n , p.10.  2.  Hempel,op.  c i t . , p.11.  7 3 . "and (we add t h i s t o N i c o d ' s s t a t e m e n t ) i t i s n e u t r a l , or i r r e l e v a n t , w i t h r e s p e c t to the h y p o t h e s i s i f i t does not s a t i s f y the a n t e c e d e n t . " The above  idea  of  confirmation,  about u n i v e r s a l c o n d i t i o n a l called  "Nicod's  d i s c o n f i r m a t i o n and  neutrality  s e n t e n c e s e x p r e s s e d by N i c o d  c r i t e r i o n " by Hempel.  is  A c c o r d i n g to Hempel,  3  i t s t a t e s e x p l i c i t l y what i s p e r h a p s t h e most common i n t e r p r e t a t i o n of the concept of c o n f i r m a t i o n . However, c r i t e r i o n to  3.  Hempel, op.  Hempel has be g i v e n  cit.,  some i m p o r t a n t c r i t i c i s m s  i n the  p.11.  next  section.  of  Nicod's  3. H E M P E L S CRITICISMS OF NICOD'S CRITERION 1  Hempel t h i n k s sible,  suffers  (I)  tional for of  Nicod's  from the  First,  tricted.  that  the  It  existential  only to  Or,  serious  the  seemingly  hypotheses of  hypotheses.  Nor d o e s i t  of  to  formulate  too  universal  res-  condi-  confirmation,  p r o v i d e any  the  plau-  shortcomings:"^  criterion is  no s t a n d a r d s  say,  standard  quantifiers  following  sentence:  You c a n f o o l a l l o f t h e p e o p l e some o f t h e t i m e and some o f t h e p e o p l e a l l o f t h e t i m e , b u t you c a n not f o o l a l l of the people a l l of the t i m e .  i n symbols,  a c c o r d i n g to Hempeli  (x)(3t)Fxt. where, In s h o r t ,  are not adequate  Fxti  you c a n f o o l  Hempel s a y s t h a t  there  person x at  account  of  ones whose  standards  shortcoming of  makes c o n f i r m a t i o n depend n o t  hypothesis  but a l s o  following  on i t s  three  1. H e m p e l : A s p e c t s 2. I b i d . . p.12.  It  is  too  of  Scientific  that  an  restricted.  Nicod's c r i t e r i o n is  o n l y on t h e  formulation.  other  any f o r m o f  content  To see  this,  hypotheses: of  t.  confirmation  obvious  c o n f i r m a t i o n would a p p l y to  Thus N i c o d ' s c r i t e r i o n i s  The s e c o n d  time  a r e many h y p o t h e s e s  c o v e r e d by N i c o d ' s c r i t e r i o n .  hypotheses. (II)  2  (3x)(t)Fxt . -(x)(t)Fxt  than u n i v e r s a l c o n d i t i o n a l  the  while  c o n f i r m a t i o n f o r any s e n t e n c e c o n t a i n i n g m i x e d  Hi  it  two  provides  s u c h a s w o u l d be n e e d e d  H:  following  a p p l i c a b i l i t y of  applies  f o r m and i t  criterion,  Explanation,  p.llff.  of  that the  consider  9 HI i  (x)(RxOBx),  H2t  (x)(-Bxr>-Rx),  H3«  (x)(Rx.-Bx:>Rx.-Rx), where Rt @ Bi® Then,  l e t u s make t h e f o l l o w i n g ai  a black  bi  a non-black  ci  a black  (ii) also  abbreviations.  raven,  non-raven. following  we h a v e ;  object b disconfirms object c is  black.  c r i t e r i o n we w o u l d have t h e  1  (i)  is  non-raven,  Now a c c o r d i n g t o N i c o d s Although  a raven,  raven,  d» a n o n - b l a c k  results*  is  HI,  H2 and H3i  n e u t r a l with respect to  and HI , H2 and H3 , we  havei  (iii)  object a confirms  and t o H3» (iv) (v)  HI  is  neutral with respect  t o H2  and  object d confirms  H2 b u t i s  n e u t r a l t o HI  and t o H 3 ,  and  no o b j e c t w h a t e v e r c a n c o n f i r m H 3 .  And y e t H I ,  H2 and H3 a r e l o g i c a l l y  we have t h e s e c o n d s h o r t c o m i n g f i r m a t i o n depends n o t a l s o on i t s  formulation  inadequate  3. Hempelt  as  Aspects  Nicod's  "which appears  of  Scientific  sentences!  criterion;  that  a hypothesis  Thus conbut  a b s u r d " ^ a s Hempel s a y s .  Hempel c o n c l u d e s  a standard  of  of  equivalent  o n l y on t h e c o n t e n t o f  From t h e above r e s u l t s on i s  but  that  Nicod's  confirmation.  Explanation,  p.13.  criteri-  4.  THE EQUIVALENCE CONDITION  In  c r i t i c i z i n g Nicod's c r i t e r i o n that  t i o n of  a hypothesis  t h e s i s but a l s o ciple  tacitly  depend n o t  on i t s  makes  o n l y on t h e  formulation there  assumed b y H e m p e l .  the  confirma-  content  is  of  the  an i m p o r t a n t  The p r i n c i p l e i s  the  hypo-  prin-  following  condition:^" Equivalence one  of  two  condition.  Clogically]  (disconfirms)  the  Only w i t h the confirmation of  its  sentences,  that  help  of t h i s  equivalence  So,  it  attention the  is to  the  equivalence  sity  of  theory (a) the  the of  reasons  equivalence  confirmation.  First,  confirms  condition is  to  lay  above  it  down  explicitly. condition  is  a d e q u a c y o f any (my i t a l i c s )  proposed  support  c o n d i t i o n i n c o n s t r u c t i n g an The two r e a s o n s of  the  a hypothesis  the  independent  equivalence  g i v e n b y Hempel t o  fulfillment  confirmation of  Otherwise,  also  a hypothesis  condition  2  two  of  nice  "a n e c e s s a r y c o n d i t i o n f o r t h e c r i t e r i o n of c o n f i r m a t i o n . " There are  (disconfirms)  other."  formulation.  thinks  equivalent  (and d i s c o n f i r m a t i o n )  Hempel c a l l s and  "Whatever c o n f i r m s  the  neces-  adequate  aret  equivalence independent  condition of i t s  makes  formulation.  a c c o r d i n g to Hempel,^  t h e q u e s t i o n as t o w h e t h e r c e r t a i n d a t a c o n f i r m s a g i v e n h y p o t h e s i s w o u l d have t o be answered by s a y i n g i " t h a t depends on w h i c h o f t h e d i f f e r e n t equivalent f o r m u l a t i o n s of the h y p o t h e s i s i s c o n s i d e r e d "  1. H e m p e l . A s p e c t s 2. I b i d . , p . 1 3 . 3. I b i d . , p . 1 3 .  of  Scientific  Explanation, p.13.  11 which appears (b)  absurd.  The s e c o n d r e a s o n i s  practice,  it  e s p e c i a l l y when h y p o t h e s e s  explanation or p r e d i c t i o n . mises  that  In  i n a d e d u c t i v e argument  this whose  does are  justice used  to  for  scientific purposes  f u n c t i o n t h e y s e r v e as conclusion  is  a  of  pre-  description  k of  the  e v e n t t o be e x p l a i n e d o r p r e d i c t e d .  And we know  that  the d e d u c t i o n i s governed by the p r i n c i p l e s o f f o r m a l l o g i c , and a c c o r d i n g t o t h e l a t t e r , a d e d u c t i o n w h i c h i s v a l i d w i l l r e m a i n so i f some o r a l l o f t h e p r e m i s e s a r e r e p l a c e d by d i f f e r e n t b u t e q u i v a l e n t statements; and i n d e e d , a s c i e n t i s t w i l l f e e l f r e e , i n any t h e o r e t i c a l reasoning i n v o l v i n g c e r t a i n hypotheses, to use the l a t t e r i n whichever of t h e i r e q u i v a l e n t formulat i o n s a r e most c o n v e n i e n t f o r t h e d e v e l o p m e n t o f his conclusions. (my i t a l i c s ) Since  the e q u i v a l e n c e c o n d i t i o n p l a y s  a controversial role  i n the development of a l l  o f c o n f i r m a t i o n , we w i l l adequacy c o n d i t i o n , of  confirmation.  4.  Ibid..  p.13.  a c r u c i a l as w e l l  l a t e r t r y to assess  major  theories  whether i t i s  a f t e r we have c o n s i d e r e d t h e m a j o r  as  a good theories  12 FUNDAMENTALS OF HEMPEL'S THEORY OF QUALITATIVE CONFIRMA-  5.  TION It tive  is  time  to  introduce  Hempel*s  c o n f i r m a t i o n and examine  s i o n and i t s  promise.  fundamental parts  of  its  However, it,  theory of  importance,  I  qualita-  its  w i l l touch only  which c o n s i s t  of  comprehent h e more  the f o l l o w i n g  three  partsi (I)  The c l a s s i c a l  s i g n a t e d as (II)  QC2} a s  its  Seven b a s i c  f o r m i n g the t h e o r y (III)  The  two-valued  underlying  basic  this  (de-  condition,  proper.  development of  the t h e o r y , adequacy  including  some r e l a t e d  conditions.  s e c t i o n we a r e g o i n g t o d e a l o n l y w i t h  concepts  logic,  logic.  c o n c e p t s and one a d e q u a c y  c o n c e p t s and some a d d i t i o n a l In  quantificational  and t h e one a d e q u a c y c o n d i t i o n ,  the  taking  seven QC2  for  granted. The s e v e n b a s i c are:  (i)  Hypothesis,  observation (vi)  concepts  report,  (ii) (iv)  disconfirmation,  lence condition. the f o l l o w i n g  and t h e one a d e q u a c y  development  of  a hypothesis,  direct confirmation,  (vii)  neutrality,  condition  and  (v)  (viii)  (iii)  confirmation, the  equiva-  They a r e d e f i n e d o r e x p l a i n e d by Hempel  in  wayi^"  Hypothesis. "We s h a l l . . . u n d e r s t a n d b y a h y p o t h e s i s any s e n t e n c e w h i c h c a n be e x p r e s s e d i n t h e assumed l a n g u a g e o f s c i e n c e , no m a t t e r w h e t h e r i t i s a g e n e r a l i z e d s e n t e n c e , c o n t a i n i n g q u a n t i f i e r s , or a p a r t i c u l a r sentence r e f e r r i n g o n l y t o a f i n i t e number o f p a r t i c u l a r o b j e c t s . " 1. C f .  Hempel: Aspects  of  Scientific  Explanation,  pp.22-37.  13  Development of a H y p o t h e s i s . "The concept of development o f a h y p o t h e s i s , H, f o r a f i n i t e c l a s s o f i n d i v i d u a l s , C, c a n he d e f i n e d p r e c i s e l y b y r e c u r s i o n ; h e n c e i t w i l l s u f f i c e t o s a y t h a t t h e d e v e l o p m e n t o f H f o r C s t a t e s what H w o u l d a s s e r t i f t h e r e e x i s t e d e x c l u s i v e l y those o b j e c t s which are elements of C." Thus, Hit for of  the development of  the  hypothesis,  (x)(FxvGx), the c l a s s the  / a , b}  i s : ( F a v G a ) . ( F b v Gb);  and t h e  development  hypothesis, (3y)Gy,  H2: for  e.g.,  the c l a s s  fb,c}  is:  Gbv Gc.  Observation Report. " A n o b s e r v a t i o n r e p o r t w i l l be c o n s t r u c t e d a s a f i n i t e c l a s s ( o r a c o n j u n c t i o n o f a f i n i t e number) o f o b s e r v a t i o n s e n t e n c e s ; and a n o b s e r v a t i o n s e n t e n c e as a s e n t e n c e w h i c h e i t h e r a s s e r t s o r d e n i e s t h a t a g i v e n o b j e c t has a c e r t a i n o b s e r v a b l e p r o p e r t y . . . o r t h a t a g i v e n sequenee o f o b j e c t s stand i n a c e r t a i n observable r e l a t i o n . " O r , e q u i v a l e n t l y , " o b s e r v a t i o n s e n t e n c e s a r e d e f i n e d as s e n t e n c e s c o n t a i n i n g no q u a n t i f i e r s . " 2  Direct Confirmation. "An o b s e r v a t i o n r e p o r t B d i r e c t l y c o n f i r m s a h y p o t h e s i s H i f B e n t a i l s t h e d e v e l o p m e n t o f H_ f o r the c l a s s of those o b j e c t s which are mentioned e s s e n t i a l l y i n B " , where " o b j e c t s w h i c h a r e m e n t i o n e d e s s e n t i a l l y i n B" means t h a t t h e o b j e c t s a r e m e n t i o n e d i n a n o n - a n a l y t i c component o f B."3  Thus l e t ' s  take  t h e above p o i n t  of  lowing  hypothesis:  H3«  (x)Px,  one example f r o m H e m p e l ' s  3.  reports.  Pa,  2. I b i d . Ibid,  Note  4 6 ,  illustrate  " e s s e n t i a l m e n t i o n " , and c o n s i d e r  and t h e two o b s e r v a t i o n BI:  to  p p . 3 7 - 3 8 .  Note 4 6 , p . 3 8 .  the  fol-  1** Pa . (Pb v - P b ) .  B2:  Now,  by d e f i n i t i o n , B l  where  for  we h a v e : B l the c l a s s  mentioned)  of  definition  those  in Bl.  of  " )(•" i s  entails  n e g l e c t e d the p a r t  i.e.  H3,  f o r we h a v e *  Pa II-Pa,  (1)  i.e.  d i r e c t l y confirms  semantic  P a , where Pa i s  o b j e c t s mentioned  relation",  the development (as  w e l l as  of H 3  essentially  d i r e c t l y confirm H3 i f  of  mentioned" played  "essentially  f o r B2 does not  in  we  the  entail  Pa.Pb,  have:  Pa . (Pb v - P b )  where P a . P b i s  entailment  B u t B 2 would n o t  direct confirmation,  we do n o t  (2)  "the  IP  Pa . Pb,  the development  o b j e c t s mentioned  however,  o f H3 f o r not  the c l a s s  essentially  of  those  mentioned  in  B2. But B l  and B 2 a r e l o g i c a l l y  equivalent.  equivalence condition discussed This sential  deficiency mention"  "Pbv-Pb"  of B2 i s  does not a p p e a r , is  not,  so  violation  in  to of  speak,  a "proper"  of  qualification  o f "es-  direct confirmation,  component  of B 2 .  "essentially" observation  "essential  d i r e c t c o n f i r m a t i o n about  velopment  the  So,  the  there.  report.  for  "b"  Hence, B2 Thus  no  t h e e q u i v a l e n c e c o n d i t i o n o c c u r s h e r e , once we  have t h e q u a l i f i c a t i o n o f of  by  the d e f i n i t i o n of  by d e f i n i t i o n ,  the  earlier.  i s remedied  an a n a l y t i c  This v i o l a t e s  a  mention"  the o b j e c t s  i n the  appearing  in  definition the  de-  hypothesis.  Confirmation.  "An o b s e r v a t i o n r e p o r t B c o n f i r m s  a hypo-  t h e s i s H i f H i s e n t a i l e d hy a c l a s s of sentences each of which i s d i r e c t l y confirmed by B . " Disconfirmation. "An o b s e r v a t i o n r e p o r t B d i s c o n f i r m s a h y p o t h e s i s H i f i t confirms the d e n i a l of H." k Neutrality. "An o b s e r v a t i o n r e p o r t B i s n e u t r a l w i t h r e s p e c t to a h y p o t h e s i s H i f B n e i t h e r confirms nor d i s c o n f i r m s H." Equivalence C o n d i t i o n . "Whatever confirms ( d i s c o n f i r m s ) one of two ( l o g i c a l l y " ) e q u i v a l e n t sentences, a l s o confirms ( d i s c o n f i r m s ) the o t h e r . " Below are some examples^ taken from Hempel to  illustrate  some of the above newly d e f i n e d c o n c e p t s . Example 1.  (x)(PxvQx)  H4J for  The development of the h y p o t h e s i s  the c l a s s -fa.bj-  is:  (Pa v Qa). (Pb v Qb).  The o b s e r v a t i o n r e -  port B3".  Pa.Pb  directly  confirms the development of H4 f o r the c l a s s  {a,b},i.e.  we havei (3)  (Pa.Pb)  | K ( P a v Qa).(Pbv Qb)).  Because of ( 3 ) , B3 a l s o c o n f i r m s H*K Example 2 . H5» is,  The development of the h y p o t h e s i s  Pc v Qc by d e f i n i t i o n ,  that hypothesis  itself.  Now the o b s e r v a t i o n  r e p o r t B3 of the above example does not d i r e c t l y because i t s development, v i z . H5 i t s e l f ,  confirms H 5 t  i s not e n t a i l e d by B3,  4. Hempel c a l l s " n e u t r a l i t y " sometimes a l s o as " i r r e l e v a n c y " , c f . i b i d , p . 5 . p . 6 and p.11. 5. I b i d , p p . 3 6 - 3 7 . However, some m i s p r i n t s contained there have been c o r r e c t e d h e r e .  16 i.e.  we do n o t  ( P a . P b ) \\- (Pc v Q c ) .  (4)  However,  entails  H4  Hence,  d i r e c t l y confirms  make. Note  in this that  we h a v e .  the  But i s n ' t i n terms capture I  it of  his  think  First,  s e c t i o n we have a n i m p o r t a n t o b s e r v a t i o n the examples  concept of  "semantic  " e n t a i l m e n t " r e l a t i o n by  that Hempel's  syntax?  If  so,  the  i n the c l a s s i c a l  "syntactic  how c a n we u s e a s e m a n t i c  logic. framed  concept  this.  two-valued q u a n t i f i c a t i o n a l  If  A 1|-B,  t h e n A |-B, (where  " (-"  is  consequence" r e l a t i o n of QC2.) If  A |-B, t h e n A ||"B.  t o g e t h e r we have t h e f o l l o w i n g  Correspondence Theorem. b y t h e above  ment r e l a t i o n and t h e  A |-B  iff  theorem. A|(-B.  t h e o r e m we know t h a t syntactic  the semantic  my  i n t e r p r e t a t i o n of Hempel's  of  the  entailment,  it  entail-  consequence r e l a t i o n of  l o g i c are e x t e n s i o n a l l y e q u i v a l e n t .  semantic  logic  theoremst  two  Theorem.  Soundness Theorem.  classical  which  theory of c o n f i r m a t i o n i s  justification is  Completeness  So,  inter-  "|f-".  the c l a s s i c a l  to  idea?  QC2 we have t h e f o l l o w i n g  So,  g i v e n above we have  e n t a i l m e n t " r e l a t i o n of  true  since  H4.  in a l l  preted Hempel's  the  i.e.,  b y d e f i n i t i o n o f c o n f i r m a t i o n , B3 c o n f i r m s H5»  Finally  is  H5»  ( x ) ( P x v Qx) ||-(Pc v Q c ) .  (5)  B3  have.  i d e a of  Hence,  "entailment"  c a n be s w i t c h e d t o  is its  the  although in  terms  corres-  to  17 ponding s y n t a c t i c  consequence r e l a t i o n any time we  Furthermore, i n t h i s and again, and i t i s e a s i e r  dissertation  like.  we g i v e examples  again  to do t h i s i n s e m a n t i c a l terms and,  hence, i n terms of the semantic entailment  relation.  18  6.  THE PARADOXES OF CONFIRMATION  One n i c e ,  neat r e s u l t  confirmation is respect three  that  it  divides  t o any c o n s i s t e n t  mutually  disconfirmed  there are a l s o  and  qualitative  hypotheses i n t o one o f  those which are those  which  are  F o r example,  the  confirmed, neutral."*"  some c o u n t e r - i n t u i t i v e r e s u l t s theory.  with  that  from the  Hypothesis,  RH» or,  possible  observation report  c a n be d e r i v e d f r o m H e m p e l * s Raven  all  exclusive classes:  those which are However,  of Hempel*s t h e o r y of  All in  RH:  ravens  are  black,  symbols, (x)(Rx=>Bx), where,  we c a n d e r i v e t h e f o l l o w i n g  results:  (A)  Any b l a c k  (B)  Any n o n - b l a c k  raven disconfirms  (C)  Any n o n - b l a c k  non-raven confirms  (D)  Anything that  is  Raven H y p o t h e s i s ; (D*)  Any b l a c k  raven confirms  and,  in  black  t h e Raven  Hypothesis;  t h e Raven  Hypothesis;  t h e Raven  Hypothesis;  or a non-raven a l s o confirms  particular,  non-raven confirms  t h e Raven  Hypothesis.  Proof. First,  let's a: b: ci di d*:  Second, 1.  Cf.  i n t r o d u c e the f o l l o w i n g a black raven, a non-black raven, a non-black non-raven, an o b j e c t t h a t i s b l a c k a black non-raven.  names:  or a non-raven,  n o t e t h a t we have t h e f o l l o w i n g  Hempel:Aspects  of  relations:  Scientific Explanation,  p.3?T  the  19 (i)  ( R a . B a ) ||- ( R a o B a ) ;  (ii)  (Rb.-Bb) ||--(Rb^Bb);  (iii)  ( - R c . - B c ) ||-(Rc r>Bc);  (iv)  ( - R d v B d ) ({-(Rd^Bd); a n d ,  (v)  (-Rd*.Bd*)  From t h e above definitions  of  respectively  five  particular,  PBd*).  results  w i t h the h e l p of  Hempel's  c o n f i r m a t i o n and  disconfirmation,  we  our d e s i r e d r e s u l t s  (D*).  of  (A),  (B),  (C),  obtain (D)  and  'QED.  The r e s u l t s (C),  II-(Rd*  in  (D)  and  How c o u l d ,  (A)  (D*)  one  and are  might  (B) not-—  ask,  a black  hole,  nothing  t o do w i t h r a v e n s ,  seemingly  are expected; but at  least  a white  or a red h e r r i n g ,  not  swan,  the  expected by  a blue  to  (and  car,  have  t h e Raven H y p o t h e s i s ?  counter-intuitive results  many.  eye, a pink  e t c . , which appear  confirm  results  These  some r e l a t e d o n e s )  are  2 called  "the paradoxes  Hempel's  calling  r e l a t e d problems has about  them.  a t t e n t i o n t o t h e p a r a d o x e s o f c o n f i r m a t i o n and p r o v o k e d an e x t e n s i v e  Some t r y t o  a few t r y t o d i s s o l v e try  to review,  t h e most  of c o n f i r m a t i o n " by Hempel.  solve  them.  diagnose,  In  the paradoxes  the paradoxes  2.  Ibid..  of  pp.l^ff.  of  and  literature  confirmation;  t h e n e x t few s e c t i o n s  and p e r h a p s  i n t e r e s t i n g and i m p o r t a n t  of  study  confirmation.  also  criticize,  solutions  and  we  will  some  of  dissolutions  20 7. MAX  BLACK'S DIAGNOSIS OF THE  Perhaps i t i s w i s e r  and more  PARADOXES OF CONFIRMATION d e s i r a b l e to f i r s t  the paradoxes of c o n f i r m a t i o n , b e f o r e coming to any dissolution.  With regard to a  c o n f i r m a t i o n , i t seems to me nent t h i n g s to say.  diagnosis  diagnose  s o l u t i o n or  of the paradoxes of  t h a t Max B l a c k has the most p e r t i -  Thus i n t h i s s e c t i o n we  to some of h i s more important  devote o u r s e l v e s  ideas.  Consider, again, the raven h y p o t h e s i s t h a t a l l ravens black.  are  A c c o r d i n g to Black common sense would h o l d t h a t i  ( i ) The the raven  e x i s t e n c e of a b l a c k raven supports  (or c o n f i r m s )  hypothesis;  ( i i ) The  raven h y p o t h e s i s i s shown to be f a l s e  firmed) by the e x i s t e n c e of a s i n g l e non-black  (or d i s c o n -  raven;  ( i i i ) Not a l l o b j e c t s bear upon the raven h y p o t h e s i s ,  posi-  t i v e l y or n e g a t i v e l y , i n t h i s way* f o r i n s t a n c e , the e x i s t e n c e of H a l l e y ' s comet n e i t h e r supports confirms) the raven  (confirms) nor f a l s i f i e s  (dis-  hypothesis.  In s h o r t , a c c o r d i n g to B l a c k "the common sense p o s i t i o n i s t h a t the e x i s t e n c e of some, but not a l l , the raven h y p o t h e s i s . " i t e d r e l e v a n c e " by  1  This i s c a l l e d  t h i n g s i s r e l e v a n t to "the p r i n c i p l e of l i m -  Black.  Thus i t f o l l o w s from B l a c k ' s terminology and d i s c u s s i o n t h a t Nicod's  c r i t e r i o n speaks f o r the common sense p o s i t i o n and  or advocates  the p r i n c i p l e of l i m i t e d  holds  relevance.  Moreover, there are three d i f f e r e n t types of the  principle  of l i m i t e d r e l e v a n c e t h a t can be d i f f e r e n t i a t e d , I t h i n k , when 1. Max B l a c k t "Notes on the "paradoxes of c o n f i r m a t i o n " " i n Asp e c t s of I n d u c t i v e L o g i c , (ed.) J . H i n t i k k a and P. Suppes, 1966, p.175. 2. I b i d . , p.175. i,  21 it  applies  are  to  raven  hypothesis!  (iii.a)  B o t h a n o n - b l a c k n o n - r a v e n and a b l a c k  neutral  to  (iii.b)  Any n o n - b l a c k n o n - r a v e n i s  pothesis;  the  while  (iii.c) sis;  the  while  raven  hypothesis.  above t h r e e one.  limited  Nicod's  relevance  simply c a l l e d for  On t h e  of  "the  other  to  also  the  first  type"; of  limited  and  following  raven  hy-  raven  hypothe-  is  the  "the  the  inter-  principle  other  relevance  most  of  two w i l l  of  be  non-Nicod's  importance.  from Hempel's  (i)  the  one  while  the  it.  w i l l be c a l l e d  of minor  hand,  besides  It  principle  they are  can d e r i v e , thesis,  neutral  types the  to  it.  a non-black non-raven confirms  e s t i n g and i m p o r t a n t  types",  neutral  a black non-raven confirms  Any b l a c k n o n - r a v e n i s  Among t h e  non-raven  (ii),  theory  of confirmation  when a p p l i e d  to  the  raven  we hy-  results.  (iv.a)  Any n o n - b l a c k  non-raven confirms  (iv.b)  Any b l a c k n o n - r a v e n a l s o c o n f i r m s  the  raven  hypothe-  sis. the  raven  hypothe-  sis. In short, the  universe  i n Hempel's either  theory  confirms  of  confirmation  or e l s e  disconfirms  any o b j e c t the  raven  in hy-  3 pothesis; nothing  is  Hempel's ory of  no o b j e c t  theory  4.  of  confirmation  relevance", 3.  neutral,  is  neutral  to  the  or i r r e l e v a n t ,  to  raven hypothesis. the  c o n f i r m a t i o n , we w o u l d holds  Since  raven hypothesis say  that Hempel's  or advocates a " p r i n c i p l e  of  in the-  universal  a l a Max B l a c k .  Here a c o m p l e t e i n f o r m a t i o n about h y p o t h e s i s i s assumed. I b i d . , p.176.  the  object  relative  to  the  In the f o l l o w i n g we w i l l t r y t o d i s c u s s some more important and  r e p r e s e n t a t i v e treatments, s o l u t i o n s o r d i s s o l u t i o n s of the  paradoxes of c o n f i r m a t i o n : and we w i l l t r y t o d i s c u s s them, i f p o s s i b l e , i n the l i g h t relevance  and u n i v e r s a l  of Max Black's relevance.  two p r i n c i p l e s o f l i m i t e d  23  8.  VON WRIGHT'S TREATMENT OF THE PARADOXES OF CONFIRMATION  Von Wright has two i n t e r e s t i n g t h i n g s t o say concerning the paradoxes of c o n f i r m a t i o n i (I)  The problem of " r e l e v a n t range".  A c c o r d i n g t o von Wright  before we t r y to c o n f i r m or d i s c o n f i r m any u n i v e r s a l hypothesis such as the raven hypothesis, we have i n the f i r s t the f o l l o w i n g q u e s t i o n i of the h y p o t h e s i s ?  p l a c e t o ask  What i s the r e l e v a n t range of a p p l i c a t i o n  Thus, i f the r e l e v a n t range of a p p l i c a t i o n of  the raven hypothesis  i s a l l o b j e c t s i n the world,  then Hempel's  d i s s o l u t i o n of the paradoxes of c o n f i r m a t i o n i s the c o r r e c t ones and,  i f the r e l e v a n t range of a p p l i c a t i o n o f the raven  i s o n l y about ravens i n the world, ven h y p o t h e s i s  hypothesis  then Nicod's answer to the r a -  i s the c o r r e c t one.  And i n between, there are o-  t h e r p o s s i b l e s o l u t i o n s i f o r i n s t a n c e , the r e l e v a n t range o f app l i c a t i o n of the raven hypothesis other animals,  etc.  c o u l d be about b i r d s , or about  Which one i s accepted  as the c o r r e c t s o l u -  t i o n t o the raven hypothesis w i l l depend upon which r e l e v a n t range of the hypothesis Thus the f i r s t  i s selected.  important  t h i n g f o r von Wright to say about  the c o n f i r m a t i o n o r d i s c o n f i r m a t i o n o f a u n i v e r s a l g e n e r a l i z a t i o n as a hypothesis  i s the f o l l o w i n g t h e s i s i^"  Von Wright's T h e s i s of Range of Relevance. A l l things i n the range o f r e l e v a n c e o f a g e n e r a l i z a t i o n may c o n s t i t u t e genuine c o n f i r m a t i o n s or d i s c o n f i r m a t i o n s o f the generalization. The t h i n g s o u t s i d e the range are i r r e l e v a n t to the g e n e r a l i z a t i o n . They cannot c o n f i r m i t g e n u i n e l y .  1.  See von Wrighti "The paradoxes o f c o n f i r m a t i o n " , i n Aspects of I n d u c t i v e L o g i c , (ed.) J . H i n t i k k a and P. Suppes, p.211.  2k The above bility  in his  a l s o has (II) Wright,  the  thesis  o f v o n W r i g h t ' s g i v e s him a c e r t a i n  treatment following  The p r o b l e m o f  of  the  second  paradoxes  of c o n f i r m a t i o n .  interesting  "natural range".  t h i n g to  flexiHe  sayi  A c c o r d i n g to von  2  "When t h e r a n g e o f r e l e v a n c e o f a g e n e r a l i z a t i o n o f t h e type t h a t a l l A are B i s not s p e c i f i e d , then the range i s . . . u s u a l l y u n d e r s t o o d t o be t h e c l a s s o f t h i n g s w h i c h f a l l u n d e r t h e a n t e c e d e n t t e r m A . The g e n e r a l i z a t i o n t h a t a l l r a v e n s a r e b l a c k , range b e i n g u n s p e c i f i e d , would n o r m a l l y be u n d e r s t o o d t o be a g e n e r a l i z a t i o n a b o u t r a v e n s and n o t a b o u t b i r d s o r a b o u t a n i m a l s o r a b o u t e v e r y t h i n g there i s . " Thus,  von Wright c a l l s the c l a s s  range  of  relevance"  With the  above  W r i g h t comes t o  of  the  of  things  that  generalization that  d e f i n i t i o n of n a t u r a l range  the  following  a r e A "the n a t u r a l a l l A are B. of relevance von  conclusion*-^  "Within the n a t u r a l range of r e l e v a n c e of a g e n e r a l i z a t i o n the c l a s s of g e n u i n e l y c o n f i r m i n g i n s t a n c e s i s determined by N i c o d ' s C r i t e r i o n . " This to  Max B l a c k ' s p r i n c i p l e o f l i m i t e d r e l e v a n c e  This  identification  paradoxes of  second c o n c l u s i o n o f von W r i g h t ' s i s ,  of  l i m i t e d relevance "the  hypothesis.  See See  ibid. ibid,  the  c o n f i r m a t i o n i n the  posed n o t i o n of  2. 3.  increases  p.2l6. p.21?.  thus,  identical  of N i c o d ' s  p l a u s i b i l i t y of s o l v i n g light  type. the  o f Max B l a c k ' s p r i n c i p l e  of N i c o d ' s type,  b y way o f v o n W r i g h t * s p r o -  n a t u r a l range  relevance"  of  of  th»  raven  However, a c c o r d i n g to Hempel there are a few f a c i n g von  Wright's treatment  difficulties  o f the paradoxes of c o n f i r m a t i o n  by c o n s i d e r a t i o n of the n a t u r a l range (or any r e s t r i c t e d range) of r e l e v a n c e .  The  difficulties  are.  (a) A c c o r d i n g to Hempel s c i e n c e i n v o l v e s no cation"  " f i e l d of a p p l i  (such as the " n a t u r a l range of r e l e v a n c e " ) , which would  have to be a r b i t r a r y a t b e s t .  For Hempel t h i n k s .  "the way i n which general hypotheses are used i n s c i e n c e never i n v o l v e s the statement of a f i e l d of a p p l i c a t i o n } and the c h o i c e of the l a t t e r i n a symbolic f o r m u l a t i o n of a g i v e n hypothesis thus i n t r o d u c e s ... a c o n s i d e r a b l e measure of a r b i t r a r i n e s s . In p a r t i c u l a r , f o r a s c i e n t i f i c hypothesis to the e f f e c t t h a t a l l P's are Q's, the f i e l d of a p p l i c a t i o n cannot simply be s a i d to be the c l a s s of a l l P's." ( m y  i  t  a  l  i  c  s  )  (b) R e s t r i c t i o n of range of r e l e v a n c e i s , a c c o r d i n g to Herapel , a c o n f u s i o n of l o g i c a l and p r a c t i c a l c o n s i d e r a t i o n s . Thus Hempel has the f o l l o w i n g to say:-* "the view ... t h a t a hypothesis of the simple form 'Every P i s a Q' a s s e r t s something about a l i m i t e d c l a s s of o b j e c t s only, namely, the c l a s s of a l l P's ... i n v o l v e s a c o n f u s i o n of l o g i c a l and p r a c t i c a l c o n s i d e r a t i o n s : Our i n t e r e s t i n the hypothesis may be focussed upon i t s a p p l i c a b i l i t y to t h a t p a r t i c u l a r c l a s s of o b j e c t s , but the hypo t h e s i s n e v e r t h e l e s s a s s e r t s something about, and indeed imposes r e s t r i c t i o n s upon, a l l o b j e c t s . " (c) The  i d e a of "a f i e l d of a p p l i c a t i o n " of a  hypothesis  cannot be d e f i n e d i n terms of syntax; b e s i d e s , the i d e a a l s o v i o l a t e s the equivalence c o n d i t i o n . hoc d e v i c e .  Hence, i t i s only an ad  These c r i t i c i s m s are made by Hempel e x p l i c i t l y or  i m p l i c i t l y i n the f o l l o w i n g paragraph:^ 4. 5. 6.  See Hempel: Aspects See i b i d , p.18. I b i d , p.18.  of S c i e n t i f i c E x p l a n a t i o n , pp.17-18.  26  "the c o n s i s t e n t use of a f i e l d of a p p l i c a t i o n i n the f o r m u l a t i o n of g e n e r a l hypotheses would i n v o l v e cons i d e r a b l e l o g i c a l c o m p l i c a t i o n s , and yet would have no c o u n t e r p a r t i n the t h e o r e t i c a l procedure of s c i ence where hypotheses are s u b j e c t e d to v a r i o u s kinds of l o g i c a l t r a n s f o r m a t i o n and i n f e r e n c e without any c o n s i d e r a t i o n t h a t might be regarded as r e f e r r i n g to changes i n the f i e l d of a p p l i c a t i o n . T h i s method of meeting the paradoxes would t h e r e f o r e amount to dodging the problem by means of an ad hoc d e v i c e which cannot be j u s t i f i e d by r e f e r e n c e to a c t u a l s c i e n t i f i c procedure." (my i t a l i c s )  27  9.  THE  The  BAYESIAN TREATMENT OF THE  Bayesian  treatment  "astonishingly popular",or as p o i n t e d out by Max denbaum,  2  J.K.  Mackie,  CONFIRMATION  of the paradoxes of c o n f i r m a t i o n i s a t l e a s t i t was  Black. 3  PARADOXES OF  1  f o r a p e r i o d of  I t i s represented  P a t r i c k Suppes,  4  by  time,  Hosiasson-Lin-  Paul Horwich,  ^  and  some  others. L e t us take Mackie's approach as an example. Mackie proposes, to b e g i n with,  the f o l l o w i n g p r i n c i p l e i  The Inverse P r i n c i p l e . A h y p o t h e s i s H i s confirmed by an o b s e r v a t i o n r e p o r t B i n r e l a t i o n t o background knowledge K i f and o n l y i f the o b s e r v a t i o n r e p o r t i s made more probable by adding the h y p o t h e s i s to the background knowledge. Then, he a v a i l s h i m s e l f of the background knowledge t h a t are l e s s ravens (and,  (and,  there  r e s p e c t i v e l y , b l a c k t h i n g s ) than non-ravens  r e s p e c t i v e l y , non-black t h i n g s ) , and  f i c a n t p r i o r p r o b a b i l i t y t o those  thus a s s i g n s  insigni-  t h i n g s under the c a t e g o r i e s of  non-black non-ravens as w e l l as b l a c k non-ravens such t h a t i t t u r n s out t h a t the t h i n g s under these r e l e v a n t to the  two  c a t e g o r i e s become i r -  hypothesis.  In t h i s approach the common s t r a t e g y i s not to r e j e c t paradoxes of c o n f i r m a t i o n , but  to render  the  them to be harmless, t o  d i s s o l v e them r a t h e r than t o s o l v e them. 1.  See Max B l a c k t "Notes on the •Paradoxes of C o n f i r m a t i o n ' " , i n Aspects of I n d u c t i v e L o g i c , (ed.) J . H i n t i k k a and P. Suppes, p . 1 9 5 . . 2. Hosiasson-Lindenbaum, J« "On c o n f i r m a t i o n " , the J o u r n a l of Symbolic L o g i c , 19^0, pp.133-168. 3. J.K. Mackie» "The paradox of c o n f i r m a t i o n " , B r i t i s h J o u r n a l f o r the P h i l o s o p h y of S c i e n c e , 1963, p p . 2 6 3 - 2 7 7 . 4. P. Suppest "A Bayesian approach to the paradoxes of c o n f i r m a t i o n " , i n Aspects of I n d u c t i v e L o g i c , (ed.) J . H i n t i k k a and P. Suppes, pp.198-207. 5. P. Horwichi P r o b a b i l i t y and Evidence, 1982.  28  Hempel has a c r i t i c i s m o f t h i s approach. about the numerical  He r a i s e s doubts  assumption f o r the p r i o r p r o b a b i l i t i e s a s -  signed by the Bayesians,  f o r he says*^  The answer depends i n p a r t upon the l o g i c a l s t r u c t u r e o f the .language o f s c i e n c e . I f a " c o o r d i n a t e language" i s used, i n which, say, f i n i t e space-time r e g i o n s f i g u r e as i n d i v i d u a l s , then the raven h y p o t h e s i s assumes some such form as 'Every space-time r e g i o n which c o n t a i n s a raven c o n t a i n s something b l a c k ' s and even i f the t o t a l number of ravens ever t o e x i s t i s f i n i t e , the c l a s s o f spacetime r e g i o n s c o n t a i n i n g a raven has the power o f the continuCu)m, and so does the c l a s s o f space-time r e g i o n s c o n t a i n i n g something nonblackj thus, f o r a c o o r d i n a t e language of the type under c o n s i d e r a t i o n . t h e above numeric a l assumption i s not warranted. Now the use o f a coord i n a t e language may appear q u i t e a r t i f i c i a l i n t h i s part i c u l a r i l l u s t r a t i o n j b u t i t may seem v e r y a p p r o p r i a t e i n many other c o n t e x t s , such as, e.g.,that o f p h y s i c a l f i e l d theories. On the o t h e r hand, on the b a s i s o f a " t h i n g language", Hempel t h i n k s t h a t the numerical here,  assumption may be j u s t i f i e d .  But even  i t remains an e m p i r i c a l q u e s t i o n .  Besides the above c r i t i c i s m made by Hempel, I t h i n k t h a t the Bayesian 1) theory?  approach a l s o c o n t a i n s the f o l l o w i n g two d i f f i c u l t i e s !  How to i n c o r p o r a t e the "background knowledge" i n t o Hempel's For Hempel r e q u i r e s t h a t any c o n f i r m a t i o n theory must be  expressible,  i n the f i r s t p l a c e , i n terms o f syntax, where there  i s no room f o r any background knowledge t o be b u i l t i n . 2)  I t seems t h a t the Bayesians  never say whether the paradoxes  of c o n f i r m a t i o n are a genuine d i f f i c u l t y o r no^fc. worked out a theory o f p r o b a b i l i t y ; but, r e a l l y have a theory o f c o n f i r m a t i o n . Max 6.  They may have  i n f a c t , they do n o t  Perhaps t h i s i s what makes  Black say that, See Hempel: Aspects note 2 5 .  o f S c i e n t i f i c E x p a l n a t i o n , pp.20-21, f o o t -  29  "On the whole, the Bayesian approach seems to me wrong i n p r i n c i p l e and i n e f f e c t i v e i n p r a c t i c e . " ' (my  italics)  7 . Max B l a c k : "Notes on the 'Paradoxes of C o n f i r m a t i o n ' " , i n A s ^ p e c t s of I n d u c t i v e L o g i c , (ed.) J . H i n t i k k a and P. Suppes, p.197.  30  QUINE'S SOLUTION OF THE PARADOXES OF CONFIRMATION  10.  Quine's described (I)  solution  as  of  the  paradoxes  of  c o n f i r m a t i o n c a n be  follows:  First,  we need  a new  conditions  The P r o . j e c t i b i l i t y C o n d i t i o n . Every p r e d i c a t e of a hypot h e s i s must be " p r o j e c t i b l e " i n o r d e r f o r t h e h y p o t h e s i s t o have a p o s i t i v e (or n e g a t i v e ) i n s t a n c e ; i n other words, the c o n f i r m a t i o n (or d i s c o n f i r m a t i o n ) of a hypothesis presupposes that every p r e d i c a t e of the h y p o t h e s i s i s i n the f i r s t p l a c e " p r o j e c t i b l e " , where " p r o j e c t i b l e " p r e d i c a t e s a r e p r e d i c a t e s P and Q whose s h a r e d i n s t a n c e s a l l do c o u n t , f o r w h a t e v e r r e a s o n , t o w a r d c o n f i r m a t i o n of " a l l P are Q". (II) are "...  Second,the  projectible, is  predicates  while  the  " . . . is  predicates  non-black" are not  a r a v e n " and " . . . i s " . . . is  projectible,  at  black"  a n o n - r a v e n " and  least  according  to  Quine. (III)  It  follows  that  we  have:  1. Any b l a c k r a v e n c o n f i r m s  the  raven  2.  Any n o n - b l a c k r a v e n d i s c o n f i r m s  3.  Any b l a c k n o n - r a v e n i s  4.  Any n o n - b l a c k n o n - r a v e n i s  neutral  hypothesis.  the to  also  raven  the  hypothesis.  raven  neutral  to  hypothesis. the  raven hypo-  thesis. Quine has  thus  "solved"  Nov/ we would l i k e lution: and  to  general,  what  is  the  Q u i n e ' s answers  it  A l l ravens is  of  confirmation.  some q u e s t i o n s c o n c e r n i n g h i s  black" are p r o j e c t i b l e  a n o n - r a v e n " and " . . . i s  but  raise  paradoxes  Why does Q u i n e t h i n k t h a t  " . . . is  (a)  the  the  predicates  while  non-black" are  c r i t e r i o n of  the not  " . . . is  predicates  a raven" "...  projectible?  projectibility  of  a  so-  is  And,  in  predicate?  are: are  not true that  of  the  same k i n d ,  a l l non-ravens  are  so of  are a l l b l a c k the  same k i n d ,  things; nor  is  31  i t the case t h a t non-black t h i n g s are of the same k i n d , and o n l y things  ( o r p r o p e r t i e s ) o f the same k i n d are p r o j e c t i b l e .  1  (b) "A p r o j e c t i b l e p r e d i c a t e i s one t h a t i s t r u e o f a l l and 2  o n l y the t h i n g s o f a k i n d . " (c) Moreover, p r o j e c t i b i l i t y , n a t u r a l k i n d , s i m i l a r i t y and s i m p l i c i t y are, a t l e a s t t o Quine, a l l o f a p i e c e . (d) And these . 4 jective.  concepts a r e i n t u i t i v e l y c l e a r , however sub-  (e) We need these e s p e c i a l l y simple nothing  J  concepts because we want t o do i n d u c t i o n ,  and s u c c e s s f u l i n d u c t i o n , and " i n i n d u c t i o n  succeeds l i k e  success."-*  Here a r e some comments on Quine*s s o l u t i o n o f the paradoxes of  confirmation! 1)  Quine's s o l u t i o n o f the paradoxes o f c o n f i r m a t i o n has some  p l a u s i b i l i t y on the one hand and on the other hand the concepts i n v o l v e d i n h i s s o l u t i o n such as " p r o j e c t i b i l i t y " , "similarity",  " s i m p l i c i t y " are s t i l l  Quine h i m s e l f  admits.  2) valence  "natural kind",  too vague o r " s u b j e c t i v e " as  Quine's P r o j e c t i b i l i t y C o n d i t i o n v i o l a t e s Hempel*s e q u i condition.  I n consequence, i t a l s o v i o l a t e s Hempel*s  s a t i s f a c t i o n c r i t e r i o n of confirmation. i s h i s c r i t e r i o n of c o n f i r m a t i o n ?  So, I would a s k i What  And the answer i s i Quine does  not r e a l l y have a theory o f c o n f i r m a t i o n J u e t as he h i m s e l f ex1. C f . Quine: " N a t u r a l k i n d s " , i n h i s O n t o l o g i c a l R e l a t i v i t y and Other Essays. 19&9* 2. I b i d , p.116. 3. C f . Quine: "Reply t o Chihara", i n Midwest S t u d i e s of Philosophy, 1981, p p . 4 5 3 - 4 5 4 . 4 . See Quine: " N a t u r a l k i n d s " , i n h i s O n t o l o g i c a l R e l a t i v i t y and Other Essays, p . l l 6 . 5. I b i d , p.129.  32  plainsi I f o l l o w t h e crowd i n c e l e b r a t i n g what i s l o o s e l y d e s c r i b e d as the h y p o t h e t i c o - d e d u c t i v e method. One s e e k s a set of hypotheses that w i l l j o i n t l y i m p l y t h e d a t a , and one c h o o s e s among s u c h s e t s w i t h a n eye t o s i m p l i c i t y and c o n s e r v a t i s m and p e r h a p s o t h e r v i r t u e s . Successful pred i c t i o n c o n f i r m s , showing as i t does the c o n t i n u i n g c o v e r a g e o f new d a t a . So Q u i n e ' s " s o l u t i o n " ignore  them,  i.e.  to  offer  to  the  paradoxes  no s o l u t i o n a t  of  confirmation is  all.  A n odd  response  by a p h i l o s o p h e r .  6.  Quine. "Reply to C h i h a r a " , 1971. p p . 4 5 3 - ^ 5 ^ .  Midwest S t u d i e s  of  to  Philosophy,  1 1 . GOODMAN'S AND S C H E F F L E R ' S  CONCEPT OF S E L E C T I V E CONFIRMATION  AND THE PARADOXES OF CONFIRMATION, AND GRANDY'S RELATED DISCUSSION Concerning the paradoxes  o f c o n f i r m a t i o n and t h a t a g i v e n o b -  j e c t which i s n e i t h e r b l a c k nor a raven can c o n f i r m the raven hypothesis,  Goodman h a s t h i s  i n t e r e s t i n g and r e m a r k a b l e t h i n g t o say:"*"  The p r o s p e c t o f b e i n g a b l e t o i n v e s t i g a t e t h e o r i e s without going out i n the r a i n i s t h a t we know t h e r e must be a c a t c h i n i t . So what i s  ornithological so a t t r a c t i v e  t h e c a t c h meant b y Goodman?  To see t h e c a t c h , l e t u s c o n s i d e r a g a i n t h e r a v e n (1)  (X)(RXPBX).  Now s u p p o s e (2)  t h a t we have t h e o b s e r v a t i o n r e p o r t i  -Rc.Bc,  where £ i s a n y b l a c k  non-raven.  Then, by Hempel's But  d e f i n i t i o n , we have t h a t  (2) c o n f i r m s a l s o t h e f o l l o w i n g  w h i c h i s , a c c o r d i n g t o Goodman, So,  we h a v e t h a t  contrary.  (1).  hypothesis*  of (1).  (2) c o n f i r m s b o t h t h e r a v e n h y p o t h e s i s and  t h a t a n a d e q u a t e c o n c e p t o f c o n f i r m a t i o n must  enable the instances contrary.  a "contrary"  T h i s i s t h e c a t c h meant b y Goodman.  Goodman t h i n k s  its  (2) c o n f i r m s  (x)(Rx3-Bx),  (3)  its  hypothesis*  of a hypothesis  T h i s i d e a o f Goodman's  Preference Condition".  to p r e f e r the hypothesis c a n be c a l l e d  Israel ScheffIer notices  to  "(Goodman's)  i t s i m p o r t a n c e and  1. Goodman, N e l s o n : F a c t , F i c t i o n , and F o r e c a s t , 3 r d e d . , 1 9 5 5 , p.70. 2. T h u s , a c c o r d i n g t o Goodman, two s e n t e n c e s a r e " c o n t r a r i e s " i f f i n a n y n o n - e m p t y domain t h e y may be b o t h t r u e b u t c a n n o t be b o t h false. Note t h i s i s n o t t h e t r a d i t i o n a l d e f i n i t i o n , w h i c h w o u l d n o t h e l p i n Goodman's and S c h e f f l e r ' s t h e o r y o f s e l e c t i v e c o n f i r m a t i o n t o solve the paradoxes of c o n f i r m a t i o n . We s h a l l emp l o y Goodman's s e n s e o f " c o n t r a r y " h e r e a f t e r .  34  develops i n t o a new concept of i s d e f i n e d as  "selective confirmation",  which  follows:^  The d e f i n i t i o n  of s e l e c t i v e c o n f i r m a t i o n .  report B s e l e c t i v e l y  An  c o n f i r m s a h y p o t h e s i s H,  and d i s c o n f i r m s the c o n t r a r y  of H,  if  observation  3 confirms H  (where both c o n f i r m a t i o n and  d i s c o n f i r m a t i o n are to be understood i n the Hempelian s e n s e s . ) Once we have t h i s new concept of s e l e c t i v e  confirmation,  the paradoxes of c o n f i r m a t i o n and "the p r o s p e c t s of indoor nithology  vanish".  To see t h i s , (i) sis.  or-  we have to c o n s i d e r the f o l l o w i n g  Any b l a c k raven s e l e c t i v e l y This  four  c o n f i r m s the raven  casest hypothe-  i s because we have: Ra • Ba  |r (Ra o3a),  and  Ra • 3 a ||- - (Ra = » - B a ) , where a i s any b l a c k r a v e n .  Thus we have any b l a c k raven c o n -  f i r m s the raven h y p o t h e s i s and d i s c o n f i r m s i t s by d e f i n i t i o n  any b l a c k raven s e l e c t i v e l y  contrary.  c o n f i r m s the  So, raven  hypothesis. (ii) This  Any n o n - b l a c k raven d i s c o n f i r m s the raven  hypothesis.  i s because we have: Rb« -Bb f|- -(Rb P B b ) ,  where b i s any n o n - b l a c k r a v e n . (iii)  Any b l a c k non-raven does not s e l e c t i v e l y  the raven h y p o t h e s i s . -Re* but we do not  confirm  T h i s i s because we have:  3 c \Y (Rc =>3c),  have: -Rc-  Be / - ( R c =>-Be), L  3 . C f . I s r a e l S c h e f f l e r : Anatomy of I n q u i r y , 4 . See Goodman: F a c t , F i c t i o n , and F o r e c a s t ,  I963. p p . 2 8 6 - 2 9 1 . 3rd, p.71.  35  where c_ i s any b l a c k non-raven. ( i v ) Any non-black non-raven does not s e l e c t i v e l y the raven h y p o t h e s i s .  T h i s i s becausei  -Rd.-Bd  although  confirm  we have:  IKRdoBd),  we do not havei -Rd.-Bd \[ -(RdD -Bd), where d i s any non-black non-raven. Although s e l e c t i v e c o n f i r m a t i o n c a n a v o i d the paradoxes of c o n f i r m a t i o n , Grandy o f f e r s three important First, tion.  objections:^  s e l e c t i v e c o n f i r m a t i o n r e j e c t s the e q u i v a l e n c e  condi-  Perhaps s e l e c t i v e c o n f i r m a t i o n can supply a reason f o r  r e j e c t i n g the equivalence h i g h t o pay f o r i t ?  condition}  but i s n ' t i t a p r i c e too  F o r , then, we would have no more r i g h t t o  make t r a n s f o r m a t i o n o f e q u i v a l e n t statements i n theory and i n p r a c t i c e without  paying a t t e n t i o n t o t h e i r forms.  Second, s e l e c t i v e c o n f i r m a t i o n cannot a p p l y t o any hypothes i s which does n o t have the form o f a g e n e r a l i z e d c o n d i t i o n a l . Thus i t i s not a p p l i c a b l e t o a h y p o t h e s i s  o f the form:  (x)Gx, which has no c o n t r a r y a t a l l . T h i r d , some p e r f e c t l y o r d i n a r y hypotheses cannot be s e l e c t i v e l y confirmed.  T h i s i s odd.  F o r i n s t a n c e , the f o l l o w i n g  hypothesis, (4)  (x)(Rx.-Bx=>Rx.-Rx),  which i s l o g i c a l l y e q u i v a l e n t t o the raven h y p o t h e s i s , and which has 5,  R i c h a r d E . Grandy: "Some comments on c o n f i r m a t i o n and s e l e c t i v e c o n f i r m a t i o n " , i n Readings i n the P h i l o s o p h y o f Science, (ed.) Baruch A. Brody, 1 9 7 0 , pp.428-432.  36 the f o l l o w i n g  contrary,  (x) ((Rx.-Bx)=> - ( R x . - R x ) ) ,  (5)  c a n n e v e r be s e l e c t i v e l y c o n f i r m e d , b e c a u s e  (5)  is a valid  sen-  t e n c e and c a n n e v e r be d i s c o n f i r m e d . Grandy's  three c r i t i c i s m s  m a t i o n c a n be f u r t h e r (a)  Grandy's  selective It  e l u c i d a t e d as  first  criticism  confirmation violates  has a t l e a s t (a.l)  to  of the theory of s e l e c t i v e  the f o l l o w i n g  T h i s would  still  confirmation  equivalence  out t h a t  Hempel's  The t h e o r y o f s e l e c t i v e  r e j e c t Hempel's  follows»  points  two  the theory of  equivalence  condition.  significances! c o n f i r m a t i o n may o f f e r a  c o n d i t i o n . But i s  remain a c o n t r o v e r s i a l i s s u e ,  of a universal  confir-  reason  i t a good r e a s o n ?  f o r the s e l e c t i v e  conditional hypothesis  such as the  f o l l o w i n g one, H:  (x)(Fx  means CH:  is  (i)  H i s c o n f i r m e d and ( i i )  disconfirmed,  o f H,  and t h i s ,  i n t u r n , means  that:  -(x)(Fx3-Gx), confirmed?  H**:  and H* i s l o g i c a l l y  to s e l e c t i v e l y confirm H i s to confirm H & H * * ;  s e l e c t i v e l y confirm the following Ki  equivalent to:  (3x)(Fx.Gx). Hence,  is  the contrary  (X)(PXD-GX),  H*: is  that  3Gx),  hypothesis,  (x)(-GxO-Fx), t o c o n f i r m b o t h K and t h e f o l l o w i n g  K**i  (3x)(-Gx.-Fx).  hypothesis!  and t o  37 T h u s w h e n we s a y does not HAH**  preserve  is  not  logically  can  Hempel's  logically  equivalent what  equally well-represented that  to  controversial  it  selective  confirmation  c o n d i t i o n we m e a n K&K**,  though  that  H is  s t i l l  K.  be  think  of  equivalence  equivalent  to  is  theory  short,  cant  but  is  whether  " a l l Fs are  Gs"  b y H o r K?  Hempel (and most  Goodman a n d  Scheffler  think  l o that  cannot. (a.2)  The f o r m o f  firmation  of  Whether in  the  In  gocians) it  that  a  sial.  form  i f  formations  of  of  practice  Anyway i t  practice  becomes  crucial  i n the  con-  such a c r u c i a l  role  hypothesis.  the  scientific  a hypothesis  a hypothesis is  would  it  is  dubious  create  s t i l l  equivalent  plays  or,  great  at  least,  again,  inconvenience  in  controverscientific  w o r k a b l e - ^ - e s p e c i a l l y i n the  sentences  and  hypotheses  and  trans-  also  in  prediction. (b) of  Grandy's  selective  which  could  Note  CK*: which K*i Since  c r i t i c i s m c l e a r l y means  confirmation is not  that  following  second  be  inadequate,  selectively  any h y p o t h e s i s  for  that  there  the  are  theory hypotheses  confirmed.  K, i n c l u d i n g (x)Gx,  c a n have  the  "contrary": (x) ((Ax v - A x ) r > - K ) ,  is  the  "contrary  o f K * " , where  K* i s :  (x) ((Ax v-Ax)=> K ) . K* i s  equivalent  always to  selectively confirmable  K*, K is  in a  sense  also  and K i s  selectively  logically confirmable  38 i f we have the equivalence  c o n d i t i o n a v a i l a b l e i n the theory of  s e l e c t i v e c o n f i r m a t i o n ; but the t r o u b l e i s , o f course, equivalence  t h a t the  c o n d i t i o n does not h o l d with r e s p e c t t o the s e l e c t i v e  confirmation of a hypothesis  i n the theory o f s e l e c t i v e  confirma-  tion. (c) Grandy's t h i r d c r i t i c i s m two  criticisms.  a hypothesis  j u s t re-emphasizes h i s f i r s t  H i s example (4) shows t h a t there are forms of  t h a t cannot be s e l e c t i v e l y confirmed, though i n the  case of (4) one o f i t s e q u i v a l e n t hypotheses, which i s the Raven Hypothesis, does t u r n out t o be s e l e c t i v e l y confirmed. l e a r n the moral a g a i n t h a t the form o f a h y p o t h e s i s c i a l role i n i t s selective •»*»  *•>*•*  Hence, we  plays a cru-  confirmation. »»*  Grandy, a f t e r the above three c r i t i c i s m s of Goodman's and S c h e f f l e r ' s s e l e c t i v e c o n f i r m a t i o n , proposes the f o l l o w i n g r e l a t i v e concept of s e l e c t i v e c o n f i r m a t i o n t o a v o i d the d i f f i c u l t i e s f a c i n g Goodman's and S c h e f f l e r ' s theory of s e l e c t i v e conf irmatiom "M s e l e c t i v e l y confirms S r e l a t i v e t o K i f S i s a member of K and M confirms S and M d i s c o n f i r m s every other member of K", where S i s a h y p o t h e s i s ,  M i s an o b s e r v a t i o n sentence and K i s  a s p e c i f i e d s e t o f hypotheses, no two o f which are e q u i v a l e n t . Now a c c o r d i n g t o Grandy h i s new d e f i n i t i o n of r e l a t i v e concept o f s e l e c t i v e c o n f i r m a t i o n r e t a i n s the equivalence at  condition,  l e a s t i n the f o l l o w i n g sense *  6 . Grandy, Richard E.: "Some comments on c o n f i r m a t i o n and s e l e c t t i v e c o n f i r m a t i o n " , i n Readings i n the P h i l o s o p h y of Science, (ed.) Baruch A. Brody, 1970, p.431.  39 G r a n d y ' s sense of e q u i v a l e n c e c o n d i t i o n . ^ If S is select i v e l y c o n f i r m e d f r o m K b y M, t h e n M w i l l s e l e c t i v e l y c o n f i r m S ' f r o m K * . where S* i s a s t a t e m e n t e q u i v a l e n t t o S and where K* i s o b t a i n e d f r o m K b y r e p l a c i n g s t a t e m e n t s b y t h e i r e q u i v a lents . Indeed,  Grandy*s r e l a t i v i z e d  sense of  c a n i m p r o v e Goodman's and S c h e f f l e r ' s mation.  BI*  Thus,  for  instance,  the  selective  theory  observation  of  confirmation  selective  report,  Ga,  selectively  confirms  (6)  (x)Gx,  instead  of  (7)  (x)-Gx,  from the  the  set  the  hypothesis*  hypothesis,  ?(x)Gx,  (x)-Gx|.  My o n l y c r i t i c i s m o f G r a n d y ' s n e w l y p r o p o s e d r e l a t i v e cept  of  selective  condition is  not  t a l k i n g about. tion  is  not  confirmation is r e a l l y the  equivalence  In other words,  Hempel's  this*  his  equivalence  c o n d i t i o n can only a p p l y to  a given  set  there  is  of  no s u c h  condition.  7. I b i d ,  h y p o t h e s e s K where  p.431.  restriction  H i s sense of c o n d i t i o n that  sense o f  condition,  valence  but  confir-  the  no two  con-  equivalence we  equivalence  are condi-  for Grandy's equi-  statements mentioned of which are  i n Hempel*s s e n s e o f  in  equivalent equivalence  40 12. ARMSTRONG'S VIEW OF THE PARADOXES OF CONFIRMATION D.M. Armstrong's  view o f the paradoxes  of c o n f i r m a t i o n can  be b r i e f l y summarized as f o l l o w s i ^ " First, RH»  i f we understand the Raven Hypothesis,  A l l ravens are b l a c k ,  as HI:  (x)(RxZ>Bx),  then Armstrong  agrees w i t h Hempel and t h i n k s t h a t the paradoxes  of c o n f i r m a t i o n are genuine. Second, Armstrong  t h i n k s i n s t e a d o f HI the Raven Hypothesis  should be understood as the f o l l o w i n g h y p o t h e s i s : H2:  I t i s a law t h a t Rs are Bs,  which i s supposed  by Armstrong  t o be s t r o n g e r than the r e g u l a r i -  t y expressed i n HI. T h i r d , i f H2 i s the c o r r e c t understanding o f the Raven Hypot h e s i s , then Armstrong  argues t h a t a t l e a s t one case of the p a r a -  doxes o f conf i r m a t i o n i s genuine.  Armstrong  argues as f o l l o w s :  ( i ) Suppose t h a t we have d i r e c t evidence b e a r i n g upon an o b j e c t which i s both an R and a B then t h i s o b s e r v a t i o n would be e x p l a i n e d by the law-hypothesis H2 (and here Armstrong  agrees  w i t h Dretske t h a t c o n f i r m a t i o n i s r o u g h l y the converse of e x p l a nation.)  So, the o b s e r v a t i o n c o n f i r m s H2.  ( i i ) Again, i f we come upon an o b j e c t which i s n e i t h e r an R nor a B, then i t can s t i l l be s a i d t h a t t h i s o b s e r v a t i o n i s ex1. D.M. Armstrong: What i s a Law of Nature? t y Press, 1983, p . 4 l f f .  Cambridge U n i v e r s i -  2. F . I . Dretske: "Law of nature", P h i l o s o p h y of Science, 44, 1977.  41 p l a i n e d by H2. since  the  F o r , g i v e n the  object  (iii)  is  Finally,  not a B i t suppose  is  not an R but i s  at  a l l for saying that  pothesis that it  does n o t  it  Thus i f  of are  the the  is  this  observation is  a l a w t h a t Rs a r e B s .  of the  an o b j e c t w h i c h  So,  it  H2.  H3:  hy-  that  H e r e we  paradoxes of c o n f i r m a t i o n .  two t o t a l l y d i f f e r e n t  right,  then three  results  c a n be  something.  genuine:  and  (3)  (2)  drawn* one  case  HI and H2  hypotheses.  i n a r g u i n g f o r case For exactly  not an R b u t i s  e x p l a i n e d by the  no c a s e  follows  law-hypothesis  paradoxes of c o n f i r m a t i o n i s  object which i s  is  e x p l a i n e d by the  Raven H y p o t h e s i s s h o u l d be u n d e r s t o o d as H2;  overlook  (iii)  a b o v e , A r m s t r o n g seems t o  i n case  ( i i i ) , when we have  a B , we have t h e  object  that  an is  following generalization:  (x)(-RxvBx),  which i s  l o g i c a l l y equivalent to H I .  object,  Hence, the  t h a t we come upon  A c c o r d i n g to Armstrong there  Armstrong i s  However,  the  a B.  H 2 , we know t h a t  c a n n o t be a n R .  i n any way c o n f i r m t h e  have one c a s e  (1)  law-hypothesis  it  which i s  is  original  also  i n t e n t i o n of Armstrong,  for  t h e y do n o t  the  object  a B, is  e x p l a i n e d b y HI and RH  are not laws nor the  is  not an R but i s  Now i t  is  that  a fact contradicting  who t h i n k s t h a t HI and H3 Raven H y p o t h e s i s ,  o b s e r v a t i o n r e p o r t which says  not an R but i s  Thus A r m s t r o n g ' s v i e w o f  obvious  e x p l a i n e d by H 3 .  proper t r a n s l a t i o n of the  e x p l a i n the  though i n t e r e s t i n g ,  is  that  a B. the  not r e a l l y  paradoxes of c o n f i r m a t i o n , a l tenable.  42 13.  SOME OTHER PROPOSED SOLUTION OF THE  PARADOXES OF CONFIR-  MATION There i s one  interesting  contemplated s o l u t i o n of the  doxes of c o n f i r m a t i o n c o n s i d e r e d by Hempel.^ s o l u t i o n i s t h i s . In A r i s t o t a l i a n l o g i c an  The  para-  contemplated  existential  import  2  i s c o n f e r r e d upon any  universal  adopt t h i s A r i s t o t a l i a n p o l i c y , can be avoided;  c o n d i t i o n a l sentence. the paradoxes of  I f we  confirmation  f o r , i n s t e a d of e x p r e s s i n g the Raven Hypothesis  as t (1)  (x)(Rx3>Bx)  we add  e x p l i c i t l y a sentence e x p r e s s i n g t h i s e x i s t e n t i a l  import  as f o l l o w s 8 (2)  (x)(Rx3Bx).(3x)Rx. Accordingly,  (3)  (x)(-Bx=>-Rx)  would  be:  (4)  (x)(-Bx3-Rx).(3x)-Bx. Now  from (2)  or (4) we  cannot d e r i v e the paradoxes of con-  f i r m a t i o n , f o r the equivalence i.e.  (2)  and  condition i s obviously  violated,  (4) are not l o g i c a l l y e q u i v a l e n t .  Hempel g i v e s three reasons to r e j e c t t h i s " A r i s t o t a l i a n lution" i  so-  3  (a) F i r s t o f a l l , because the A r i s t o t a l i a n p o l i c y would i n 1. Hempel» Aspects of S c i e n t i f i c E x p l a n a t i o n , pp.16-17. 2. C f . R u s s e l l . H i s t o r y of Western Philosophy. p . 2 0 6 f f . 3. Hempel, op. -ci-t.. pp.16%-17.  ^3  v a l i d a t e many l o g i c a l i n f e r e n c e s t h a t appear to be  valid.  (b) Second, i n e m p i r i c a l s c i e n c e the customary of g e n e r a l  hypotheses does not r e a l l y c o n t a i n an  c l a u s e : nor does i t determine such a c l a u s e For example, c o n s i d e r  formulation  existential  unambiguously.  t h i s example g i v e n by Hempeli I f a person  a f t e r r e c e i v i n g an i n j e c t i o n of a c e r t a i n t e s t substance has p o s i t i v e s k i n r e a c t i o n , he has p o r t the e x i s t e n t i a l c l a u s e ?  diphtheria. Should we  Where should we  a im-  import i t as r e f e r r i n g  to persons? to persons a f t e r r e c e i v i n g the i n j e c t i o n ? or to  per  sons who  a f t e r r e c e i v i n g the i n j e c t i o n show a p o s i t i v e s k i n r e -  action?  According  b i t r a r y , and  to Hempel any  d e c i s i o n made here i s q u i t e  ar  each d e c i s i o n g i v e s a d i f f e r e n t i n t e r p r e t a t i o n to  the hypothesis,  and  none of them seems c o r r e c t .  (c) F i n a l l y , many u n i v e r s a l hypotheses cannot be imply an e x i s t e n t i a l c l a u s e .  s a i d to  Hempel s t a t e s h i s view as  follows  " i t may happen t h a t from a c e r t a i n a s t r o p h y s i c a l theory a u n i v e r s a l h y p o t h e s i s i s deduced concerning the c h a r a c t e r of the phenomena which would take p l a c e under c e r t a i n s p e c i f i e d extreme c o n d i t i o n s . A hypot h e s i s of t h i s kind need not (and, as a r u l e , does not) imply t h a t such extreme c o n d i t i o n s ever were or w i l l be r e a l i z e d ; i t has no e x i s t e n t i a l import." The  l a s t p o i n t g i v e n by Hempel seems l e s s c o n v i n c i n g ,  for  a h y p o t h e s i s t h a t cannot or w i l l never be r e a l i z e d would cause people to doubt i t s p l a u s i b i l i t y .  4.  Ibid,  p.17.  44 14. HEMPEL'S VIEW OF THE PARADOXES OF CONFIRMATION Hempel's view of the paradoxes of c o n f i r m a t i o n i s d i f f e r e n t from a l l o f the views so f a r we have examined.  H i s view i s i n  p a r t i c u l a r d i f f e r e n t from the views o f von Wright's, Quine's, S c h e f f l e r ' s and Goodman's. Hempel sees n o t h i n g wrong with the paradoxes of c o n f i r m a t i o n . In other words, he does not t h i n k t h a t the paradoxes of confirmat i o n a r e genuine.  He i s an upholder o f the p r i n c i p l e o f u n i v e r s a l  r e l e v a n c e ( s o - c a l l e d by M. Black), which i s q u i t e d i f f e r e n t from the p r i n c i p l e of l i m i t e d relevance  of Nicod's type as upheld  by Quine,  S c h e f f l e r , Goodman, e t c . There are s e v e r a l reasons t h a t make Hempel come to the c o n c l u s i o n t h a t the paradoxes of c o n f i r m a t i o n are not genuine* First,  he t h i n k s t h a t we need the E q u i v a l e n c e  Condition, f o r  1  " F u l f i l l m e n t of t h i s c o n d i t i o n makes the c o n f i r m a t i o n of a hypothesis independent of the way i n which i t i s formulated; and no doubt i t w i l l be conceded t h a t t h i s i s a necessary c o n d i t i o n f o r the adequacy o f any proposed c r i t e r i o n of c o n f i r m a t i o n . " Second. Hempel r e j e c t s the A r i s t o t a l i a n type o f s o l u t i o n of the paradoxes of c o n f i r m a t i o n by c o n f e r r i n g on any u n i v e r s a l c o n d i t i o n a l sentence an e x i s t e n t i a l import.  He has g i v e n three reasons  for  discussed  t h i s r e j e c t i o n , which  we have j u s t  i n the p r e v i o u s  section. T h i r d , Hempel a l s o r e j e c t s v o n Wright's i d e a of "the n a t u r a l range of r e l e v a n c e " of a u n i v e r s a l c o n d i t i o n a l h y p o t h e s i s by s a y i n g t h a t the way i n which general hypotheses are used i n science never i n v o l v e s the statement of a " f i e l d of a p p l i c a t i o n " or any " n a t u r a l range of r e l e v a n c e " , 1. Hempel: Aspects o f S c i e n t i f i c E x p l a n a t i o n , 2. I b i d , p.17.  p. 13.  4 After method o f  examining the representing  conditionals,  Hempel  above  two a l t e r n a t i v e s  general  concludes!  to  5  the  customary-  of  universal  h y p o t h e s e s b y means 3  " n e i t h e r o f them p r o v e d a n a d e q u a t e means o f p r e c l u d i n g the paradoxes of c o n f i r m a t i o n . We s h a l l now t r y t o show t h a t what i s wrong does n o t l i e i n t h e c u s t o m a r y way o f c o n s t r u i n g and r e p r e s e n t i n g g e n e r a l h y p o t h e s e s , b u t r a t h e r i n o u r r e l i a n c e on a m i s l e a d i n g i n t u i t i o n i n t h e m a t t e r . The i m p r e s s i o n o f a p a r a d o x i c a l s i t u a t i o n i s n o t o b j e c t i v e l y founded; i t i s a p s y c h o l o g i c a l i l l u s i o n . " (niy i t a l i c s ) According to (a) thesis  Hempel t h e r e  One s o u r c e of  the  of  simple  are  two  sources  misunderstanding  is  of  the  view  that  a hypo-  something  about a  form,  H:  Every P i s  s u c h as  " a l l sodium s a l t s b u r n y e l l o w " , a s s e r t s  a Q,  certain limited class This  misunderstanding!  of  idea, according to  objects only,  Hempel, i n v o l v e s  viz.,  the  class  of  all  P's.  (as a l r e a d y m e n t i o n e d a b o v e )  3  a c o n f u s i o n o f l o g i c a l and p r a c t i c a l c o n s i d e r a t i o n : Our i n t e r e s t i n t h e h y p o t h e s i s may be f o c u s s e d upon i t s a p p l i c a b i l i t y to t h a t p a r t i c u l a r c l a s s of o b j e c t s , but the h y p o t h e s i s n e v e r t h e l e s s a s s e r t s s o m e t h i n g a b o u t , and i n d e e d i m poses r e s t r i c t i o n s upon, a l l o b j e c t s . (b)  A second  c e r t a i n cases of lowing  source  of  the  appearance  confirmation is  considratiom  exhibited  of  paradoxicality  b y Hempel b y t h e  in fol-  4  Suppose t h a t i n s u p p o r t o f t h e a s s e r t i o n ' A l l s o d i u m s a l t s b u r n y e l l o w ' somebody were t o adduce a n e x p e r i m e n t i n w h i c h a p i e c e o f p u r e i c e was h e l d i n t o a c o l o r l e s s f l a m e and d i d not t u r n the flame y e l l o w . T h i s r e s u l t would c o n f i r m the assertion, ' W h a t e v e r does n o t b u r n y e l l o w i s no sodium s a l t ' and c o n s e q u e n t l y , b y v i r t u e o f t h e e q u i v a l e n c e c o n d i t i o n , i t would c o n f i r m the o r i g i n a l f o r m u l a t i o n . Why d o e s t h i s 3. 4.  Ibid, Ibid,  p.18. p.19.  46 impress us as p a r a d o x i c a l ? The reason becomes c l e a r when we compare the p r e v i o u s s i t u a t i o n with the case where an o b j e c t whose chemical c o n s t i t u t i o n i s as yet unknown to us i s held i n t o a flame and f a i l s to t u r n i t yellow, and where subsequent a n a l y s i s r e v e a l s i t to c o n t a i n no sodium s a l t . T h i s outcome... i s what was to be expected on the b a s i s of the h y p o t h e s i s t h a t a l l sodium s a l t s burn y e l l o w . . . Now the o n l y d i f f e r e n c e between the two s i t u a t i o n s here c o n s i d e r e d i s t h a t i n the f i r s t case we are t o l d beforehand the t e s t substance i s i c e , and we happen to "know anyhow" t h a t i c e c o n t a i n s no sodium s a l t ; t h i s has the consequence t h a t the outcome of the flamec o l o r t e s t becomes e n t i r e l y i r r e l e v a n t f o r the c o n f i r m a t i o n of the hypothesis and thus can y i e l d no new evidence f o r us. Indeed, i f the flame should not t u r n yellow, the h y p o t h e s i s r e q u i r e s t h a t the substance c o n t a i n no sodium salt and we know beforehand t h a t i c e does not; and i f the flame should t u r n yellow, the h y p o t h e s i s would impose no f u r t h e r r e s t r i c t i o n s on the substance: hence, e i t h e r of the p o s s i b l e outcomes of the experiment would be i n accord with the h y p o t h e s i s . Thus Hempel comes to a general p o i n t from the a n a l y s i s of the above example:^ In the seemingly p a r a d o x i c a l cases of c o n f i r m a t i o n , we are o f t e n not a c t u a l l y judging the r e l a t i o n of the g i v e n evidence E alone to the h y p o t h e s i s H ...; i n s t e a d , we t a c i t l y i n t r o d u c e a comparison of H with a body of e v i dence which c o n s i s t s of E i n c o n j u n c t i o n with a d d i t i o n a l i n f o r m a t i o n t h a t we happen to have at our d i s p o s a l ; i n our i l l u s t r a t i o n , t h i s i n f o r m a t i o n i n c l u d e s the knowledge (1) t h a t the substance used i n the experiment i s i c e , and (2) t h a t i c e c o n t a i n s no sodium s a l t . I f we assume t h i s a d d i t i o n a l i n f o r m a t i o n as g i v e n , then, of course, the outcome of the experiment can add no s t r e n g t h to.the hypothes i s under c o n s i d e r a t i o n . But i f we are c a r e f u l co a v o i d t h i s t a c i t r e f e r e n c e to a d d i t i o n a l knowledge... i t i s c l e a r t h a t ... the paradoxes v a n i s h . So f a r , i n (b), what Hempel has c o n s i d e r e d mainly i s the  type  of p a r a d o x i c a l case which i s i l l u s t r a t e d by the a s s e r t i o n t h a t any non-black non-raven c o n s t i t u t e s c o n f i r m i n g evidence hypothesis  ,"all  ravens  are b l a c k " .  According  f o r the  to Hempel, " 6  Other p a r a d o x i c a l cases of c o n f i r m a t i o n may be d e a l t with analogously. Thus i t t u r n s out t h a t the paradoxes of con5. 6.  Ibid, Ibid,  p.19. p.20.  47  f i r m a t i o n ... are due to a misguided i n t u i t i o n i n the matter r a t h e r than to a l o g i c a l flaw i n the ... s t i p u l a t i o n s from which they were d e r i v e d . (my i t a l i c s ) Thus Hempel t h i n k s , i n c o n c l u s i o n , t h a t the paradoxes of  con  f i r m a t i o n are not r e a l l y genuine. By the way, Goodman, and  P r o f . Stewart a t t r i b u t e s the f o l l o w i n g view to  the view i s . Goodman t h i n k s t h a t the paradoxes of  c o n f i r m a t i o n are indeed not genuine as Hempel t r i e s to say and  here,  Goodman's i d e a of s e l e c t i v e c o n f i r m a t i o n i s o n l y one way  show how  the i l l u s i o n a r y mechanism works i n our mind.  Unfortu-  n a t e l y , I have not found these c l a i m s o f Goodman's i n the a t u r e , and,  consequently  t h i s i n t e r e s t i n g view.  to  liter-  cannot o f f e r a more c a r e f u l a n a l y s i s of  48 1 5 . A NEW VIEW OF THE PARADOXES OF CONFIRMATION From o u r p r e v i o u s of  confirmation  a lot  of  (I) ine.  especially  I  believe that  psychological (II)  all  I  agree  proposed  l i m i t e d r e l e v a n c e of least,  more  for  I  a  that  type  on t h e o t h e r h a n d , of c o n f i r m a t i o n  satisfactory they support  is,  I  to b e l i e v e .  or,  that  contain even,  b e l i e v e , correct or,  von W r i g h t ' s  a great  ornithology"  following  Cf.  of at  All  concept of  "natural  proposed range  as  difficulty:  it  of  the  paradoxes  commits  p o i n t e d o u t b y Goodman." ' 1  I n d o o r Raven ravens  dissolution  the  "fallacy  Thus  consider  Hypothesis:  i n my o f f i c e a t  UBC a r e  black,  symbols, (x) (Rx . Ox =>Bx),  1.  sim-  a principle  s o m e t h i n g we c a n l e a r n f r o m t h e  b e l i e v e that Hempel's  indoor  or i n  totally  us  are  hypothesis.  of  1  opinions:  c o n f i r m a t i o n are genu-  Hempel p e r s u a d e s opinion,  However,  there is  confirmation faces  IRH :  of  paradoxes  and d i s s o l u t i o n  To sum up my own  the paradoxes  Nicod*s  of  the  of  instance,  r e l e v a n c e " of (III)  the  plausible.  Nevertheless solutions,  as  and a r e n o t  satisfactory.  of  and o b j e c t i v e l y g r o u n d e d and t h e y  illusions  solutions  solutions  the paradoxes  w i t h Hempel*s  some d i f f i c u l t i e s p l y not  their  t h i n g s c a n be l e a r n e d .  They a r e l o g i c a l l y  not  of  r e v i e w and d i s c u s s i o n s  Goodman: F a c t ,  where R: (T)  is  a raven,  0:(J)  is  i n my o f f i c e a t  B: (J)  is  black.  Fiction,  and F o r e c a s t ,  3rd  ed.,  UBC,  70-71.  49  Now i n my o f f i c e a t UBC there are one desk, two c h a i r s and some books. IRH^  I t f o l l o w s t h a t i n Hempel's theory of c o n f i r m a t i o n  i s confirmed  by the o b s e r v a t i o n r e p o r t about the development  of IRH.^ w i t h r e s p e c t to the r e f e r e n c e c l a s s of i n d i v i d u a l s I, which are the desk, the two c h a i r s and the books i n my o f f i c e a t UBC. So a r e the f o l l o w i n g hypotheses confirmed  by the same c l a s s o f  individuals I: IRH «  A l l ravens i n my o f f i c e a t UBC are white.  IRH^:  A l l ravens i n my o f f i c e a t UBC are r e d .  IRH^:  A l l ravens i n my o f f i c e a t UBC are both b l a c k and white.  2  In f a c t the f a l l a c y o f indoor o r n i t h o l o g y i s not c o n f i n e d t o indoor nor only to o r n i t h o l o g y .  So c o n s i d e r the f o l l o w i n g L o c a l  Panda Hypothesess LPH^:  Each panda i n A l a s k a has one eye.  LPH :  Each panda i n A l a s k a has two eyes.  LPH^:  Each panda i n A l a s k a has 1,001 eyes.  LPH^:  Each panda i n A l a s k a has both three and f o u r eyes.  2  Now each of the above L o c a l Panda Hypotheses i s a l s o i n Hempel's theory of c o n f i r m a t i o n ,  confirmed  simply because there i s not  a panda i n A l a s k a . That Hempel's theory of c o n f i r m a t i o n has committed the f a l l a c y of "indoor o r n i t h o l o g y " as shown i n the above examples always depends the same d e v i c e : the development o f each o f the above hypotheses always has a f a l s e I t i s important  antecedent.  to r e a l i z e  the Raven Hypothesis.  t h a t the " d e v i c e " i s a l s o used i n  Thus when people say t h a t i t i s absurd t o  50  say t h a t ( i ) a non-black non-raven, or ( i i ) a b l a c k non-raven confirms the Raven Hypothesis,  the reason i s simply t h a t  development of the Raven Hypothesis a l s o has a f a l s e antecedent. us any  w i t h r e s p e c t to t h a t e n t i t y  In consequence, Hempel cannot show  o b j e c t which s a t i s f i e s both the antecendent  quent of I R H  lt  IRH  ?i  IRHy  IRH^,  LPH^  LPH , L P H 2  and the conseor LPH^  3  what he can show to us i s something t h a t s t r i k e s us as In s h o r t , there  is  one  important  of "indoor o r n i t h o l o g y " : there are some s c i e n t i f i c  LPH^  such as the c o n f i r m a t i o n of IRH^-  criticism fallacy  fictions  IRH^  b u i l t i n t o Hempel's theory of c o n f i r m a t i o n .  theses are confirmed  or e l s e  irrelevant.  p o i n t made by the  t h a t Hempel's theory of c o n f i r m a t i o n has committed the  mythologies  the  and  or LPH^-  These hypo-  i n Hempel's theory of c o n f i r m a t i o n and,  he cannot show us even one  yet,  such o b j e c t (a raven o r a panda) t h a t  s a t i s f i e s any of the hypotheses (under d i s c u s s i o n ) . (IV) I b e l i e v e t h a t von Wright has an important p o i n t to make when he says t h a t the " n a t u r a l " r e a d i n g of the Raven i s c o n f i n e d to ravens and n o t h i n g e l s e .  Hypothesis  In other words h i s n o t i o n  of "the n a t u r a l range of r e l e v a n c e " of a u n i v e r s a l c o n d i t i o n a l h y p o t h e s i s makes senset although the standard r e a d i n g of the Raven Hypothesis^,as  about e v e r y t h i n g i n the u n i v e r s e  including  ravens  and non-ravens, a l s o make sense. (V) I b e l i e v e f i n a l l y t h a t both readings of the Raven Hypothes i s can o n l y be accommodated i n a l o g i c other than c l a s s i c a l In other words, the whole p o i n t of t h i s i s to suggest  logic.  t h a t a change  of l o g i c may  h e l p s o l v e the Raven Paradox and the paradoxes of con-  firmation.  Thus we  come to the s u g g e s t i o n t h a t the source of the  paradoxes of c o n f i r m a t i o n and the f a l l a c y of "indoor o r n i t h o l o gy" may  d e r i v e from c l a s s i c a l l o g i c .  Since I am not the  to c h a l l e n g e the a b s o l u t e s t a t u s of the c l a s s i c a l l o g i c ,  first i n the  f o l l o w i n g I s h a l l have to t u r n to a review of the c r i t i c i s m s conc e r n i n g the c l a s s i c a l l o g i c made by  others.  52 16.  CHALLENGES TO THE CLASSICAL LOGIC  The  classical  logical like  enjoyed  o f E u c l i d e a n geometry  world,  lenged  has  the  unique p o s i t i o n  w o r l d f o r more t h a n two t h o u s a n d y e a r s .  that  tific  logic  o r e v e n more s o .  i n the  ning of  this  i n the  E u c l i d e a n g e o m e t r y was  1 8 t h c e n t r y and r e j e c t e d  by E i n s t e i n a t  was n o t  challenged  until  tion  still this  challengers (I)  believed  section to  late  by the  we w i l l  classical  laws i n v o l v e so-called  features J (i)  ator.  the  the  excluded-middle  infinite  three  sets  point of  "intuitionistic  and t h e  of view,  beginas  the  logic  unique  most  posi-  important  ideas. the  law of  universal double-  especially  (mathematical) l o g i c " we have  chal-  when  entities. these  Thus  prominent  1  The d o u b l e - n e g a t i o n of  l a w does n o t  a sentence  is  not  hold;  more  equivalent  specifically, to  the  sen-  is  concerned  itself.  (ii)  The law o f  (II)  C . I . Lewis' challenge.  m a i n l y w i t h the  1.  law o f  double n e g a t i o n  tence  first  logicians.  Brouwer c h a l l e n g e s  from a c o n s t r u c t i v i e t  the  the  m a j o r i t y of  and t h e i r b a s i c  Brouwer's challenge.  negation  his  great  logic  of  in  scien-  classical  c e n t u r y and i t s  describe  validity  the  last  while  the  somewhat  c e n t u r y when he a d o p t e d Riemannean g e o m e t r y theory of r e l a t i v i t y ,  In  is  m a t h e m a t i c a l and  proper tool f o r his  is  It  in  excluded-middle  i n t e r p r e t a t i o n of  He t h i n k s t h a t  it  is  does not  hold.  Lewis' challenge the  wrong t o  classical  "horseshoe"  i n t e r p r e t the  C f . A . H e y t i n g : I n t u i t i o n i s m , 1966, t i o n to Metamathematics. p p . 4 6 - 5 3 .  0  r  oper-  "horseshoe"  S.C. Kleenei  Introduc-  53 as  "implication",  of  the  classical  and he logic  lists  as  1.  P 5 ( Q PP)  2.  -P 3(P3(J)  3.  ( P . Q ) 3(P=>Q)  4.  (P.Q) 3 ( Q 3 P )  5.  ( - P . - Q ) ^> ( P ^ Q )  6.  (-P.-Q)3(Q3P)  7.  ( - P . Q ) =>(P^Q)  8.  - ( P 3 Q ) =>P  9.  -(POQ)a-Q  "the  10.  -(P3Q)P(P5-Q)  11.  -(P?Q)P(-P3Q)  12.  - ( P P Q ) ^ (-P3-Q)  13.  -(PPQ)?(Q5P)  The m a i n r e a s o n shoe" as ables  C . I . Lewis  "implication"  is  the  following  paradoxes  thinks  wrong i s  of  that  this:  thirteen  theorems  implication"*  interpreting  the  2  "horse-  interpretation  two i n t r i n s i c a l l y ( o r c a u s a l l y ) u n r e l a t e d d i s j u n c t s  of  a  encer-  t a i n k i n d to enjoy a l o g i c a l l y i n t r i n s i c r e l a t i o n of " i m p l i c a t i o n " . His challenge S1-S5. tion"  results  Unfortunately, s u c h as (i)  (ii)  the  some o f  following  est  systems  2. See, 3. I b i d ,  of  "paradoxes  new m o d a l of  strict  systems, implica-  two j  (P.-PMQ short  his  the  series  Q-J(Pv-P)  (where " A - ? B " i s of  i n a whole  modal  for  "Cj(A=>B)"),  even i n the  weak-  SI.  C . I . L e w i s and C . H . L a n g f o r d * p.504.  appear,  Symbolic  Logic,  pp.86-88.  54 (III)  -Lukasiewicz's challenge.  classical His  logic  intention  is,  is  at  to  once,  solve  Lukasiewicz's challenge  b o t h v e r y modern and v e r y  the  Aristotalian  problem  to  ancient.  of  future  contingents. The A r i s t o t a l i a n p r o b l e m o f brief,  this.-'  sentence  What i s  such as  the  "there  the  future contingents  truth-value  following  of  i s , t o be  a future  very  contingent  one,  w i l l be a s e a - b a t t l e  tomorrow"?  Or i n L u k a s i e w i c z ' s modern v e r s i o n . ^ "I c a n assume w i t h o u t c o n t r a d i c t i o n t h a t my p r e s e n c e i n Warsaw a t a c e r t a i n moment o f n e x t y e a r , e . g . a t n o o n on 21 December, i s a t t h e p r e s e n t t i m e d e t e r m i n e d n e i t h e r p o s i t i v e l y n o r n e g a t i v e l y . Hence i t i s p o s s i b l e , b u t n o t n e c e s s a r y , t h a t I s h a l l be p r e s e n t i n Warsaw a t t h e the g i v e n t i m e . On t h i s a s s u m p t i o n t h e p r o p o s i t i o n *I s h a l l be i n Warsaw a t n o o n on 21 December o f n e x t y e a r ' , c a n a t t h e p r e s e n t t i m e be n e i t h e r t r u e n o r f a l s e . F o r i f i t were t r u e now, my f u t u r e p r e s e n c e i n Warsaw w o u l d have t o be n e c e s s a r y , w h i c h i s c o n t r a d i c t o r y t o t h e a s sumption. I f i t w e r e f a l s e now, on t h e o t h e r h a n d , my f u t u r e p r e s e n c e i n Warsaw w o u l d be i m p o s s i b l e , w h i c h i s a l s o c o n t r a d i c t o r y to the a s s u m p t i o n . Therefore the p r o p o s i t i o n c o n s i d e r e d i s a t t h e moment n e i t h e r t r u e n o r f a l s e and must p o s s e s s a t h i r d v a l u e . . . " From t h e system of  above  three-valued  According to problem of (i) valence (ii) 4. 5. 6. 7.  line  future  one  logic  contingents  rejected  the  thought  law o f  in  Lukasiewicz invented  a logic  i n the  c a n be  with which I agree, solved,  such that  logic,  a new  1920.  interpretation,  we c a n i n v e n t is  of  the  the  if p r i n c i p l e of  bi-  while  excluded-middle  is  retained  i n the  logic,  C f . L u k a s i e w i c z , J i "On 3 - v a l u e d l o g i c " , i n M c C a l l , S . , P o l i s h L o g i c . 1920-1939. A r i s t o t l e . De I n t e r p r e t a t i o n e , C h a p t e r 9 . H a a c k , S» D e v i a n t L o g i c , p.73. T a y l o r , R: " F a t a l i s m " , P h i l o s o p h i c a l R e v i e w ( 7 1 ) , 1962.  (ed.),  55  In other words, the problem o f f u t u r e c o n t i n g e n t s can be s o l v e d i f we have» (i*)tf(T  p  v F ) , and p  ( i i * ) |(-(Pv-P),  where "T" and "F" a r e r e s p e c t i v e l y the p r e -  d i c a t e s "... i s t r u e " and "... i s f a l s e " . I t t u r n s out t h a t Lukasiewicz's  t h r e e - v a l u e d l o g i c L3 i s ,  indeed, not b i - v a l e n t : b u t , u n f o r t u n a t e l y , the law of excludedmiddle does not h o l d e i t h e r . Thus Lukasiewicz tingents.  does not s o l v e the problem o f f u t u r e con-  However, t h a t does n o t mean t h a t h i s c h a l l e n g e does  not make sense.  56  17. THE  PLAUSIBLE INADEQUACY OF THE  CLASSICAL LOGIC  Besides the p l a u s i b l e p o i n t s made by Brouwer, Lewis and •Lukasiewicz  there are a t l e a s t three other p o i n t s t h a t w i l l  show the p l a u s i b l e inadequacy (I) I t f a i l s to express  of c l a s s i c a l  some important  In n a t u r a l language such as Chinese, d i s t i n c t senses  of n e g a t i o n such  ( i ) Negation  logic. senses of n e g a t i o n .  E n g l i s h , e t c . there are  as  as c o n t r a r y ,  ( i i ) n e g a t i o n as complementation,  and  ( i i i ) n e g a t i o n as c o n t r a d i c t i o n . But  i n c l a s s i c a l l o g i c a l l of them are taken c a r e o f by a s i n g l e  n e g a t i o n o p e r a t i o n . How  many d i f f i c u l t i e s and paradoxes i n l o g i c  are d e r i v e d from t h i s awkward "squeezing"?  In c o n t r a s t , i n c e r -  t a i n t h r e e - v a l u e d l o g i c there are, t h e o r e t i c a l l y , d i f f e r e n t senses  of n e g a t i o n t h a t can be  ( I I ) I t f a i l s to express the concept  26 p o s s i b l e  differentiated. of p r e s u p p o s i t i o n , which  r e q u i r e s the a d d i t i o n a l t r u t h - v a l u e " n e i t h e r t r u e nor  false",  Consider the well-known example of Russell's» (1)  The present k i n g of France  i s bald.  A c c o r d i n g to R u s s e l l ' s theory of d e f i n i t e d e s c r i p t i o n s i t should be t r a n s l a t e d a s i (2)  1  (3x)(Kx.Bx.(y)(Ky3x=y)), where Ki (I)is Bt Q  a present k i n g of France,  i s bald.  Since t h e r e i s no present k i n g of France,(2) 1. Cf R u s s e l l i  "On  denoting", Mind, 1905,  i s false.  PP. 79- 93. k  k  Hence,  57  (1) i s f a l s e a c c o r d i n g t o R u s s e l l ' s theory o f d e f i n i t e d e s c r i p tions . But a c c o r d i n g to P.F. Strawson, ing  (1) presupposes the f o l l o w -  statement:  (3)  The present  k i n g o f France  exists,  or, i n symbols, (3*)  (3x)Px, (I)  where P:  i s the present  k i n g o f France.  Then, a c c o r d i n g t o Strawson's a n a l y s i s a statement i s " n e i 2 t h e r t r u e nor f a l s e " i f i t s p r e s u p p o s i t i o n i s not f u l f i l l e d . Thus i n the present nor f a l s e " ,  case the t r u t h v a l u e  of (1) i s " n e i t h e r t r u e  s i n c e (3) i s not f u l f i l l e d .  So, whose a n a l y s i s i s the c o r r e c tfitmore s a t i s f a c t o r y , one? My present understanding  i s this:  ( i ) R u s s e l l ' s theory o f d e f i n i t e d e s c r i p t i o n s seems t e n a b l e , ( i i ) On the other hand, the t r u t h - v a l u e of (1) should be " n e i t h e r t r u t h nor f a l s i t y " , f o r  (1) indeed presupposes (3) and here  Strawson appears t o me t o be r i g h t t o say t h a t a sentence i s " n e i t h e r t r u e nor f a l s e " when i t s p r e s u p p o s i t i o n i s not f u l f i l l e d . I f my understanding then c l a s s i c a l l o g i c  i s r i g h t o r o f f e r s another p l a u s i b l e view  i s l i k e l y t o be inadequate f o r both the con-  c e p t s o f " n e i t h e r t r u t h nor f a l s i t y " and " p r e s u p p o s i t i o n " cannot be expressed  i n classical  logic.  ( I l l ) I t f a i l s t o express the concept o f " c a u s a l i t y " . 2. C f . Strawson: "On r e f e r r i n g " , Mind. 1950. 3. My i m p l i c i t s o l u t i o n o f t h i s problem i s c o n t a i n e d below i n sect i o n 11, P a r t I I .  Let's  follow  Then,  there  a  David are  causes event But  this  consider  If  logicians b" w i t h  "0a«^0b".  following  the  "a  is  an  event"  as  who h a v e p r o p o s e d t o i d e n t i t y  identification  the  (5)  L e w i s and w r i t e  is  "Oa". "event  k  not  acceptable.  To see  this,  example*-'  match  is  struck  then  match  is  struck  and  it  will  light.  Therefore, if  the  will Or,  in  Os^Ol  Now  it  is  acceptable; because  /  /.  it  s:  (Os.Od)^Ol  the  s t r i k i n g of  the  1: t h e  match  being  light,  d*  match  being  dunked  the  intuitively clear  but  the  in  water.  inference  (6)  inference  of  i n c l a s s i c a l l o g i c we h a v e  the  following  law*  Weakening*  In c l a s s i c a l  logic,  from  (5)  of  the  Law o f  in classical  that  match,  logic  The  i s un-  holds,  ( P ^ Q ) we  can  (P.R)DQ.  infer:  Thus  the  logic  plausible history  5. 6.  then  light.  where  4.  i n water  symbols,  (6)  sical  dunked  of  concept i n the  and  of  causality  above  definable  philosophy  proposed  cannot  defined  i n the  clas-  and  no  o t h e r way t h a t  in classical logic  is  known i n the  and  way,  be  is  long  logic  C f . Sosa, E r n e s t ( e d . ) : C a u s a t i o n and C o n d i t i o n a l s , e s p e c i a l l y the Introduction. C f . v a n F r a a s s e n , B a s C : The S c i e n t i f i c I m a g e . p.ll ff. Sosa, op. c i t . , e s p e c i a l l y the Introduction. k  59 18.  A\  T  AXIOMATIC REVIEW OF THE  CLASSICAL SENTENTIAL LOGIC  Since I t h i n k t h a t c l a s s i c a l l o g i c paradoxes of c o n f i r m a t i o n and  i s l i k e l y the source of the  s i n c e there are a few  who  t r y to c h a l l e n g e i t s adequacy, i t would be q u i t e  for  us to have a b r i e f review b e f o r e we  pronounce our f i n a l The was  t r y to announce and  f i r s t modern r i g o r o u s treatment of the c l a s s i c a l  logic  I t c o n t a i n s both the s e n t e n t i a l c a l -  c u l u s and the p r e d i c a t e c a l c u l u s .  I t s s e n t e n t i a l p a r t has  six  In terms of the c u r r e n t n o t a t i o n s they are:''" OP)  Fl.  P3(Q  F2.  (P3(Q3R)) ((Paa)3(P5R)))  F3.  (Ps» (Q=>R)) => (Q =>(?=>R))  ?4.  (?»Q)  F5.  -Pa  ?6.  P => - - P  5  »(-Q»-P)  P  In t h e i r monumental work P r i n c i p i a in  proper  judgement.  g i v e n by Frege i n 1879.  axioms.  logicians  1910, Whitehead and  R u s s e l l reduced the number of the axioms  of the s e n t e n t i a l c a l c u l u s to f i v e . PM1.  (Pvp)=>?  PM2.  Q3(Pv3)  PM3.  (P v Q) = (Qv P)  PM4.  PM5. I t was  Mathematica, p u b l i s h e d  They a r e :  2  ( Q 3 R ) 3 ( ( P V Q ) 3 ( P V R ) )  ( P v ( Q v R ) ) a (Qv (P v R)) found i n 1928  by H i l b e r t and  Ackermann, among others,  t h a t the f i f t h axiom of PM5 i s redundant. In the f o l l o w i n g we 1. Frege, G.: E e g r i f f s c h r i f t . 2. Cf. R u s s e l l , B. and A.M. Whitehead: P r i n c i p i a f-lathematica, v o l . 1. Note t h a t i n t h e i r system • and 'v' are the p r i m i t i v e o p e r a t o r s and, hence, 'PaQ' i s short f o r U P v Q ' ,  60  give i n outline below f o r l a t e r I.  the axiomatic 3 references  system of H i l b e r t  and Ackermann  Formation r u l e s . 1. P r i m i t i v e 2.  operators:  A u x i l i a r y symbols:  -,  (,  3. S e n t e n t i a l l e t t e r s : 4. Rules of  v  .  ).  X,Y,Z,...  .  wffs.  5. Axioms: HA1.  (XvX)3X  HA2.  Xo(XvY)  HA3-  (XvY)^(YvX)  HA .  (X=>Y) =» ((Z v X) => (Z v Y))  k  II.  Transformation r u l e s . 1.  2.  Definitions: Df 1.  X-Y  =df  -(-Xv-Y)  Df 2.  X =>Y  =df  -Xv Y  Df 3.  X= Y  =df  (X3Y)-(Y3X)  Rule of i n f e r e n c e :  3. Rules of  modus ponens.  substitution.  In 1929 -Lukasiewicz d i s c o v e r e d t h a t the c l a s s i c a l c a l c u l u s needs only three axioms i f primitive  operators.  JL1.  (-P»P)=>P  JL2.  pa(-PDQ)  JL3.  In  we adopt -  and  sentential  z> as the  L u k a s i e w i c z ' s three axioms a r e :  ( P 3 Q ) 3 ( ( Q 3 R ) 3 ( P 3 R ) )  fact,  if  we express them i n P o l i s h n o t a t i o n , we do not  even need the p a r e n t h e s e s . are s u p e r f l u o u s ,  In other words,  as shown by the f o l l o w i n g  the parentheses compromised  Polish  3. See, H i l b e r t D. and W. Ackermann: Mathematical L o g i c , p . 2 7 f f . Chelsea, 1950. 4. -Lukasiewicz, J . : Elements of Mathematical L o g i c , Warsaw, I 9 2 9 .  61  notations JL1*.  3^-PPP  JL2*.  ^P^-PQ  JL3*.  ^^PQ=>aQR-»PR  However, as e a r l y as i n 1913,H.M. S h e f f e r d i s c o v e r e d the c l a s s i c a l  s e n t e n t i a l l o g i c needs, i n f a c t , one  p r i m i t i v e operator.  One  and  that  only  such operator, denoted as "|" and  one  called  " S h e f f e r ' s s t r o k e " , i s d e f i n e d i n the f o l l o w i n g ways-*  Q  D e f i n i t i o n of S h e f f e r ' s s t r o k e . are not both t r u e . Thus the two  axiomatic  p r i m i t i v e operators  -P =df  Then, i n 1916 l o g i c can be  and  Ackermann's de-  followss  P |(Q|Q)  J.G.P.Nicod found t h a t the c l a s s i c a l s e n t e n t i a l  axiomatized  with  terms of S h e f f e r ' s s t r o k e .  Nil  of H i l b e r t ' s and  P  PIP  Df b. Pr>Q =df  following  i s true i f f  system f o r the c l a s s i c a l s e n t e n t i a l c a l c u l u s can be  f i n e d by S h e f f e r ' s stroke as Df a.  PIQ  j u s t one  The  axiom  single  i f we  frame i t i n  axiom of Nicod's i s the  onei^ ( P I ( Q I R ) ) l ( ( ( T | T ) | T ) I ( ( S | Q ) |((P|S) i ( P | S ) ) ) )  However, i n Nicod's a x i o m a t i z a t i o n we  need the f o l l o w i n g r u l e of  s t r o n g modus ponens i f NI i s the only axioms P  ,  P|(R|Q)  /  .*. Q  5. S h e f f e r , H.Ms "A s e t of f i v e independent p o s t u l a t e s f o r Bool e a n a l g e b r a " , T r a n s a c t i o n of the American Mathematical Soc i e t y , 1913, pp.481-488. 6. Nicod, J.G.Ps "A r e d u c t i o n i n the number of the p r i m i t i v e prop o s i t i o n s of l o g i c " , Proceedings of the Cambridge P h i l o s o p h i c a l S o c i e t y . 1916, pp.32-40.  62  Again i t was L u k a s i e w i c z who d i s c o v e r e d t h a t Nicod's can be improved.  T h i s time i t t u r n s out t h a t not the number o f the  axioms, but the l e n g t h of the s i n g l e axiom Nicod's s i n g l e axiom f o r s e n t e n t i a l l o g i c t i a l letters gle  c a n be  shortened. I n  there are f i v e  senten-  i n v o l v e d ; but Lukasiewicz found t h a t we need a s i n -  axiom w i t h o n l y f o u r s e n t e n t i a l l e t t e r s .  t e r axiom i s t h i s . Lit  system  L u k a s i e w i c z ' s shor-  n  (PI(Q|R))|((SI ( S I S ) ) l ( ( S | Q ) I ((PIS) I ( P I S ) ) ) )  Or, i n compromised P o l i s h n o t a t i o n t LI* i  IIP IQRIIS I SSIISQU  So f a r , i t seems t h a t a t l a s t we have the s i m p l e s t system f o r the c l a s s i c a l  s e n t e n t i a l c a l c u l u s , a t l e a s t i n the sense of the  l e a s t number of p r i m i t i v e o p e r a t o r s , the number o f axioms, and the number of s e n t e n t i a l l e t t e r s  i n v o l v e d i n the s i n g l e axiom.  Our review of the c l a s s i c a l l o g i c w i l l be ended h e r e .  7.  C f . Borkowski,  Lt Jan Lukasiewicz* S e l e c t e d Works.  63  PART  II.  A  SYNTACTIC  THREE-VALUED SEMANTICS  AND  FORMALLY  CONFIRMATION  SEMANTIC  THEORIES,  APPROACH AND  A  TO  THREE-VALUED  64  1. A BRIEF INTRODUCTION In t h i s P a r t we w i l l s t a r t to c o n s t r u c t s e v e r a l "formal  t h e o r i e s of c o n f i r m a t i o n " t h a t are intended  three-valued to solve  the  paradoxes of c o n f i r m a t i o n . However, i n the end,  i n each case,  the formal theory of con-  f i r m a t i o n t h a t i s supposed to s o l v e the paradoxes of c o n f i r m a t i o n t u r n s out to c o n t a i n some v e r s i o n of the paradoxes of tion.  confirma-  T h i s becomes c l e a r when, i n each case, we work out a com-  p l e t e semantics f o r the u n d e r l y i n g q u a n t i f i c a t i o n a l Thus we  l e a r n a g a i n the o l d l e s s o n t h a t s y n t a c t i c , or f o r -  mally  semantic, adequacy of a c o n f i r m a t i o n theory may  to be  s e m a n t i c a l l y inadequate.  One  logic.  r e l a t e d l e s s o n i s t h i s : on the one  t u r n out  hand s y n t a c t i c a l  or  f o r m a l l y s e m a n t i c a l p o s s i b i l i t y i s broader than semantical p o s s i bility:  on the other hand semantical  c o n s i d e r a t i o n i s more d e c i -  s i v e than s y n t a c t i c a l or f o r m a l l y s e m a n t i c a l c o n s i d e r a t i o n , at l e a s t t h i s i s so i n the case of c o n s t r u c t i n g an adequate  theory  of c o n f i r m a t i o n . So,  the r e s u l t s of our endeavour i n t h i s P a r t are l a r g e l y  g a t i v e a t the end,  except t h a t we  have worked out a complete con-  f i r m a t i o n semantics f o r t h r e e - v a l u e d or without,  identity.  t i c s would have i t s own  ne-  q u a n t i f i c a t i o n a l l o g i c s with,  T h i s newly worked-out t h r e e - v a l u e d independent v a l u e .  seman-  65 2.  A DEFINITION  A  "confirmation logic"  lying logic But, in  of  infallible tatively, subject  an adequate  then,  order to  OF "CONFIRMATION LOGIC" c a n be b r i e f l y d e f i n e d confirmation  what c o n d i t i o n s  be e n t i t l e d c r i t e r i a of  future  "adequate"? such k i n d  revision  "the  under-  theory".  must a c o n f i r m a t i o n t h e o r y  and o n l y t e n t a t i v e l y ,  to  as  Since  there  are,  fulfill  I think,  of  adequacy,  let  the  following  c o n d i t i o n s , which are  i n c a s e we f i n d  that  us l a y  no  they  down  are not  ten-  jus-  tifiable t Condition 1. tuitions and,  It  such t h a t  above  all,  and g i v e  C o n d i t i o n 3. tax  or formal  for  the  is  It  must be a b l e  the It  solution  systematizable,  solve  the  a reasonable  must be a b l e  It  consistent  to  paradoxes  of  con-  explanation.  be f r a m e d i n t e r m s  can provide a d e f i n i t i o n  derivable,  an e x t e n s i o n  concept  are  to  a n d / o r " p r o b a b i l i t y " so  probability is  theory  intuitions  our c o n f i r m a t i o n i n -  of  syn-  semantics.  C o n d i t i o n 4. firmation"  these  i n accord with  justifiable.  C o n d i t i o n 2. firmation  must be  of  of  for  that it  the  of  "degree o f  s t a n d a r d axiom  seems c l e a r  confirmation theory  that  i n the  "degree o f c o n f i r m a t i o n " p r e s u p p o s e s  con-  system  probability same way  the  that  concept  of  "confirmation". Since of  above  theories se  seems t h a t  conditions  is  classical  logic  f r o m what we have  of confirmation i n Part I,  a confirmation l o g i c .  there to  it  such a t h i n g .  search f o r  However,  may be is  I n what f o l l o w s  i n three-valued  logics,  fulfill  reviewed  it it  cannot  about thus  the  all  not  hopefully,  start  proposed  i n our  a n open q u e s t i o n we w i l l  second  sen-  whether  tentatively  a confirmation  logic.  66  However, to search f o r o r t o c o n s t r u c t a l l a t once such a c o n f i r mation l o g i c would be too b i g a. "task.  So, we l i m i t our e f f o r t i n  what f o l l o w s by l o o k i n g f o r only c e r t a i n  l o g i c s which f u l f i l l  the  second of the above c o n d i t i o n s . Thus the f o l l o w i n g two d e f i n i t i o n s : Definition  o f a minimal c o n f i r m a t i o n t h e o r y .  A confirmation  theory i s a minimal c o n f i r m a t i o n theory i f f i t has the f o l l o w i n g properties * i) ii) iii)  I t employs (some form o f ) Hempel*s seven b a s i c concepts; I t s a t i s f i e s some v e r s i o n of Hempel *s equivalence c o n d i t i o n ; I t s o l v e s the paradoxes o f c o n f i r m a t i o n .  Definition  o f a minimal c o n f i r m a t i o n l o g i c .  A logic i s a  minimal c o n f i r m a t i o n l o g i c i f f i t i s the u n d e r l y i n g l o g i c f o r a minimal c o n f i r m a t i o n  theory.  So d e f i n e d , a minimal c o n f i r m a t i o n l o g i c a confirmation logic  and  i s not n e c e s s a r i l y  a minimal c o n f i r m a t i o n theory i s not  n e c e s s a r i l y an adequate c o n f i r m a t i o n t h e o r y . However, i t i s the first  step towards a c h i e v i n g a c o n f i r m a t i o n l o g i c .  a lesser  task and, hence, e a s i e r t o f u l f i l l ,  we w i l l s t a r t  search f o r a c o n f i r m a t i o n l o g i c by l o o k i n g , f i r s t , confirmation  logic.  Since i t i s our  f o r a minimal  67  3.  AN EXAMPLE OF MINIMAL CONFIRMATION LOGIC  We give an example o f a minimal c o n f i r m a t i o n The minimal c o n f i r m a t i o n  logic  t i o n a l l o g i c t o be denoted  l o g i c below.  i s a three-valued q u a n t i f i c a -  as QMC3 » whose  sentential logic  w i l l be denoted as MC3. The three t r u t h - v a l u e s  of MC3 (and QMC3) a r e denoted a s . t ,  f , n: and they a r e r e s p e c t i v e l y t o be i n t e r p r e t e d " f a l s i t y " and " n e i t h e r  as " t r u t h " ,  t r u t h nor f a l s i t y " .  MC3 has three p r i m i t i v e o p e r a t i o n s i ~ , v , and — » ; and they are  i n t e r p r e t e d r e s p e c t i v e l y as "the ( e x t e r n a l )  "complementation"), " d i s j u n c t i o n " and  "implication".  t r u t h r u l e s are shown by the f o l l o w i n g t r u t h p V Q t f n  P —> Q t f n  ~P  t t t t f n t n n  t f n t t t n n t  f t t  n e g a t i o n " (or Their  tablesi  MC3 has two other s e n t e n t i a l c o n n e c t i v e s ! & , and f-» ; and  they a r e i n t e r p r e t e d  "equivalence".  r e s p e c t i v e l y as " c o n j u n c t i o n "  They have the f o l l o w i n g t r u t h  *\  t f n  P & Q t f n  P <-> Q t f n  t f n f f f n f n  t f n f t n n n t  and  tablesi  The c o n j u n c t i o n and the equivalence o f MC3 (and QMC3) a r e s u p e r f l u o u s , f o r they can be d e f i n e d erations  as shown belowi  by i t s three p r i m i t i v e op-  68  Def  1.  P&Q=df  ((P-»~P) v (Q->~Q))-»-((P-»~P) v ( Q - * ^ ) )  Def  2.  P**Q =df  (P-»Q) & (Q-»P)  (Note t h a t although Def  the c o n j u n c t i o n i n the definiendum of  2 i s not a p r i m i t i v e o p e r a t i o n , i t can be converted  a composition The  of p r i m i t i v e o p e r a t i o n s with the h e l p of Def  q u a n t i f i c a t i o n a l l o g i c QMC3 w i l l then be  e x i s t e n t i a l and  MC3  plus  the u n i v e r s a l q u a n t i f i c a t i o n s , which w i l l  r e s p e c t i v e l y the g e n e r a l i z a t i o n s of the d i s j u n c t i o n and c o n j u n c t i o n of We  i f we  add  to QMC3 the E q u i v a l e n c e  " o b s e r v a t i o n r e p o r t " , "hypothesis", "direct confirmation",  f i r m a t i o n " and The  the be  the  theory  Condition  and Hempel's seven b a s i c concepts of a c o n f i r m a t i o n  hypothesis",  1.)  MC3.  w i l l then have our f i r s t minimal c o n f i r m a t i o n  denoted as MGT,  viz.,  into  theory,  "development of a  "confirmation",  "discon-  "neutrality".  f o l l o w i n g two  explanations about the minimal  confirma-  t i o n theory are i n order: ( i ) The  "negation"  involved  i n the d e f i n i t i o n of  conf i r m a t i o n " w i l l be understood n a t u r a l l y as the  "external  of QMC3•  negation"  ( i i ) The  "entailment"  r e l a t i o n involved  t i o n of " d i r e c t c o n f i r m a t i o n " and  i n the  defini-  " c o n f i r m a t i o n " of MCT  be the f o l l o w i n g " q u a s i - c l a s s i c a l entailment as "  "dis-  which i s q u i t e p a r a l l e l to the  will  r e l a t i o n " denoted  "classical  entailment  relation": D e f i n i t i o n of the q u a s i - c l a s s i c a l entailment "P IJ-Q"  i s t r u e i f f "Q"  must be t r u e i n case "P"  relation. i s true.  (So,  69 given  that  "P" i s t r u e  haves  "P |J-Q" i s  false;  Now we make t h e Theorem I .  but  M  Q" is  otherwise  following  QMC3 i s  anything other "P |HQ" i s  than true,  we  true.)  claims  a minimal confirmation  logic.  Proof. This  is  due t o  the  fact  that  whose u n d e r l y i n g l o g i c  is  m a t i o n as  following  shown b y t h e  Lemma ( 1 ) .  as  Lemma ( 3 - D .  of  a  is  black,  is  a raven,  B a & ~ R a )(• (Ra - » B a )  Lemma ( 4 . 1 ) .  ~ B a & ~ R a ^(Ra-^Ba)  Lemma ( 4 . 2 )  ~Ba&-~Ra^-(Ra-^Ba)  (1)  above  /r  lemmas a r e a p p l i e d t o  are b l a c k ,  the  Raven H y p o t h e s i s  they w i l l y i e l d the  development  of  the  following  Raven H y p o t h e s i s  (4) As mas, to  desired r e is  considered  The R a v e n H y p o t h e s i s i s c o n f i r m e d b y and o n l y b y b l a c k  Any b l a c k n o n - r a v e n i s  Hypothesis; so to  that  "a"s  (2) A n y n o n - b l a c k r a v e n w i l l d i s e o n f i r m t h e (3)  confir-  raven.)  Ba&~Ra <'"'(Ra-•Ba)  an o b j e c t  of  lemmass  Lemma (3-2).  s u l t s when t h e  paradoxes  ~ B a & Ra K C R a - ^ B a )  Lemma ( 2 ) .  ravens  t h e o r y MOT  Ba & Ra {- ( R a - » B a )  Ri Q  When t h e  confirmation  QMC3 s o l v e s t h e  (where B i (I)  all  the  ravens.  Raven H y p o t h e s i s .  neutral with respect  to  the  Raven  and is  any n o n - b l a c k n o n - r a v e n .  the  we h a v e  derivations the  demonstrate  following  them,  and n o n - d e r i v a b i l i t i e s truth tables  assuming t h a t  of  the  above  and p a r t i a l t r u t h  we have  the  lem-  tables  Correspondence The-  70 orem o f MC3»  Lemma (1) Ba Ra  t t t f . t n f t f f f n t n f n n  Lemma (2)  Ba&Ra lr (Ra - » B a )  re t t t t  t  f n f f f  n  f  n  t n f t  n n  t  n  f f f f f f t t t f t n t t t f t n  t  t t  ~Ba&Ra  t t  Lemma (3.1) Ba Ra  tt  ©  ~Ba&~Ra  f f n  t  n  tt  t  n  Thus t h e p a r a d o x e s  n  a s we c l a i m e d .  t t  t t t t  t  t f t  n n n t  t  t  ||- — ( R a —> Ba)  (f) f  ~Ba & - R a  t  t  tt  |f- ~ ( R a -^>Ba)  (f) f  of confirmation are avoided  c o n f i r m a t i o n t h e o r y MCT a n d , h e n c e , logic  t  Lemma (4.2)  ||-(Ra—>Ba)  (D  f f t t f t t f f  ft  Ba & ~Ra  Lemma (4.1) Ba Ra  ~(Ra -*Ba)  Lemma (3»2)  B a & ~Ra \[ (Ra - » Ba)  t f t n  II-  QED.  QMC3 i s  t i n the minimal  a minimal  confirmation  4.  THE AXIOMATIZATION OF MC3  In the f o l l o w i n g we g i v e an axiom system f o r MC3 f o r  future  reference * I.  Formation r u l e s . 1. P r i m i t i v e o p e r a t o r s : ~ - , v, 2.  Auxiliary  symbols* (,  3. Sentence l e t t e r s : 4.  ).  P,Q,R,...  Formation r u l e s of w f f s a) A sentence l e t t e r b)  If  —  P,Q are w f f s ,  (or  . sentences):  i s a wff. so are ~P, P v Q ,  and P->Q.  5. Axioms: Ax 1 ) . Ax 2 ) . Ax 3 ) . Ax  4).  P 3(Q3P) (PDQ) 0 ( ( Q OR) =>(P3R)) ( - Q 3 - P ) 3 (P3Q) ((P3-P)^P)^P  (where " P D Q " i s shorthand f o r "-P" is for " P - » ~ P " . ) a  II.  " ( P - » Q ) Q " ' and V  Transformation r u l e s . 1. D e f i n i t i o n s :  2.  Def 1.  P & Q =df  ((P-*~P) v  Def 2 .  P <-» Q =df  (Q-»~Q)  (P-*Q) & (Q-*P)  Rule of i n f e r e n c e : modus ponens, P ,  )->-((P-»—P) v (Q-»~Q))  i.e.,  P->Q.//.Q  3. Rule of uniform s u b s t i t u t i o n :  the r e s u l t  r e p l a c i n g any sentence l e t t e r fey any wff itself  a theorem.  of  uniformly  i n a theorem i s  72  IN SEARCH OF MORE MINIMAL CONFIRMATION LOGICS  5.  Let's  call  the s e n t e n t i a l  logic  of a minimal  l o g i c , which i s always a q u a n t i f i c a t i o n a l minimal confirmation l o g i c " . confirmation logic  if  logic, i t s "sentential  Thus MC3 i s t h e s e n t e n t i a l  minimal  o f QMC3 .  MC3 i s n o t t h e o n l y m i n i m a l have  confirmation  confirmation logic,  f o r we c a n  i n f i n i t e l y many s u c h s e n t e n t i a l m i n i m a l c o n f i r m a t i o n  we f o l l o w  logics  t h e p a t t e r n o f t h e c o n s t r u c t i o n o f MC3 i n m - v a l u e d  f o r any m ^ 3 .  However,  i n what  valued minimal (i)  (ii)  follows  we w i l l  confirmation logics  First,  the m-valued  three-valued logic  logics,  concentrate  only  on t h r e e -  f o r two r e a s o n s : i s the simplest  kind  among  when m^-3.  Among a n y c h o i c e s  i t seems  that a three-valued  is  the only proper choice f o r a p l a u s i b l e  of  confirmation,  for I  believe  a proper three-valued l o g i c three  logics  confirmational  that  seem,  states:  and a d e q u a t e  logic theory  the three t r u t h - v a l u e s  of  i n a way, t o c o r r e s p o n d t o t h e  confirmation,  d i s c o n f i r m a t i o n , and  neutrality. I n what  follows  we a r e g o i n g t o c o n s t r u c t  s e n t e n t i a l minimal c o n f i r m a t i o n l o g i c s However,  a s we have m e n t i o n e d t h a t  many-valued of  them.  structing of  t h e r e a r e i n f i n i t e l y many  ourselves  we c a n n o t  only  candidates  study a l l  to the task  three-valued minimal c o n f i r m a t i o n l o g i c s  possible  three-valued  i n a more s y s t e m a t i c way.  minimal c o n f i r m a t i o n l o g i c s ,  E v e n i f we l i m i t  more  of con-  t h e number  c o u l d be a c e l e s t i a l o n e , f o r t h e r e a r e  (3 -D-3 -(3 -l)-(3 -2)-(3 -3) 3  possible  9  9  9  three-valued sentential  9  logics,  i f we assume  that  73  each such l o g i c c o n t a i n s a negation,  a d i s j u n c t i o n , a conjunc-  t i o n , a c o n d i t i o n a l (or i m p l i c a t i o n ) and a b i - c o n d i t i o n a l (or equivalence). So, a t t h i s p o i n t i t i s not a bad p o l i c y f o r us to l a y down some c r i t e r i a  t h a t are i n t u i t i v e l y p l a u s i b l e and t h e o r e t i c a l l y  p r a c t i c a l i n order t o p i c k out those more promising minimal c o n f i r m a t i o n l o g i c s . few  three-valued  To begin with we have the f o l l o w i n g  criteria: C r i t e r i o n I.  The three t r u t h - v a l u e s w i l l be denoted a s : t ,  f, and n; and they w i l l be i n t e r p r e t e d as " t r u t h " tion"), falsity"  "falsity"  (or "confirma-  ( o r " d i s c o n f i r m a t i o n " ) and " n e i t h e r t r u t h nor  (or " n e u t r a l i t y " ) .  Criterion II.  Any s e n t e n t i a l minimal c o n f i r m a t i o n l o g i c  must c o n t a i n a t l e a s t a negation,  a d i s j u n c t i o n , a conjunction,  a c o n d i t i o n a l (or i m p l i c a t i o n ) and a b i - c o n d i t i o n a l  (or equiva-  lence) . Criterion III.  Since any minimal c o n f i r m a t i o n l o g i c must  be able t o s o l v e the paradoxes o f c o n f i r m a t i o n when i t i s employed as the u n d e r l y i n g l o g i c of a c o n f i r m a t i o n theory, i t s negation (denoted  i f i t has o n l y one  as " ~ " ) i n order to be able t o express  "complement" o f a c l a s s  or property  non-raven, e t c . O f f i c i a l l y peat,  must be the e x t e r n a l n e g a t i o n the concept of  such as non-black, and  the e x t e r n a l n e g a t i o n obeys, to r e -  the f o l l o w i n g t r u t h t a b l e :  P II ~P t f n C r i t e r i o n IV.  f t t  Any t h r e e - v a l u e d minimal c o n f i r m a t i o n l o g i c  74 must c o n t a i n  the  "standard  junction"  its  disjunction  as  to repeat,  are given  cations,  "standard Their truth  V  Q  t  f n  & Q P t f n  t f n  t t t t f n t n n  t f n f f f n f n  t h e i r adoption for  is  this:  the e x i s t e n t i a l  They g i v e  and u n i v e r s a l  w h i c h c a n t h e n be i n t e r p r e t e d r e s p e c t i v e l y  alizations  of the  contables,  belows  P  generalization  and t h e  and c o n j u n c t i o n .  PX  The i n t e n t i o n o f tural  in  disjunction"  standard  disjunction  and o f  the  as  us  a  na-  quantifithe  standard  genercon-  junction. C r i t e r i o n V. tion")  of  Then,  its  rally  as  L e t us  bi-conditional  (or  "conditional"  confirmation  equivalence)  will  (or  logic  "implicaas  be d e f i n e d  natu-  follows: (i).  P-3»Q =df  W i t h t h e above  five  any t h r e e - v a l u e d m i n i m a l  (P-i*Q)&(Q  c r i t e r i a as  -^>P).  the minimal  confirmation  logic  requirements  we c a n p r o v e  the  for fol-  theorem:  Theorem I I . tion  the  any t h r e e - v a l u e d m i n i m a l  Def  lowing  denote  There are  960 t h r e e - v a l u e d m i n i m a l  confirma-  show t h a t  Theorem II  construct  logics. Proof. (A)  First,  to  960 c o n d i t i o n a l s tion,  the  such t h a t  standard  bi-conditional  each of  disjunction,  forms  true  them p l u s  the  a sentential  is  standard  minimal  is  the  to  external  nega-  c o n j u n c t i o n and  confirmation  logic.  the  75  (B) Secondly,  t o show t h a t the above f i v e o p e r a t i o n s form  a s e n t e n t i a l minimal c o n f i r m a t i o n l o g i c  i s t o show t h a t they con-  t a i n the f o l l o w i n g few theses, whieh the avoidance of the paradoxes of c o n f i r m a t i o n are mainly Thesis  due tot  ( l ) i Br & Rr \\-Rr -i*Br , where Bi (J) i s b l a c k , R: (l) i s a raven, r : a raven.  Thesis  (2):  Thesis  (3.1)*  Thesis  (3.2):  Thesis  (4.1):  ~ B r & - R r ^ R r -i»Br  Thesis  (4.2):  ~ B r & ~ R r ^ ~ ( R r -i*Br)  ~ B r & Rr  (Rr -^*Br) #  Br & ~ R r > R r - i * B r Br&-RrJJMRr^Br)  (G) Since each c o n d i t i o n a l o f the minimal c o n f i r m a t i o n l o g i c s t h a t we are c o n s t r u c t i n g has t o observe a l l o f the above theses, we c a n c o n s t r u c t each of the c o n d i t i o n a l s i n l i g h t o f the  theses  and argue i n the f o l l o w i n g ways: a) F i r s t ,  Thesis  (1) f o r c e s us t o adopt the f o l l o w i n g p a r t i a l  t r u t h t a b l e f o r a l l o f the c o n d i t i o n a l s t h a t we a r e c o n s t r u c t i n g : \Br Rr^  ( 2 ) f o r c e s us t o adopt the f o l l o w i n g p a r t i a l  b) Then, T h e s i s  t r u t h t a b l e f o r the c o n d i t i o n a l s :  where y€ {f,n\  R r \  f n  t  y y  .  c) T h i r d l y , the p a i r o f theses T h e s i s  (3.1)  and T h e s i s ( 3 . 2 )  f o r c e s us t o adopt e i t h e r one o f the f o l l o w i n g t r u t h t a b l e s :  Br f n  t t y  t * y t  where y € {f ,n}. d) L a s t l y , Theses (4.1) and ( 4 . 2 ) f o r c e us t o adopt one o f the f o l l o w i n g twelve p o s s i b l e p a r t i a l t r u t h t a b l e s : fn fn fn fn fn fn fn fn fn fn fn fn f n  t y y t nn nn n f n f f n f n f f f f t t t t XX X X t x y t t x y t t x y t t x y t yx t y  where x e {t,f,n} and  ye{f.n}.  (D) B y simple c a l c u l a t i o n , then, i t t e l l s us t h a t  there  are 2 v 2 - 2 - 2 « ( 2 * 3 * 3 + 2 ' 3 ' 3 + 3 + 2 + 3 + 2 + 3 + 2 + 3 + 2 + 2 + 2 ) 960 p o s s i b l e combinations f o r a l l x's and y * s . we have e x a c t l y 9 6 0 c o n d i t i o n a l s p o s s i b l e combinations. mal  confirmation  logics.  =  I n o t h e r words  (or i m p l i c a t i o n s ) out o f the  Hence, there  are 9 6 O three-valued QED.  mini-  77 6 . NEO-CLASSICAL MINIMAL CONFIRMATION LOGICS We have 960 minimal c o n f i r m a t i o n l o g i c s now, but the number is  still  too l a r g e f o r us to have a d e t a i l e d study of them.  importantly,  not every one of them i s as promising as the  So, we are going to impose some more u s e f u l r e s t r i c t i o n s  More  others. on them  i n order to e l i m i n a t e the l e s s p r o m i s i n g ones. Thus the f o l l o w i n g two new  restrictions:  Criterion cerned,  VI.  As f a r as the t r u t h - v a l u e s  t and f are c o n -  each c o n d i t i o n a l of any of the 960 minimal c o n f i r m a t i o n  l o g i c s must comply with those of the c l a s s i c a l Criterion  VII.  " P - i » P " must h o l d ,  conditional.  for this  most obvious theorem t h a t we can expect from any Any minimal c o n f i r m a t i o n l o g i c restrictions logic",  i s one of  the  logic.  t h a t obeys the above two new  w i l l be c a l l e d a " n e o - c l a s s i c a l minimal c o n f i r m a t i o n  s i n c e each t r u t h t a b l e of any s e n t e n t i a l connective  the c l a s s i c a l l o g i c mation l o g i c  of  and t h a t of any n e o - c l a s s i c a l minimal c o n f i r -  are i d e n t i c a l as f a r as the t r u t h - v a l u e s  t's  and  f s  are concerned. We admit t h a t the i n t e n t i o n of imposing the above two new restrictions In t h i s  i s not r e a l l y  confirmation-theoretical  sense we need not to impose them.  serves one of our i n t e l l e c t u a l d i f f e r e n c e between the  classical  minimal c o n f i r m a t i o n l o g i c ? l e t us t r y  curiosity: logic  However, Is  oriented. the  intention  there any  important  and the  neo-classical  Before we can answer t h i s  question,  to c a l c u l a t e the number of the n e o - c l a s s i c a l minimal  confirmation l o g i c s .  It  turns  out t h a t we have the  following  result: Theorem I I I .  There are 32 n e o - c l a s s i c a l minimal c o n f i r -  mation l o g i c s . Proof. A f t e r the imposing of the two new r e s t r i c t i o n s any c o n d i t i o n a l P -^*Q  of each o f the n e o - c l a s s i c a l minimal  l o g i c s can take o n l y one of the f o l l o w i n g  t f n  form 1  form 2  t f n  t f n  t f y t t X y y t  t f y t t y y t t  confirmation  two forms:  where x € { t , f , n } and y € {f,n}. So, there are 2-3*2- 2 + 2-2-2 confirmation l o g i c s .  = 32 n e o - c l a s s i c a l QED.  minimal  79 7. QUASI CONFIRMATION LOGICS AND QUASI CONFIRMATION THEORIES We have only 32 n e o - c l a s s i c a l minimal c o n f i r m a t i o n l o g i c s f o r the p o s s i b l e condidates still  of a c o n f i r m a t i o n l o g i c .  However, i t i s  the case t h a t not each of them i s as promising  as the o t h e r s ,  f o r not each of them can a v o i d Goodman's paradox:'*' and t h i s i s because to a v o i d Goodman's paradox the f o l l o w i n g two 2 must not h o l d i n l i g h t of Leblanc's a n a l y s i s : •  Cond 1.  2.  Cond  (  •  equivalences  •  Ds((D&R)vH^R)))w(Di-R)  (  ((D-i*~D)-M(D& R) v (-D & ~R)) ) A (D -i*-~R), where R: a c e r t a i n o b j e c t i s a marble, D: an o b j e c t i s drawn on a c e r t a i n day.  A f t e r checking  through each o f the 32 n e o - c l a s s i c a l minimal  c o n f i r m a t i o n l o g i c s we f i n d t h a t Cond 1 and Cond 2 do not hold i n o n l y 20 of the 32 n e o - c l a s s i c a l minimal c o n f i r m a t i o n  logics.  These 20 n e o - c l a s s i c a l minimal c o n f i r m a t i o n l o g i c s w i l l be f u r ther c a l l e d  "quasi c o n f i r m a t i o n l o g i c s " and denoted as Q C 3 ( i ) t  where l ^ i ^ 2 0 ,  and each QC3(i) has the f o l l o w i n g r e s p e c t i v e con-  d i t i o n a l s , denoted as  t f n  6^  t f n  t f n  t f n  t f n t t n f n t  t f n t t f n n t  t f n t t f f n t  12,  14  t f n t t f f f t  -12, t f n t t n n t t 20,  t f n t t f n t t  t f f t t n f t t  t f f t t f f t t  3,  s.  t f n  t f n  t f n  t f n t t t n n t  t f n t t t f n t  t f n t t n n n t  8, t f n  5,  2  1> t f n  f or each i = 1 , 2 , 3 , . . . ,20:  9,  t f n t t t f f t  t f n t t n n f t  t f n t t n f f t  t f n t t f n f t  15,  16  17,  18  t f n t t n f t t  t f n t t f f t t  t f f t t n n t t  t f f t t f n t t  7. — ? t f n t f n t t t n f t  7-  1. Goodman's paradox w i l l be d i s c u s s e d i n P a r t IV below. 2. Leblanc, Hi "That p o s i t i v e i n s t a n c e s a r e no h e l p " , pp.453-462.  80  So, as  the  "quasi  if  we employ any one  underlying logic confirmation  paradoxes of  of  of  the  20 q u a s i  a new c o n f i r m a t i o n  theory",  c o n f i r m a t i o n but  we c a n ,  confirmation theory  h o p e f u l l y , not  a l s o Goodman's  to  be  logics called  only avoid  paradox.  the  8. SOME PROPERTIES OF THE QUASI CONFIRMATION LOGICS The  20 q u a s i c o n f i r m a t i o n l o g i c s have some  outstanding  properties: (A) F i r s t , as we know, each o f the 20 q u a s i  confirmation  l o g i c as the u n d e r l y i n g l o g i c o f a q u a s i c o n f i r m a t i o n  theory  can a v o i d the paradoxes o f c o n f i r m a t i o n and (we s h a l l  later  show) Goodman's paradox. (3) The second o u t s t a n d i n g p r o p e r t y o f each o f t h e 20 q u a s i c o n f i r m a t i o n l o g i c s i s the f o l l o w i n g one: Theorem IV.  Each o f the 20 q u a s i c o n f i r m a t i o n l o g i c s i s  " t r u t h - f u n c t i o n a l l y incomplete"; i n o t h e r words, t h e r e a r e t r u t h - f u n c t i o n s t h a t cannot be expressed i n each o f the 20 quasi confirmation  logics.  Proof. To g i v e one example o f t r u t h - f u n c t i o n s t h a t cannot be expressed i n each of the 20 q u a s i c o n f i r m a t i o n l o g i c s , c o n s i d e r the f o l l o w i n g t r u t h p t f n  just  tablei  g(P) n n t  We c l a i m now t h a t there i s no t r u t h f u n c t i o n i n any  of the  20 q u a s i c o n f i r m a t i o n l o g i c s t h a t c a n express the t r u t h - v a l u e s of g ( P ) .  T h i s i s becausei  ( i ) a l l s e n t e n t i a l operations of  each o f the 20 q u a s i c o n f i r m a t i o n l o g i c s a r e " n o r m a l " o r 1  "quasi-classical",  i . e . , i f the t r u t h - v a l u e s o f the domain  1. Cf Rescher, N i c h o l a s : Many-valued l o g i c ,  1 9 ^ 9 , PP.55-57.  82  of a  truth  f u n c t i o n are the  classical  t r u t h - v a l u e s t or f ,  then the t r u t h v a l u e s of the range of the t r u t h f u n c t i o n w i l l a l s o be of the c l a s s i c a l t r u t h - v a l u e s j ( i i ) the of the composition normalj we  truth-values  of any number o f normal o p e r a t i o n s are  ( i i i ) but the t r u t h f u n c t i o n g(P)  again  i s not normal. Hence,  have t h a t g(P) cannot be d e f i n e d i n each of the 2 0 q u a s i con-  firmation logics.  QED.  (C) Although each o f the 20 t r u t h - f u n c t i o n a l l y incomplete,  quasi c o n f i r m a t i o n l o g i c s i s i t i s not a d e f e c t as an under-  l y i n g l o g i c of a c o n f i r m a t i o n theory.  Nevertheless,  each o f  the 20 q u a s i c o n f i r m a t i o n l o g i c s has the f o l l o w i n g n i c e  pro-  perty: Each of the 20 q u a s i c o n f i r m a t i o n l o g i c s i s  Theorem V. "normally  complete" (or " q u a s i - c l a s s i c a l l y complete"),  i.e.,  each of them i s " t r u t h - f u n c t i o n a l l y complete as f a r as c l a s s i c a l t r u t h - v a l u e s t and any  three-valued  two  f are concerned"s i n other words  normal o p e r a t i o n s can be expressed  d e f i n a b l e ) i n each of them.  the  (or are  In p a r t i c u l a r , the f o l l o w i n g  t r u t h t a b l e s of a l l p o s s i b l e unary and b i n a r y normal  o p e r a t i o n s can be expressed (and are the 20  quasi confirmation  logics:  h(P,Q) t f n where x€  [t,f}  X  X y  X X  y  y y y  and y g / t , f , n } •  k(P) X X  y  d e f i n a b l e ) i n each of  Note t h a t Theorem V i s a consequence of the f o l l o w i n g more g e n e r a l theorem: Each of the 20  Theorem VI. an  "extended  quasi confirmation l o g i c s  has  ( d i s j u n c t i v e ) normal form theorem", which says  t h a t each sentence o f any can be expressed  o f the 20  quasi confirmation  logics  as a c o n j u n c t i o n of d i s j u n c t i o n s , where each  d i s j u n c t of every d i s j u n c t i o n i s one of the f o l l o w i n g "molecules" i T  , F(i)  p  p  (where l ^ i ^ 2 0 ) ,  N  p  and  mP  such t h a t these molecules have the f o l l o w i n g t r u t h t a b l e s and  they can be d e f i n e d i n each of the 20  l o g i c s as shown belowi P t  X  n Def  a.  Def  b.  Def  c.  Def  d.  mP  T t f f  =df  p  F(i) f t f  P&-P  p  N  f  f t  p  mP f  f n  quasi  confirmation  84 9 . SOME MORE PROPERTIES OF THE QUASI CONFIRMATION LOGICS Before we c a n g i v e some more i n t e r e s t i n g  p r o p r e t i e s about  the 20 q u a s i c o n f i r m a t i o n l o g i c s , we need the f o l l o w i n g few definitions: Definition logics",  o f Isomorphic L o g i c .  Two l o g i c s a r e "isomorphic  i f a l l p r i m i t i v e o p e r a t o r s o f one l o g i c a r e d e f i n a b l e  by the p r i m i t i v e operators Definition  o f the other l o g i c and v i c e v e r s a .  of Equivalent Logic.  "equivalent l o g i c s " ,  Two l o g i c s L I and L 2 a r e  i f the f o l l o w i n g two c o n d i t i o n s a r e s a t i s -  fied: ( i ) They a r e isomorphic  logics;  ( i i ) they have t h e same "semantical i . e . , we have: P j £ Q Definition  i f fP ^  o f Twin L o g i c .  i f they a r e isomorphic  entailment  relation",  Q. Two l o g i c s  a r e "twin  l o g i c s , but not e q u i v a l e n t  logics",  logics.  Now we c a n c l a i m the f o l l o w i n g few theorems about the 20 quasi confirmation l o g i c s Theorem V I I .  Each o f the 20 q u a s i c o n f i r m a t i o n l o g i c s i s  another's isomorphic Theorem V I I I .  again:  logic.  A l l 20 q u a s i c o n f i r m a t i o n s l o g i c s a r e twin  l o g i c s t o one another. Theorem IX.  Lukasiewicz's  three-valued  s e n t e n t i a l l o g i c L3  and any o f t h e 20 ( s e n t e n t i a l ) q u a s i c o n f i r m a t i o n l o g i c a r e twin l o g i c s , b u t L 3 and any of the 20 ( s e n t e n t i a l ) q u a s i c o n f i r m a t i o n a l l o g i c s a r e not e q u i v a l e n t  logics.  85  We know t h a t  any of the 20  t r u t h - f u n c t i o n a l l y incomplete.  quasi  confirmation  However,  l o g i c s are  i f we add t o any of them  the f o l l o w i n g o p e r a t i o n denoted as "n" and i n t e r p r e t e d as "the n e u t r a l i z e d o p e r a t i o n " which has the t r u t h t a b l e g i v e n p  nP  t f n  n n n  belowi  then each o f the 20 q u a s i c o n f i r m a t i o n l o g i c s i s t r u t h - f u n c t i o n a l l y complete. Theorem X. adding  I n other words, we make the f o l l o w i n g c l a i m : Each of the 20 q u a s i c o n f i r m a t i o n l o g i c s  after  the n e u t r a l i z e d o p e r a t i o n t o i t i s t r u t h - f u n c t i o n a l l y com-  plete . Proof. We know t h a t i f we add S l u p e c k i ' s T - o p e r a t i o n three-valued  s e n t e n t i a l l o g i c L 3 , then we w i l l have a t r u t h - f u n c -  t i o n a l l y complete s e n t e n t i a l l o g i c pecki's T-operation  denoted as  i s n o t h i n g but the n - o p e r a t i o n  former i s u n i n t e r p r e t e d while differently,  to Lukasiewicz's  otherwise  Now except  (i) Slut h a t the  the l a t t e r i s i n t e r p r e t e d and denoted  they obey e x a c t l y the same t r u t h t a b l e ; and  ( i i ) L3 and any of the 20 ( s e n t e n t i a l ) q u a s i c o n f i r m a t i o n are twin l o g i c s as i n d i c a t e d by Theorem IX.  logics  Hence, i f L^ i s t r u t h -  f u n c t i o n a l l y complete, so must be each o f the 20 ( s e n t e n t i a l ) q u a s i c o n f i r m a t i o n l o g i c s a f t e r adding  t o the n - o p e r a t i o n .  1. C f . , Rescher, N» Many-Valued L o g i c , 1 9 6 9 , p.335.  QED.  86  10. QUASI  THE AXIOMATIZATION OF A TRUTH-FUNCTIONALLY COMPLETE (SENTENTIAL)  CONFIRMATION LOGIC  Since each of the after  adding to  20 q u a s i  (sentential)  it  the  n-operation is  we c a n p i c k  out  one  simple reason that  all  other quasi  plete,  logics  c a n have  The  CQSC3.  as of"  we w i l l  n-operation.  Since i t  troduce  into  study f o r  (sentential)  is  it  p i c k out f o r  This  the  confirmation  logic  study i s  will  t r u t h - f u n c t i o n a l l y complete,  the  and r e a d as  special  sentential  following  "the  w h i c h have  we w i l l  "not" and "the  respectively  p  -P  'P  t f n  f t n  f t f  as  the  and "the  following  QC3(1)  be d e n o t e d  two n e g a t i o n s , d e n o t e d  i n t e r n a l negation"  CQSC3 c a n be a x i o m a t i z e d  I.  t r u t h - f u n c t i o n a l l y com-  them f o r s p e c i a l  " - " and ' " " and i n t e r p r e t e d a s  gation",  logics  similar properties.  one t h a t  w i t h the  of  confirmation  also  P r i m i t i v e operators*  contradictory i n t r i n s i c ne-  truth  tables*  follows*  2.  Auxiliary  3-  Sentence  4.  Formation rules (i)  letters:  (,  If  P,  of wffs  Q are wffs,  Axioms* Ax 1 .  P ^ ( Q OP)  n.  ).  P,Q,R,...  A sentence l e t t e r  (ii) 5.  symbols*  v,  (or  is  a  . sentences). wff.  so a r e - P ,  in-  respectively  Formation r u l e s . 1.  as  P - * Q , P v Q, and n P .  87 Ax  2.  Ax 3Ax  4.  (P3Q)3((Q5R)o(PPR))  ( - Q o - P ) r ? (Pt>Q) ( ( P 3 - P ) 3 P ) 3 P  Ax 5.  n P o -nP  Ax 6.  -nPr>nP  (Note t h a t " O " i s a shorthand f o r  "(P-*Q) v Q" and i s  p a r t of the p r i m i t i v e b a s i s of the II.  not  language.)  Transformation r u l e s . 1.  2.  Definitions. Df 1 .  P & Q =df - ( - P  Df 2.  P<-»Q =df  Df 3.  - P =df  Df 4.  'P =df  v-Q)  (P->Q) & ( Q - » P ) P-*-P  *  P  Rule of i n f e r e n c e : modus ponens, P , P-*Q  3.  f-  i.e.,  Q  Rule of uniform s u b s t i t u t i o n : p l a c i n g any sentence l e t t e r  the r e s u l t  by any wff  of uniformly r e -  i n a theorem i s  it-  s e l f a theorem. Note t h a t the j u s t i f i c a t i o n CQSG3 i s due to the f a c t t h a t i t  of the above a x i o m a t i z a t i o n f o r and  are twin l o g i c s .  Note a l s o that the q u a n t i f i c a t i o n a l g e n e r a l i z a t i o n of CQSC3 w i l l be denoted as CQQC3,  (which can be read as "complete, quan-  t i f i c a t i o n a l quasi confirmation  logic")  88 11. CONFIRMATIONAL VALUATION AND A COMPLETE SEMANTICS FOR ANY THREE-VALUED QUANTIFICATIONAL LOGIC WITH IDENTITY Up t o t h i s p o i n t we have not c o n s i d e r e d  the v a l u a t i o n of the  atomic and t h e g e n e r a l sentences o f any q u a s i c o n f i r m a t i o n with, or without, i d e n t i t y .  logic  I n t h i s s e c t i o n we w i l l g i v e a f u l l y  developed semantics, which i s complete, f o r any t h r e e - v a l u e d  quan-  t i f i c a t i o n a l l o g i c with i d e n t i t y from the view p o i n t o f c o n f i r m a tional valuation. T r u t h t a b l e s as a matrix language t e l l us only the v a l u a t i o n s of molecular  sentences.  So, how do we evaluate  the atomic sentences  and, i n g e n e r a l , the atomic as w e l l as the g e n e r a l sentences o f any  o f the q u a s i c o n f i r m a t i o n l o g i c s ?  Before we can do t h i s , l e t  us g i v e b r i e f l y the grammar of CQQC3 and some r e l e v a n t p r e l i m i nary  concepts. I.  1  The grammar o f CQQC3 with i d e n t i t y .  The e x p r e s s i o n s  of the  language o f CQQC3 w i t h i d e n t i t y are any f i n i t e l e n g t h o f s t r i n g s of symbols o f the f o l l o w i n g two kinds* (1) V a r i a b l e s .  The i n d i v i d u a l v a r i a b l e s are the lower case  i t a l i c l e t t e r s "x" through " z " with, o r without, numeral s u b s c r i p t s . (2) Constants. (i) (ii)  They f a l l i n t o two major kinds*  The l o g i c a l  constants,  The n o n - l o g i c a l c o n s t a n t s . a) P r e d i c a t e  These i n c l u d e *  letters.  b) I n d i v i d u a l c o n s t a n t s . A s e r i e s o f fundamental n o t i o n s a r e , then, d e f i n e d as f o l l o w s * A p r e d i c a t e l e t t e r of degree n ( o r an n-ary, o r n-place  pre-  d i c a t e l e t t e r ) i s a p r e d i c a t e l e t t e r having as s u p e r s c r i p t s a nul.Here i n t h i s r e s p e c t I t r y t o f o l l o w Mates' approach expressed i n h i s Elementary Logic as c l o s e l y as p o s s i b l e , although there are i n e v i t a b l y some fundamental d i f f e r e n c e s .  89  meral f o r the p o s i t i v e i n t e g e r n, which w i l l u s u a l l y be being  omitted,  understood.  An i n d i v i d u a l symbol i s e i t h e r an i n d i v i d u a l v a r i a b l e  or an  i n d i v i d u a l constant. An atomic  formula i s e i t h e r a sentence l e t t e r which i s de-  f i n e d as a 0-place p r e d i c a t e t e r followed positive  l e t t e r or an i - p l a c e p r e d i c a t e  by a s t r i n g o f i i n d i v i d u a l symbols,  let-  where i i s any  integer.  A formula i s e i t h e r an atomic  formula o r e l s e d e f i n e d  recur-  s i v e l y by the u s u a l f o r m a t i o n r u l e s . Furthermore,  an occurrence of an i n d i v i d u a l v a r i a b l e v i n a  formula A i s bound. i f i t i s w i t h i n mula o f the form  (3v)B  an occurrence i n A of a f o r -  or of the form(\/v)B; otherwise i t i s a f r e e  occurrence. A sentence  (or w f f ) i n our o f f i c i a l  sense i s a formula i n  which no i n d i v i d u a l v a r i a b l e o c c u r s f r e e . Finally, general,  a formula t h a t  i s not an atomic  i f i t begins w i t h a u n i v e r s a l  otherwise i t i s c a l l e d m o l e c u l a r . of the s e n t e n t i a l c a l c u l u s if  formula i s c a l l e d  or e x i s t e n t i a l q u a n t i f i e r ;  And, a sentence  or s e n t e n t i a l l o g i c  is a  sentence  (say, of CQQC3),  i t c o n t a i n s no i n d i v i d u a l symbol. I I . Some a d d i t i o n a l t e r m i n o l o g y and s e m a n t i c a l c o n c e p t s .  need the f o l l o w i n g  a d d i t i o n a l terminology and s e m a n t i c a l concepts  b e f o r e we can d i s c u s s  about c o n f i r m a t i o n a l  valuations!  (a) For any formula A, i n d i v i d u a l v a r i a b l e v, and symbol w,  A  v / / w  We  i s the r e s u l t of r e p l a c i n g  of v i n A by occurrences of  a l l free  individual  occurrences  w.  (b) Given a sentence A of CQQC3 with or without i d e n t i t y , an  90 i n t e r p r e t a t i o n c a n be c h a r a c t e r i z e d r o u g h l y f i r s t later)  i n two  (1)  "c" h a s a d e n o t a t i o n ,  (2)  steps*  is  an i n d i v i d u a l  if  c does n o t e x i s t ,  (where of  g i v e n domain D o f is  1  2  c,  iff  c exists,  sentence  the  the  going  to  have  is  an a b s o l u t e  empty  to  classical valuation.  more p r e c i s e l y ,  cate  letters  2.  3.  4. 5. 6.  c a n be c a r r i e d  to p r e d i c a t e l e t t e r s  (or,  1.  any  proper  subset  which  set.  a l l have d e n o t a t i o n s  sets of  o f degree  or n o n - e n t i t i e s , ^  semantics,  an i n t e r p r e t a t i o n i n our c o n f i r m a t i o n a l v a l u a t i o n  stants  Then,  one  constants).  Thus t h e v a l u a t i o n o f any a t o m i c s e n t e n c e  the  "c"  truth-value n,  c o n t a i n i n g "c" a s  a l w a y s assumed t o be t h e a c t u a l w o r l d ^ o r i t s  includes  (where  and  t h e n " A ( c ) " has  an a t o m i c  individual  S i n c e what we a r e  namely,  constant ):  "A(c)" i s  its  (and p r e c i s e l y  2 it  and so  of  degree  entities assigns  out  1 it  whose  individual  i n the  assigns  similar  way  "properties"^  or n o n - e n t i t i e s ) .  binary  con-  " r e l a t i o n s " of  To p r e d i entities  on.  What a n i n d i v i d u a l c o n s t a n t d e n o t e d i s c a l l e d " i n d i v i d u a l " , " o b ject" or "entity". Since a p r e c i s e d e f i n i t i o n of c o n f i r m a t i o n r e q u i r e s reference to some d e f i n i t e " l a n g u a g e o f s c i e n c e " L a s Hempel s a y s , we c a n assume t h a t a n y i n d i v i d u a l c o n s t a n t "c" a p p e a r s i n L has a m e a n i n g a n d , h e n c e , w h e t h e r "c" e x i s t s ( i n t h e a c t u a l w o r l d ) o r n o t c a n be d e c i d e d w i t h t h e h e l p o f t h e m e a n i n g o f "c" i n L . A s e c o n d way t o d e v e l o p o u r c o n f i r m a t i o n a l v a l u a t i o n i s t o assume t h a t any g i v e n domain o f a n i n t e r p r e t a t i o n i s a w o r l d of p o s s i b l e e n t i t i e s a n d t h a t we a l s o have a p r e d i c a t e "A" m e a n i n g " . . . i s a c t u a l i z e d ( i n the a c t u a l w o r l d ) " . T h e n , when i t i s n o t t h e c a s e t h a t A c , we have c a s e ( 2 ) a b o v e ; o t h e r w i s e , we have c a s e ( 1 ) . T h i s i s q u i t e d i f f e r e n t from the c l a s s i c a l v a l u a t i o n . I f a n y s e t c o n t a i n s some n o n - e n t i t i e s , t h e s e t w i l l be c o n s i d e r e d as n o t r e a l l y a " p r o p e r t y " . S i m i l a r l y , f o r any p - a r y " r e l a t i o n " . A t h i r d way t o d e v e l o p o u r c o n f i r m a t i o n a l v a l u a t i o n i s n e v e r t o l e t any n o n - e n t i t i e s e n t e r o u r b a s i c v o c a b u l a r i e s . T h u s , any n o n - e n t i t y i s n o t i n a n y g i v e n d o m a i n n o r i n a n y r e l a t i o n . And when we e n c o u n t e r some ( c o m p l e x ) t e r m s o f n o n - e n t i t i e s , we c a n a n a l y z e them b y our theory of d e f i n i t e d e s c r i p t i o n s . T h u s , when no u n i q u e e n t i t y c a n s a t i s f y t h e d e s c r i p t i v e p h r a s e " ( t h e x ) D x " , we have t h a t t h e t r u t h ^ v a l u e o f "Dc" i s n (where " ( t h e x ) " i s o u r i n f o r m a l n o t a t i o n f o r the d e s c r i p t i v e o p e r a t o r . )  91 And  t o sentence l e t t e r s i t a s s i g n s t r u t h - v a l u e s o r one of the  confirmation-statuses,  v i z . , confirmation,  d i s c o n f i r m a t i o n o r neu-  trality.^ (c) L e t I n t and I n t * he two i n t e r p r e t a t i o n s of CQQC3 with o r without i d e n t i t y , and l e t k be an i n d i v i d u a l c o n s t a n t .  Then, I n t  i s a k - v a r i a n t o f I n t * e x a c t l y when I n t and I n t * are the same or d i f f e r only i n what they a s s i g n t o k. (d) A sentence i s v a l i d  ( o r l o g i c a l l y t r u e ) , i f i t i s t r u e un-  der every i n t e r p r e t a t i o n s otherwise i t i s i n v a l i d . (e) A sentence A i s a consequence o f a s e t o f sentences S, which can a l s o be symbolized as "S||-A", i f f there i s no i n t e r p r e t a t i o n under which a l l sentences o f S a r e t r u e and A i s not t r u e . ( f ) A s e t o f sentences S i s c o n s i s t e n t ( o r s a t i s f i a b l e ) i f f there i s an i n t e r p r e t a t i o n under which a l l sentences o f S are t r u e . F i n a l l y , we can g i v e the c o n f i r m a t i o n a l v a l u a t i o n i n our o f f i cial  sense below. III.  mational  Confirmational  valuation.  A t l a s t we can g i v e the c o n f i r -  v a l u a t i o n p r e c i s e l y as f o l l o w s :  (1) L e t us s t a r t from the s i m p l e s t case.  Consider  the 1-place  atomic sentence: Ga. Case 1.  Suppose t h a t ( i ) under an i n t e r p r e t a t i o n I n t the i n -  d i v i d u a l constant  "a" denotes no e n t i t y i n a domain D o r ( i i ) the  p r e d i c a t e l e t t e r "G" r e f e r s t o no p r o p e r t y under the I n t . we make the assignment: V ( G a ) = n , where " V ( of  8.  )" means "the v a l u a t i o n  ( )" o r "the t r u t h - v a l u e o f ( ) " . Case 2.  7.  Then,  Suppose t h a t the "a" under the I n t denotes some en-  T h i s i s another fundamental d i f f e r e n c e o f the c o n f i r m a t i o n a l v a l u a t i o n and the c l a s s i c a l v a l u a t i o n . Note t h a t here we do not adopt the second or the t h i r d way o f dev e l o p i n g our c o n f i r m a t i o n a l v a l u a t i o n .  92  tity,  say, a i n domain D, and a l s o suppose t h a t the "G  same I n t r e f e r s t o some p r o p e r t y , cases t o be f u r t h e r Subcase 2.1. Int.  say, G.  under the  H  Then, there are two sub-  considered: Suppose t h a t the a has the p r o p e r t y  G under the  Then, we make the assignment: V(Ga)= t under the I n t . Subcase 2 . 2 .  Suppose t h a t the a f a i l s t o possess the p r o p e r -  t y G under the I n t . (2)  Now c o n s i d e r  Then, we have: V(Ga) = f under the I n t . the g e n e r a l case o f a sentence of j - p l a c e p r e -  c a t e l e t t e r , where j > l , such as t h i s one: Ga^a2...a... Case 1.  Suppose t h a t  ( i ) under i n t e r p r e t a t i o n I n t one or more  or a l l o f the i n d i v i d u a l c o n s t a n t s  "a^","a ",...,"aj" f a i l 2  t o denote  any e n t i t y i n a g i v e n domain D, o r ( i i ) the j - p l a c e p r e d i c a t e "G" r e f e r s t o no p r o p e r t y  o r no r e l a t i o n under the I n t .  letter  Then, we  have: VfGa^ag...a^)= n under the I n t . Case 2 .  Suppose t h a t each o f the j i n d i v i d u a l c o n s t a n t s de-  notes some e n t i t y , say, a-^.a^,. •. »a.. i n a domain D under an i n t e r p r e t a t i o n I n t , and a l s o t h a t the j - p l a c e p r e d i c a t e some p r o p e r t y  or r e l a t i o n ,  ( i ) V(Ga^a2 • • .a^ ) = t  u n c  *  e r  N  G  M  say G under the same I n t .  r e f e r s to Then, we have:  "the I n t , i f <a^,a ,... » j > a  n  a  2  p e r t y o r r e l a t i o n G under the I n t : otherwise  s  the p r o -  ( i i ) V(Ga a .•.a^) = f 1  2  under the I n t . (3)  The v a l u a t i o n s o f ~ A , A v B, A & B, A-»B, A+*B, e t c . c a n be  c a r r i e d out a c c o r d i n g tence  to t h e i r t r u t h t a b l e s .  (So can be any sen-  i n v o l v i n g some new s e n t e n t i a l c o n n e c t i v e s  whose t r u t h t a b l e s  are once g i v e n and added t o CQQC3-) (4)  Suppose t h a t A=(\/v)B.  t a t i o n I n t e x a c t l y when V ( B (ii) V(A)=f  v / / k  Then,  (i) V(A)=t  under i n t e r p r e -  ) = t on every k - v a r i a n t o f the I n t ;  under the I n t , e x a c t l y when V ( B  V / / k  ) = f f o rat least  93 one k - v a r i a n t o f t h e the  Suppose  pretation the  Int  Int;  that  A = (3v)B.  e x a c t l y when V ( B  ( i i ) V(A)=f  every k - v a r i a n t o f the the  this  "d"  Finally,  one:  d=e  We  evaluate  or  "e"  tation  V /  under  Int;  V(A)=n  Then,  (i)  V(A)=t  ^ ) = t  for  at  C  the  Int  under  least  one  an  under  e x a c t l y when V ( B  ( i i i ) i n a l l other  cases,  inter-  k-variant V / / k  ) =f  for  V(A)= n  Int,  that  V ( d = e) = t the  (7) tional  an i d e n t i t y sentence  (where  are  it  " d " and  i n the  we h a v e :  " d " and  under  ?  how d o we e v a l u a t e  o r "both d e n o t e  both  "e"  Int;  denote under  the  are  will  H  have  the  a  sentences w i l l  of  hence,  at  mantics for  last  the  under  they  such as  same  the  V(d = e ) = f  term.  n i f the  stand  logics  the  an  interpre-  Int,  where  we  have  entity  in D  Under our  the  Int.  confirma-  "(Vx)Hx",  " K a ' \ "a = a"  g i v e n domain i s  empty  the  Such  (relative  with  its  to  set).  developed or complete  including  our c o n f i r m a t i o n a l v a l u a t i o n as  when what  f o r no e n t i t i e s . )  we h a v e p r o v i d e d a f u l l y  our three-valued  constants.)  under  be c a l l e d "an empty s e n t e n c e " domain i s  as  ( i i ) i n the case  denote  "(3x)Gx",  truth-value "b"  Int;  such  case  the  any t h r e e - v a l u e d q u a n t i f i c a t i o n a l l o g i c adopt  I n the  i n D under  a useful  " a " and  (i)  i n a domain D under  under  i f  two i n d i v i d u a l  way*  ( i i i ) we h a v e :  i n t e r p r e t a t i o n when the So,  not  Int  sentences  (and,  kind  following  Now we c a n i n t r o d u c e  "a = b  e"  some e n t i t i e s  otherwise  valuation,  B  V(d = e ) = n  empty s e t  the  cases,  Int. (6)  and  ( i i i ) i n a l l other  Int. (5)  of  Int;  se-  CQQC3 ( a s  well  as  identity i f  i t  will  semantics.)  for  94 12. THE FAILURE OF THE 20 QUASI-CONFIRMATION TISFY A MONOTONICITY CONDITION It  LOGICS TO SA-  1  appears t h a t there are 20 semantic and s y n t a c t i c  solu-  t i o n s of the paradoxes of c o n f i r m a t i o n from the semantics p r o v i d e d i n the l a s t s e c t i o n .  However,  these 20 s o l u t i o n s seem  to c o n f l i c t w i t h a monotonicity c o n d i t i o n . The i n t u i t i v e that i t  i d e a of what might be c a l l e d m o n o t o n i c i t y  should be the case that i f  to o b s e r v a t i o n r e p o r t s  c o n f i r m a h y p o t h e s i s are added some o b s e r v a t i o n r e p o r t s  is  that  that  cannot d i s c o n f i r m that h y p o t h e s i s the r e s u l t i n g o b s e r v a t i o n r e p o r t s would s t i l l However,  this  c o n f i r m the h y p o t h e s i s . i s not the case f o r any of the 20 q u a s i - c o n -  f i r m a t i o n l o g i c s t h a t are claimed to s o l v e the paradoxes of confirmation.  Thus c o n s i d e r an o b s e r v a t i o n r e p o r t  0 that  con-  f i r m s the raven h y p o t h e s i s ! H:  Ct/x) (Rx-*Bx) ;  f o r example, l e t 0 be the (1)  conjunction  (Ra & B a ^ & . . . & (Ra & B a ) 1  R  R  .  The development of the raven h y p o t h e s i s r e l a t i v e v a t i o n i s D, the (2)  Now, l e t  1  obser-  conjunction  ( R a - » B a ) & . . . & (Ra ~*Ba ) 1  to t h i s  n  n  .  0^ be the claimed o b s e r v a t i o n r e p o r t based on an o b j e c t  whose name i s  " b " such t h a t "Bb" i s true and "Kb" i s n e u t r a l ; 0^,  1. T h i s s e c t i o n was r e - w r i t t e n a f t e r my o r a l defence when a s e r i ous mistake was d i s c o v e r e d by P r o f . Richard E . Robinson.  95  then, i s (3  )  Bb & ~Rb .  The development of H r e l a t i v e t o 0 ^ i s the c o n d i t i o n a l  (k)  Rb -»Bb ,  where "-*" i s any c o n d i t i o n a l of the 2 0 q u a s i - c o n f i r m a t i o n  logics.  Not o n l y does 0^ n e i t h e r s e m a n t i c a l l y e n t a i l s  nor s e m a n t i c a l l y  entails  r e p o r t 0* ob-  the negation  of D^, but the o b s e r v a t i o n  t a i n e d by c o n j o i n i n g 0 and 0 ^ (5)  ( R a & B a ) & ... & ( R a & B a ) & (Rb &Bb) 1  1  n  n  n e i t h e r s e m a n t i c a l l y e n t a i l s the development D* (6 ) nor  (Ra -tBa^^) & ... & ( x  R a  ~ * a ) & (Rb-*Bb) B  n  n  s e m a n t i c a l l y e n t a i l s the n e g a t i o n Thus, 0 * n e i t h e r confirms  of D*.  nor d i s c o n f i r m s  the raven hypo-  t h e s i s H. However, i s t h i s s i t u a t i o n r e a l i s t i c g i v e n i n the l a s t  under the semantics  section?  I f "Rb" i s n e u t r a l , e i t h e r what "b" names i s not a bona f i d e o b j e c t or the p r e d i c a t e  "R" i s i n t e r p r e t e d by something t h a t i s  not a bona f i d e p r o p e r t y tics.  according  t o our c o n f i r m a t i o n a l seman-  But i f "b" names something t h a t i s not a bona f i d e  object,  then "Bb" w i l l not be t r u e ; and i f the i n t e r p r e t a t i o n of "R" i s not a bona f i d e property,  then "Ra^" w i l l not be true  to our conf i r m a t i o n a l semantics, f o r any i such t h a t Thus, a c c o r d i n g  according  1 4i4n.  t o our c o n f i r m a t i o n a l semantics g i v e n i n the  96  l a s t s e c t i o n the problem o u t l i n e d above would not happen.  Un-  f o r t u n a t e l y , the above semantics i s not the only one p o s s i b l e . There i s a r e a l problem of f i n d i n g  a logic  t h a t both s o l v e s the  paradoxes o f c o n f i r m a t i o n and s a t i s f i e s the monotonicity t i o n i n t u i t i v e l y presented  condi-  above.  Note t h a t i t might appear t h a t Hempel's s o l u t i o n or d i s s o l u t i o n of the paradoxes of c o n f i r m a t i o n runs a f o u l of the monot o n i c i t y c o n d i t i o n as w e l l .  Thus, l e t 0 and D be as above,  and  l e t 0_2 be the o b s e r v a t i o n r e p o r t based on an o b j e c t whose name is  " c " such t h a t "Rc" i s true and no t r u t h - v a l u e i s assigned t o  "Be".  Then, p_ i s simply 2  hypothesis (7)  r e l a t i v e to  R C D B C  "Rc" and the development of the raven  is:  .  However, the o b s e r v a t i o n r e p o r t 0 # obtained by c o n j o i n i n g 0 and p_2  is:  (8)  ( R a . B a ) " ... • ( R a . B a ) ' R c 1  1  n  n  which n e i t h e r s e m a n t i c a l l y e n t a i l s (Ra 3Ba )  (9)  1  1  the development D#  (Ra =>Ba ). (Rc P B C )  nor s e m a n t i c a l l y e n t a i l s  n  n  the n e g a t i o n  Thus, 0 # n e i t h e r confirms  of D#.  nor d i s c o n f i r m s the raven  hypothe-  sis . However, t h i s i s not a problem f o r Hempel because i f "Be" is  f a l s e , then the hypothesis  w i l l be d i s c o n f i r m e d ,  should be the case t h a t 0 # n e i t h e r confirms 2.  and so i t  nor d i s c o n f i r m s the  T e c h n i c a l l y 0 and D r e s t a t e d i n c l a s s i c a l l o g i c  notation.  97 raven h y p o t h e s i s .  T h i s i s q u i t e d i f f e r e n t from the case f o r  the 20 s o l u t i o n s d i s c u s s e d above. In s h o r t , the 20 s o l u t i o n s of the paradoxes of c o n f i r m a t i o n d i s c u s s e d above are to be r e j e c t e d , but t h i s r e j e c t i o n need not apply to Hempel's  theory.  98  PART  III.  A  CONFIRMATION  PARADOXES  OF  LOGIC  AND A  CONFIRMATION  PLAUSIBLE  SOLUTION  OF  THE  99 1. A BRIEF INTRODUCTION In t h i s P a r t we  s t a r t a f r e s h by paying more a t t e n t i o n to se-  m a n t i c a l problems to c o n s t r u c t a " q u a s i - c l a s s i c a l l o g i c " , which i s v e r y c l o s e t o the c l a s s i c a l Then, we  three-valued  logic.  strengthen the q u a s i - c l a s s i c a l t h r e e - v a l u e d  logic  to a " c o n f i r m a t i o n l o g i c " by i n t r o d u c i n g i n t o i t some new t e n t i a l c o n n e c t i v e s . Among them t h e r e are an i n t e r n a l  sen-  negation  which i s d i f f e r e n t from the e x t e r n a l n e g a t i o n , an i m p l i c a t i o n which i s d i f f e r e n t from the c o n d i t i o n a l , and an equivalence  which  i s d i f f e r e n t from the b i - c o n d i t i o n a l . F i n a l l y , we employ the c o n f i r m a t i o n l o g i c as the u n d e r l y i n g l o g i c of a new c o n f i r m a t i o n theory c a l l e d the " i n t e r n a l  confir-  mation t h e o r y " and go on to s o l v e the paradoxes of c o n f i r m a t i o n i n the f o l l o w i n g way« On the one hand the paradoxes o f c o n f i r m a t i o n are d e r i v a b l e as the n a t u r a l outcome of the Raven s i s , which i s commonly understood  as a " c o n d i t i o n a l law"; on the  other hand the paradoxes o f c o n f i r m a t i o n are avoided Hypothesis  Hypothe-  i f the Raven  i s framed as a d e - n e u t r a l i z e d I m p l i c a t i o n a l Raven  p o t h e s i s , which i s claimed t o r e p r e s e n t an " i m p l i c a t i o n a l  Hy-  law".  Only i n the l a t t e r sense o f the d e - n e u t r a l i z e d I m p l i c a t i o n a l Raven Hypothesis  can we say t h a t we have a p l a u s i b l e s o l u t i o n of  the paradoxes of c o n f i r m a t i o n .  100 2. A THREE-VALUED QUASI-CLASSICAL LOGIC Consider the f o l l o w i n g t h r e e - v a l u e d matrix language* ( i ) I t s three t r u t h - v a l u e s are* t (or 1 ) , f (or - 1 ) , and n (or  0 ) . And they are i n t e r p r e t e d r e s p e c t i v e l y as " t r u t h " (or  "confirmation"), " f a l s i t y " t r u t h nor f a l s i t y "  (or " d i s c o n f i r m a t i o n " ) and " n e i t h e r  (or " n e u t r a l i t y " ) .  ( i i ) The matrix language has the f o l l o w i n g f i v e connectives  sentential  (or operations)*  ~  (the e x t e r n a l n e g a t i o n , or complementation),  v  (disjunction),  & (conjunction), -» ( c o n d i t i o n a l ) , and (bi-conditional) * and they are r e s p e c t i v e l y to be read and i n t e r p r e t e d as* non, or, and, if,  then  i f and only i f (which i s a b b r e v i a t e d as* i f f ) . ( i i i ) The f i v e s e n t e n t i a l c o n n e c t i v e s obey the f o l l o w i n g t r u t h r u l e s d e s c r i b e d by the t r u t h t a b l e s as shown below*  t f n  P & Q t f n  P -4 Q t f n  P *-* Q t f n  ~P  t t t t f n t n n  t f n f f f n f n  t f n t t t t t t  t f n f t t n t t  f t t  p  V  Q  Now we w i l l have a decent noted  p  t h r e e - v a l u e d s e n t e n t i a l l o g i c de-  as L~ as w e l l as i t s q u a n t i f i c a t i o n a l e x t e n s i o n  (with o r  101 without  identity)  language of  its  fied  of  2  QLj,  H e r e we w i l l  if  we add t o  semantics,  s e n t e n c e s as w e l l  sentences.  Part  as  a fully-developed  atomic  semantics  denoted  as  the  especially  the  the v a l u a t i o n s  s i m p l y add t o i t  confirmational valuations  the  above  of  matrix  valuations its  quanti-  fully-developed  given i n section  11  of  II. Thus we c l a i m t h a t we have a d e c e n t  logic  2  L - and a d e c e n t  three-valued  sentential  2  three-valued quantificational logic QL~.  102 3. AN AXIOMATIZATION OF I  2  F o r the sake o f l a t e r r e f e r e n c e s and d i s c u s s i o n s we w i l l 2  give  here,  f o r the time being, a l s o an axiom system f o r i t  as f o l l o w s ! I.  Formation  rules.  1. P r i m i t i v e o p e r a t o r s : 2.  A u x i l i a r y symbolsi  ~ , v, &.  (,  ).  3. Sentence l e t t e r s : P,Q,R,... . 4.  Formation  r u l e s o f wffs  a) A sentence  (or s e n t e n c e s ) .  l e t t e r i s a wff.  b) I f A, B a r e w f f s ,  so are ~A, A v B , and A & B .  c) These a r e the o n l y r u l e s o f w f f s . Axioms. Ax 1.1 Ax 1 . 2  (A v A)-»A A - » ( A V B )  - * (B v A )  Ax 1.3  (A v B  Ax 1.4  (A-»B)-*((CvA)-»(CvB))  Ax 2.1 Ax  2.2  A  )  — » ( A & A )  (A&B  )-»A  Ax 2.3  ( A & B ) ^ ( B & A )  Ax 2.4  ( A - > B )->((C & A )->(C & B ) )  Ax 3.1  (AvB)-> - ( ~ A & ~ B )  Ax 3 - 2  ( A & 3)—). v(-Av-B)  I I . Transformation  rules.  1. D e f i n i t i o n s . Df 1.  P->Q =df ~P v  Q  103  Df 2.  P^Q  =df  (P-*Q) & (Q-*P)  2. Rule o f i n f e r e n c e : modus ponens, i . e . , A , A-»B 3.  I-  B  Three laws o f i n t e r c h a n g e . a) Law o f double-negation  (abbr. as DN).  Suppose t h a t a wff A occurs i n a theorem.  Then, A  can be r e p l a c e d a t any o f i t s occurrences by  A,  and v i c e v e r s a . b) Law of t r a n s p o s i t i o n (abbr. as T r a n s p ) . Any occurrence of a wff A & B  (respectively,  i n a theorem can be r e p l a c e d by B& A BvA),  AvB)  (respectively,  and v i c e v e r s a .  c) Law o f iderapotence (abbr. as I d ) . Any occurrence o f a wff A i n a theorem can be r e p l a c e d by A v A  (or, by A & A ) , and v i c e v e r s a .  4. Rule o f uniform s u b s t i t u t i o n : the r e s u l t o f u n i f o r m l y r e p l a c i n g any sentence l e t t e r i n a theorem by any wff i s i t s e l f a theorem.  104 4. SOME PROPERTIES CF THE THREE-VALUED SENTENTIAL LOGIC L Note t h a t there are a few important  d i f f e r e n c e s of the axiom  system o f the t h r e e - v a l u e d s e n t e n t i a l l o g i c L^ last  2  given  in  the  s e c t i o n and the axiom system f o r the c l a s s i c a l two-valued  s e n t e n t i a l l o g i c C2 as i t i s g i v e n by H i l b e r t and Ackermanni (1)  1  T h e i r axiom system f o r C2 needs o n l y two p r i m i t i v e oper-  a t o r s , v i z . the c l a s s i c a l n e g a t i o n —r and the c l a s s i c a l d i s j u n c -  2 t i o n v; while i n our axiom system o f L^ we need three p r i m i t i v e operators. (2)  T h e i r axiom system has f o u r axioms, v i z . HA  1.  (A v A ) OA  (where "A 3 B " i s d e f i n e d as HA  2.  " T A V B " . )  A o ( A v B)  HA 3-  (A v B) O (B v A)  HA 4.  (AvB)o ((CvA)o(CvB))  2  But i n our axiom system f o r L^ we need something more i f the three laws o f interchange a r e omitted f o r the c o n j u n c t o r of  2  Lj i s a p r i m i t i v e operator which i s n o t d e f i n a b l e by other p r i -  2  m i t i v e o p e r a t o r s o f L^. (3) I n C2 the c o n j u n c t o r can be d e f i n e d by i t s n e g a t i o n and d i s j u n c t o r as followsi p.Q  =df  -7 (7PY-7Q)>  where "•" i s the " c o n j u n c t o r "  of C2, such t h a t the f o l l o w i n g two c o n d i t i o n s a r e s a t i s f i e d * (i)  (p.Q) =  ^(^Pv-7Q),  and  1. C f . H i l b e r t D. and W. Ackermanm P r i n c i p l e s of Mathematical L o g i c , t r . by L.M.Hammond, G.G.Leckie and F . S t e m h a r d t , p . 2 7 f f . . Chelsea, 1950.  105  ( i i ) V(P.Q) = V(-»(->Pv-»Q)),where " = " i s the " b i - c o n d i t i o n a l " of C2  (and  "V(A)" i s , to repeat,  In other words, we Theorem 1.  "the v a l u a t i o n of  A".)  have the f o l l o w i n g theorem* (1)  In c l a s s i c a l s e n t e n t i a l l o g i c C2,  V(P.Q) = Vt>(-rPv->Q)), i t f o l l o w s t h a t  since  ( i ) (P.Q)s-n>Pv-*Q)  hence, i t i s l e g i t i m a t e to make the f o l l o w i n g d e f i n i t i o n ! P.Q  =df ->(-rPv-Q): and,  i t follows that  (iii)  and, (ii)  d u a l l y , (2) s i n c e V(P v Q) = V(->(--P. -»Q)),  ( P v Q)= -»(- P.~'Q) and,  hence, i t i s a l s o  r  l e g i t i m a t e to make the d e f i n i t i o n i  (iv) PvQ  =df  -r(-^P,-»Q).  2 On  the other hand, i n  weaker, and  yet very  we  important,  can o n l y have the  following  theoremt 2  Theorem 2. i s not  In the t h r e e - v a l u e d  true that V(P&Q) = V H - P v - Q ) ) .  l e g i t i m a t e to make the d e f i n i t i o n * 2 ( i i ) i t i s the case t h a t i n  we  P&Q  PvQ  follows that =df  Hence, ( i ) i t i s not  =df ~ ( ~ P v ~ Q ) ,  although  ( i i i ) V ( P v Q ) = V(~ (~P &~Q)),  ( i v ) i t i s not l e g i t i m a t e to make the  ~(~P &~Q),  (1) i t  always have* P&Q-f*~(~P v ~ Q ) ;  a l s o , d u a l l y , (2) i t i s f a l s e t h a t it  s e n t e n t i a l l o g i c L^,  definition* 2  although (v) i t i s always t r u e i n  to have*  P v Q<-»~(~P&~Q). (The when we We  importance of the above two l a t e r d i s c u s s the q u e s t i o n *  list  l y the  two  what i s a "deviant  logic"?)  here some of the more important theorems f o r the  pose of l a t e r Theorem 3 f o r any  theorems w i l l become c l e a r  reference* (The completeness theorem). 2  w f f s A and  "semantical  B of L^.where "|(-" and  entailment  relation"  and  I f A|f-B, " (-" are  pur-  then Al-B, respective-  the " s y n t a c t i c a l  106 consequence r e l a t i o n " of L^.) Theorem 4 (The soundness theorem). 2 any two wffs A and B of L y  I f A(-B , then A||-B , f o r  Theorem 5 (The d e d u c t i o n theorem). that of  I f A|-B , then i t f o l l o w s  }-(A-*B), where A i s any f i n i t e s e t of wffs and B i s any wff Ly  2  Theorem 6 (The converse d e d u c t i o n theorem).  In  , i f |-(A->B),  then A |-B , f o r any two w f f s A and B. 2 The next theorem i s about the r e l a t i o n s h i p between L 2 that  justifies calling  tial  logic"t Theorem 7. ( i ) A|-B  and C2  a " t h r e e - v a l u e d q u a s i - c l a s s i c a l senten-  holds i n C2 i f f A |-B holds i n  ( i i ) A|(-B holds i n C2 i f f A |(-B holds i n The l a s t theorem, by the way, same n o t a t i o n f o r the  2  semantical  also  : and  .  justifies  entailment  the use of the  r e l a t i o n s and the 2 s y n t a c t i c a l consequence r e l a t i o n s r e s p e c t i v e l y of C2 and of L~.  107  5.  A QUASI-HEMPELEAN CONFIRMATION THEORY  We  w i l l have a quasi-Hempelean e x t e r n a l c o n f i r m a t i o n  i f the f o l l o w i n g three c o n d i t i o n s are (I) I t s u n d e r l y i n g l o g i c t i o n a l l o g i c QL^  (without  satisfiedi  i s the q u a s i - c l a s s i c a l q u a n t i f i c a -  identity).  ( I I ) I t has a l s o Hempel's E q u i v a l e n c e t h a t the "equivalence" of  theory,  Condition  1  provided  here simply means the " b i - c o n d i t i o n a l "  QL|. ( I I I ) I t has a l s o Hempel*s seven b a s i c concepts of a c o n f i r -  mation theory, namely, ( i ) o b s e r v a t i o n r e p o r t , ( i i i ) development of a h y p o t h e s i s , confirmation,  (iv) d i r e c t confirmation, 2  ( v i ) d i s c o n f i r m a t i o n , and  v i d e d t h a t the "entailment"  ( i i ) hypothesis,  ( v i i ) neutrality,  ment r e l a t i o n " \ [ " and  pro-  i n v o l v e d i n the d e f i n i t i o n s of  l a t t e r f o u r concepts w i l l be understood as the s e m a n t i c a l  (v)  the entail-  the " d i s c o n f i r m a t i o n " of our new  confir-  mation theory w i l l be d e f i n e d i n terms of the e x t e r n a l  negation  2  of QL^.  Thus i n our new  "B lhH " and d  c o n f i r m a t i o n theory  "B d i s c o n f i r m s H"  "B confirms H" i s  i s "B ||-~H", where H d  d  i s "the  de-  velopment of H w i t h r e s p e c t to the i n d i v i d u a l s whose names occur essentially in  B".^  Since the new  c o n f i r m a t i o n theory i s so c l o s e to Hempel*s  theory of c o n f i r m a t i o n and  since i t s disconfirmation i s defined 2  i n terms of the e x t e r n a l n e g a t i o n  of QL^ , we  will call  i t "the  quasi-Hempelean e x t e r n a l c o n f i r m a t i o n t h e o r y " or, simply, e x t e r n a l c o n f i r m a t i o n theory". 1. Cf. Hempelt Aspects of S c i e n t i f i c E x p l a n a t i o n , 2. I b i d , p.37 or, b e t t e r , s e c t i o n 5, P a r t I . 3. I b i d , p.36 or, b e t t e r , s e c t i o n 5, P a r t I .  p.31.  "the  1 6. PARADOXES 0? CONFIRMATION REGAINED In the newly c o n s t r u c t e d quasi-Henpe l e a n e x t e r n a l c o n f i r m a t i o n t h e o r y a l l "paradoxes of c o n f i r m a t i o n " can be r e - d e r i v e d , 2 m a i n l y because i n QL^ we have the f o l l o w i n g theorems: Th 1.  (~Ra&~Ba) |J-(Ra-»-Ba)  Th 2.  (-Ra&Ba) ||-(Ra-**Ba)  Th 3-  ~fla ||-(Ra->Ba)  Th 4.1  (-RavBa) ||-(Ra-»Ba)  Th 4,2  (Ra->3a) i f f (~Ba->~Ra)  where R: (1) i s a raven, B: (D  i s black,  a: any  entity.  These t r u t h s can be v e r i f i e d by t h e i r r e s p e c t i v e t r u t h t a b l e s w i t h the help of the meaning of the s e m a n t i c a l e n t a i l ment r e l a t i o n .  With the h e l p of our d e f i n i t i o n s o f c o n f i r m a -  t i o n and d i s c o n f i r m a t i o n i t f o l l o w s from the above r e s u l t s that s ( A ) Any b l a c k raven c o n f i r m s the Raven Hypothesis t h a t a l l ravens are b l a c k or, i n symbols, (B)  Any non-black  (Vx) (Rx-»Bx).  non-raven a l s o confirms the Raven Hypo-  thesis. (C) Any b l a c k non-raven c o n f i r m s the Raven Hypothesis, i n particular, ( C . l ) Any non-raven confirms the Raven Hypothesis, (C-2)  and  Anything b l a c k a l s o confirms the Raven Hypothesis.  (D) The Raven Hypothesis i s d i s c o n f i r m e d by and only by a non-black  raven.  I t i s indeed v e r y odd t h a t we  have the r e s u l t s  (B), (C),  109  ( C . l ) and regained theory.  (C.2)  they are "the paradoxes of c o n f i r m a t i o n "  i n our new quasi-Hempelean e x t e r n a l c o n f i r m a t i o n  110  7. AN The  ANALYSIS OF THE  three-valued  FAILURE  l o g i c s t h a t we  have c o n s t r u c t e d  up to  this  p o i n t a l l f a i l as c o n f i r m a t i o n l o g i c s , f o r the paradoxes of conf i r m a t i o n are r e - d e r i v a b l e i n the c o n f i r m a t i o n t h e o r i e s which employ these The of  logics.  reason they f a i l  mistake.  The  i s t h a t they repeat  the  same p a t t e r n  p a t t e r n of t h i s mistake i s t h a t a l l of them con-  t a i n " q u a s i - c l a s s i c a l c o n d i t i o n a l s " , by which we  mean any  formi  t i o n a l s whose t r u t h t a b l e s are of the f o l l o w i n g  P  PX  t  t  t  f  t  _k^  condi-  Q  f f  u  t  V  n  X y z  n  where u,v,x,y,z * f t , f n \ . t  In other words, a q u a s i - c l a s s i c a l  d i t i o n a l of a t h r e e - v a l u e d  logic  t a b l e i s i d e n t i c a l to the one  con-  i s a b i n a r y o p e r a t i o n whose t r u t h  of the c l a s s i c a l c o n d i t i o n a l as f a r  as the t r u t h - v a l u e s t ( t r u t h ) and  f ( f a l s i t y ) f o r P and  Q are  con-  cerned . Any  c o n f i r m a t i o n theory whose u n d e r l y i n g l o g i c has  such a  q u a s i - c l a s s i c a l c o n d i t i o n a l w i l l c e r t a i n l y c o n t a i n the paradoxes of c o n f i r m a t i o n with the minimal h e l p of Hempel s concepts of 1  ( i ) the development of a h y p o t h e s i s , a q u a s i - c l a s s i c a l negation,and  ( i i ) confirmation, ( i i i )  ( i v ) a q u a s i - c l a s s i c a l conjunc-  t i o n , which have the f o l l o w i n g t r u t h t a b l e s t P P\  t f n  where x  e{t,f,n}  .  t t f n  and f f f f  Q  n n f n  negation  f t X  of  P  Ill  To see t h a t the paradoxes of c o n f i r m a t i o n can be  derived  i n such a q u a s i - c l a s s i c a l c o n f i r m a t i o n theory i s to see t h a t i n such a l o g i c we (1)  have»  f-i^f t  and  fit  (2)  t and  so, by d e f i n i t i o n of c o n f i r m a t i o n  (with the h e l p of the  concept of development of a h y p o t h e s i s ) (A) Any and  t h a t we  non-black non-raven confirms  havei  the Raven  (B) any non-raven which i s b l a c k a l s o confirms  Hypothesis; the Raven  Hypothesis. Therefore  , we  must not repeat t h i s same p a t t e r n .  However, to r e a l i z e t h a t we  should not repeat the same p a t -  t e r n of mistake does not mean t h a t we c l a s s i c a l c o n d i t i o n a l , otherwise of s u r p r e s s i n g  i n our new  should  s u r p r e s s the  quasi-  we would commit another mistake  theory the e x p r e s s i o n of the paradoxes  of c o n f i r m a t i o n , i n s t e a d of s o l v i n g them. Here i t seems t h a t we ( i ) we  are trapped  i n a predicament! E i t h e r  s u r p r e s s any q u a s i - c l a s s i c a l c o n d i t i o n a l from our  l o g i c and,  hence we  new  s u r p r e s s the e x p r e s s i b i l i t y of the paradoxes  of c o n f i r m a t i o n , or e l s e ( i i ) we  introduce a q u a s i - c l a s s i c a l  c o n d i t i o n a l i n t o our new  d e r i v e the paradoxes of con-  f i r m a t i o n i n our new  and  l o g i c and  theory of c o n f i r m a t i o n .  In the l a t t e r case we  have the paradoxes of c o n f i r m a t i o n  i n the former case we  do not even have a chance to solve  the paradoxes of c o n f i r m a t i o n .  112  8. CONDITIONAL LAW  VS. IMPLICATIONAL  I t seems t h a t the only way c r i b e d i n the l a s t  LAW  to get out the predicament des-  section i s t h i s i  ( i ) F i r s t , we admit t h a t we need a q u a s i - c l a s s i c a l c o n d i t i o n a l and, hence, we have a l s o had t o admit t h a t the paradoxes of c o n f i r m a t i o n  are not s o l v a b l e i f we frame the Raven Hypothe-  s i s i n terms of such a q u a s i - c l a s s i c a l c o n d i t i o n a l as a u n i v e r sal  ( q u a s i - c l a s s i c a l ) c o n d i t i o n a l law. ( i i ) But t h a t does not mean we cannot frame the Raven Hypo-  t h e s i s i n a d i f f e r e n t way,  say, as a u n i v e r s a l i m p l i c a t i o n a l  law such t h a t the paradoxes of c o n f i r m a t i o n  are s o l v a b l e .  The Raven Hypothesis as a c o n d i t i o n a l law i s framed,  say,  i n QL-j i n the f o l l o w i n g way* Hi  (VX)(RX^BX),  which w i l l be c a l l e d now  "the C o n d i t i o n a l Raven H y p o t h e s i s " from  on. Suppose t h a t we have a l s o an " I m p l i c a t i o n a l Raven  sis"  H*i  Hypothe-  (or "Causal Raven Hypothesis") which i s symbolized as  (Vx)(Rx ==>Bx),  and read s t i l l the f o l l o w i n g  as " a l l ravens are b l a c k " and y e t understood i n sensei  $.  H^t (or, H;>«  That x i s a raven i m p l i e s t h a t x i s b l a c k , i n a sense, a l s o as That x i s a raven c a u s a l l y i m p l i e s t h a t x i s b l a c k . )  113 Now we ask: c a n we f i n d s u c h a new d e r s t o o d as " i m p l i c a t i o n "  c o n n e c t i v e " = £ " to be  i n t h e s e n s e of H]_ (and, perhaps,  un-  also  * i n t h e s e n s e o f H 2 ) such t h a t the paradoxes of c o n f i r m a t i o n are not  derivable  i n the proposed new  form of " I m p l i c a t i o n a l  Raven Hy-  pothesis"? I f there i s such an i m p l i c a t i o n i n a t h r e e - v a l u e d l o g i c , then we w i l l have a ( c a u s a l l y ) i m p l i c a t i o n a l s o l u t i o n of the paradoxes of c o n f i r m a t i o n , ditional  though we do not have any  ( q u a s i - c l a s s i c a l l y ) con-  solution.  However, t h i s i s good enough f o r us to get out of the p r e d i cament of s o l v i n g or not s o l v i n g the paradoxes of  confirmation.  Anyway, t h i s i s almost a l a s t chance f o r us to solve radoxes of c o n f i r m a t i o n . implication  i n the next  So, we w i l l section.  the pa-  s t a r t to search f o r such an  114 9.  IN SEARCH OF AN IMPLICATION I  A few c o n d i t i o n s must be observed  i f there i s such an i m p l i -  cations (A) C o n d i t i o n 1. classical.  The i m p l i c a t i o n  should  not be  quasi-  That i s , i t should not have the f o l l o w i n g p a r t i a l  truth tableJ P=*-Q t f n t f t t  £^4 t f n  Otherwise, as we argue i n s e c t i o n 7 the paradoxes o f c o n f i r m a tion  w i l l be r e - d e r i v a b l e . (B) C o n d i t i o n 2.  The i m p l i c a t i o n should be " s e m i - c l a s s i c a l " ,  by which we mean i t should c o n t a i n the f o l l o w i n g p a r t i a l  truth  tables P =#Q t f n t f  t f n  Otherwise, a b l a c k raven won't be able t o c o n f i r m the I m p l i c a tional  Raven Hypothesis  disconfirm i t either. So,  i t i s reasonable  and a non-black raven won't be able to T h i s would be t o t a l l y c o u n t e r - i n t u i t i v e .  to have C o n d i t i o n 2.  By C o n d i t i o n 1 and C o n d i t i o n 2 there a r e s t i l l p o s s i b i l i t i e s f o r a b i n a r y o p e r a t i o n denoted as possible implication! *  t f n  Q n t f yl x l x2 y2 y3 y ^ y5  the f o l l o w i n g to be the  115  where x l , x 2 € f f , n } and y l , y 2 , y 3 , y 4 , y 5 e{t t,nj. t  .  So, we have i = 2 ' 2 - 3 - 3 - 3 - 3 - 3 = 972 , i . e . , we have 972 possible  candidates f o r such an i m p l i c a t i o n . 3«  (C) C o n d i t i o n the f o l l o w i n g  The p o s s i b l e  i m p l i c a t i o n cannot have  "counter-classical" p a r t i a l truth table  eitheri  i. t f n t f n Otherwise,  f f  ( i ) a b l a c k non-raven  raven would d i s c o n f i r m  or  ( i i ) a non-black non-  the I m p l i c a t i o n a l  Raven Hypothesis.  T h i s would be as c o u n t e r - i n t u t i t i v e as both o f them c o n f i r m i n g the I m p l i c a t i o n a l  Raven Hypothesis.  Thus we have o n l y the f o l l o w i n g p o s s i b i l i t i e s f o r the possible  implication: P P\ t f n t fyl t f n n y2 n y3y4y5  where y l , y 2 , y 3 , y 4 , y 5 e f t , f , n } . So f a r we have i = 3^ = 243, i . e . , we have 243 p o s s i b l e c a n d i d a t e s f o r the p o s s i b l e (D) C o n d i t i o n  4.  implication.  The p o s s i b l e  i m p l i c a t i o n cannot have the  f o l l o w i n g p a r t i a l t r u t h table» P=^»Q t f n t f n Otherwise, i f the t r u t h v a l u e n a p p l i e s  i n the case when the e n t i t y  110 does not e x i s t then a n o n - e n t i t y confirm  such as a centaur, e t c . , would  the I m p l i c a t i o n a l Raven Hypothesis, f o r we would have 2  the  following r e l a t i o n held i n Q L y ~Rc&~Bc  |[-(Rc=^Bc)  where " c " i s "a c e n t a u r ( o r , a u n i c o r n , surd f o r any n o n - e n t i t y thesis.  "to c o n f i r m  e t c . ) " . And i t i s ab-  the I m p l i c a t i o n a l Raven Hypo-  We t h i n k t h a t any n o n - e n t i t y  i s neutral with  t o the I m p l i c a t i o n a l Raven H y p o t h e s i s .  This i n t u i t i o n  4 as w e l l as the f o l l o w i n g  f o r us C o n d i t i o n (E) C o n d i t i o n  5-  respect justifies  conditions  The p o s s i b l e i m p l i c a t i o n cannot have the  following p a r t i a l t r u t h tables  P  t f n t f n  f  Otherwise, as we have s a i d t h a t i t would enable any to d i s c o n f i r m t h e  non-entity  I m p l i c a t i o n a l Raven Hypothesis, and t h a t i s  counter-intuitive. Condition result,  k plus Condition  5 g i v e us, w i t h the  previous  the f o l l o w i n g p o s s i b l e t r u t h t a b l e f o r the p o s s i b l e  implications Q  t f n where y l , y 2 , y 3 , y  k  P  i  t  f  n  t n  f n y  yl y2 n  y3  k  6}t,f,n} .  Thus f a r we have i = 3 , i . e . , we have 81 more p o s s i b l e c a n d i d a t e s f o r the p o s s i b l e i m p l i c a t i o n .  117 However,  the assignment of V(y5) = n would c r e a t e f o r us  the f o l l o w i n g d i f f i c u l t y : Any n o n - e n t i t y w i l l d i s c o n f i r m the I m p l i c a t i o n a l Raven Hypothesis,  although  i t won't c o n f i r m the  I m p l i c a t i o n a l Raven Hypothesis,  i f the u n d e r l y i n g l o g i c of our  2 new c o n f i r m a t i o n theory i s QL^ p l u s t h i s i m p l i c a t i o n . The o n l y way t o overcome t h i s new d i f f i c u l t y i s : 2 2 ( i ) B e s i d e s adding the would-be i m p l i c a t i o n to QL^ (and L^) 2 we a l s o have t o add the f o l l o w i n g " i n t e r n a l n e g a t i o n " p -P (and ): f t f t n n  to QL-^  where "-" i s t o be read as "not" and i n t e r p r e t e d as " o p p o s i t i o n " . ( i i ) A l s o change the " d i s c o n f i r m a t i o n * theory  1  of our new  to "with r e s p e c t to the i n t e r n a l n e g a t i o n " .  new c o n f i r m a t i o n t h e o r y  confirmation Thus i n our  "B d i s c o n f i r m s H" i s d e f i n e d as  "B^-lfi".  Thus, the above d i f f i c u l t y w i l l be overcome, f o r under the new d e f i n i t i o n of d i s c o n f i r m a t i o n any n o n - e n t i t y i s n e i t h e r conf i r m i n g nor d i s c o n f i r m i n g the I m p l i c a t i o n a l Raven  Hypothesis.  In other words, under the new d e f i n i t i o n o f d i s c o n f i r m a t i o n any n o n - e n t i t y  i s simply n e u t r a l t o the I m p l i c a t i o n a l Raven Hy-  pothesis. I f e v e n t u a l l y we can f i n d the i m p l i c a t i o n we want, then 2 p l u s the i m p l i c a t i o n and the i n t e r n a l n e g a t i o n w i l l be de2 noted as C3 and QC3, and thus w i l l be QL^ p l u s the i m p l i c a t i o n and  the i n t e r n a l n e g a t i o n .  The new c o n f i r m a t i o n theory, whose  u n d e r l y i n g l o g i c i s QC3,will be c a l l e d  "the i n t e r n a l  confirma-  t i o n theory", f o r i t s d i s c o n f i r m a t i o n i s d e f i n e d i n terms o f the internal  negation.  118  10.  IN SEARCH OF AN IMPLICATION I I  Up to t h i s p o i n t we s t i l l possible  implication.  have 81 p o s s i b i l i t i e s f o r the  T h i s number i s d e r i v e d from the  p r o v i d e d from the f i v e c o n d i t i o n s the  Implicational  section.  on i m p l i c a t i o n to i n t e r p r e t  Raven Hypothesis l a i d down i n the p r e v i o u s  We would not have any more i n s i g h t of t h i s type, f o r  we have exhausted a l l p o s s i b l e we c o u l d  truth-value  ground the I m p l i c a t i o n a l  (1)  t r u t h r u l e f o r sentences of  P=»Q, form:  Ga=^Hb  where P,Q,G,H,a and b are a l l u n - i n t e r p r e t e d ; i n search of the t r u t h r u l e o f the f o l l o w i n g (3)  Remember  f o l l o w i n g g e n e r a l form:  or, i n a way, of the f o l l o w i n g (2)  combinations on which  Raven H y p o t h e s i s .  t h a t we a r e i n s e a r c h of a p o s s i b l e the  insight  and we are not form:  Rc =*Bc ,  where R and B are a l l i n t e r p r e t e d as f o l l o w s : R: (l) i s a raven, Bi  ®  i s black.  Thus the p o s s i b l e combinations o f t r u t h - v a l u e s the  of (2) of  f o l l o w i n g form .Hb t f n  Ga =*Hb t  f  n yi  y3 y^+  (where y l , y 2 , y 3 , y 4 6 ( t , f , n \ ) are r u l e d out from the p o s s i b l e  119  considerations, of  (3)  because the imposed i n t e r p r e t e d r e s t r i c t i o n s  t h a t G = R,  ( i . e . R) and  H = B,  a = c=b,  b l a c k n e s s ( i . e . B)  and  there  are indeed ravens  i n the world.  Thus the c o n f i r -  m a t i o n a l v a l u a t i o n r u l e s out the p o s s i b l e t r u t h - v a l u e  combina-  t i o n s of  V(Hb)=n,  ( i ) V ( G a ) = t and  ( i i i ) V ( H b ) = t and  V(Hb)=n,  V ( G a ) = n , and  ( i i ) V(.Ga) = f and  ( i v ) V ( H b ) = f and  In other words, the f o r m a l l y p o s s i b l e t r u t h - v a l u e of tic  (i), (ii),  ( i i i ) and  ( i v ) are impossible  p o i n t of view of c o n f i r m a t i o n a l  r e s p e c t i v e l y i n t e r p r e t e d as "... i s black" So, not  now  V(Ga)=n. combinations  from the  seman-  v a l u a t i o n when G and  H  are  i s a raven" ( i . e . R) and  "...  ( i . e . B).  from now  (1) nor  on I w i l l c o n c e n t r a t e my  (2),  ble truth-value implication.  f o r i n s p i r a t i o n about the i n s i g h t of the combinations of the t r u t h r u l e of the  Once we  (3),  a t t e n t i o n on form  have t h i s understanding, we  possi-  possible  can have the  f o l l o w i n g more i n s i g h t s : (F) C o n d i t i o n v a l u e s of y l , y2, be:  n.  6.  (the N e u t r a l i z e d  y3 and  In other words, we  t a b l e f o r the p o s s i b l e  The  yk  Condition).  The  truth-  of the l a s t p a r t i a l t r u t h t a b l e should have the f o l l o w i n g p a r t i a l  truth  implication!  reason f o r t h i s i s : s i n c e the f o r m a l sentence ( 2 ) ,  Ga=^Hb, once i n t e r p r e t e d as sentence have the above t r u t h - v a l u e considerations!  should  (3)»  assignments due  v i z Rc=»Bc,  we  viz. must  to the f o l l o w i n g  two  120  "...  ( i ) Since both R and B are now i n t e r p r e t e d  ( r e s p e c t i v e l y as  i s a raven" and "... i s b l a c k " ) and there  a r e such p r o p e r -  t i e s i n the a c t u a l world, the only chance f o r both Rc and Be t o have the t r u t h - v a l u e  n (as g i v e n ) a c c o r d i n g  t o the c o n f i r m a t i o n a l  v a l u a t i o n i s t h a t " c " denotes n o t h i n g i n the a c t u a l world. ( i i ) I f " c " denotes n o t h i n g i n the a c t u a l world, i t i s outs i d e the " n a t u r a l range o f r e l e v a n c e " In such case,  according  t o von Wright.  ( i . e . , when "Ga=*Hb" i s i n t e r p r e t e d as "Rc=*Bc")  we should b e t t e r a s s i g n Rc =*Bc the t r u t h - v a l u e  n, i n order t h a t  we have a chance of s o l v i n g the paradoxes o f c o n f i r m a t i o n  al a  von Wright's treatment o f the paradoxes v i a i t s " n a t u r a l range of relevance If  of a p p l i c a t i o n " .  the above  r e a s o n i n g i s c o r r e c t , then the i m p l i c a t i o n  t h a t we are l o o k i n g f o r should have the f o l l o w i n g t r u t h t a b l e t P Q t f n t f n  t f n n n n n n n  121  11.  JUSTIFICATION OF THE TRUTH RULE OF THE IMPLICATION I  C.I. Lewis r e f u s e s to i n t e r p r e t the c l a s s i c a l horseshoe operator shoe  3  as an "implication",because  he t h i n k s the  o p e r a t o r does n o t g i v e us any reasonable,  c o n n e c t i o n between the antecedent  say, c a u s a l  and the consequence.  For example, i n c l a s s i c a l l o g i c we c a n i n f e r (1)  horse-  from  The sky i s b l u e and the sky i s not b l u e  to the c o n c l u s i o n : (2)  UBC i s i n Vancouver. T h i s i n f e r e n c e i s v a l i d because i n c l a s s i c a l l o g i c we have  the f o l l o w i n g theoremt (3)  P and not-P D Q. C.I. Lewis r e f u s e s to i n t e r p r e t the horseshoe o p e r a t o r as  " i m p l i c a t i o n " f o r the reason t h a t there i s no " i n t r i n s i c " , say, c a u s a l , connection, and  i n our present example, between (1)  (2).  So, he c a l l s  (3) a "paradox o f i m p l i c a t i o n " i f we i n t e r p r e t  the horseshoe o p e r a t o r as an " i m p l i c a t i o n " . 13 such paradoxes o f i m p l i c a t i o n * 1.  P3(Q3P)  2.  - , P 5 ( P 3 Q )  3.  (P.Q)»(PSQ)  4.  (-,P.-,Q)r>(PDQ)  5.  (-P.Q) =>(Pt?Q)  6.  -r(P=>Q)OP  1.  He l i s t s a t l e a s t  1  C f . Lewis, C.I. and C.H. Langfordt Dover, I 9 3 2 , 1 9 5 9 .  Symbolic  Logic,p.86ff.  122  7.  -?(PoQ)o-Q  8.  - r ( P o Q ) => ( P 3 - » Q )  9.  - » ( P 3 Q ) 5 (~»P 3 Q )  10.  11.  -r(POQ)^ -7  (P^Q)  r>(Q =>P)  12.  Po(Qv-7Q)  13.  (P.^P)QQ The  (TP3-»Q)  above sentences r e p r e s e n t i n g the paradoxes of i m p l i c a p  tion  ( i n the c l a s s i c a l l o g i c ) a r e a l l i n v a l i d i n  we s u b s t i t u t e a l l occurrences  o f the horseshoe by  plus  , if  and under-  stand the n e g a t i o n , d i s j u n c t i o n and c o n j u n c t i o n of the c l a s s i c a l l o g i c as the e x t e r n a l n e g a t i o n (or any negations to be i n t r o d u c e d 2  into  such as the i n t e r n a l and the i n t r i n s i c n e g a t i o n s ) , d i s 2  j u n c t i o n and c o n j u n c t i o n of L^. T h i s would c o n s t i t u t e our f i r s t  p l a u s i b l e j u s t i f i c a t i o n of  the t r u t h r u l e o f the i m p l i c a t i o n . We understand t h a t t h i s j u s t i f i c a t i o n i s s t i l l  too b l e a k ,  f o r the avoidance of the paradoxes o f i m p l i c a t i o n c o u l d be j u s t a favor of luck.  123  12.  JUSTIFICATION OF THE  Since the  TRUTH RULE OF  j u s t i f i c a t i o n of the  THE  IMPLICATION I I  t r u t h r u l e of the  implica-  t i o n g i v e n i n the p r e v i o u s s e c t i o n i s not  rigorous,  time I was  fat conditional"  "the  content simply to c a l l i t "the  rigorous  j u s t i f i c a t i o n i f I could  p l a c e , what C.I.  Lewis means by  f i r s t l o g i c i a n who the c l a s s i c a l  is  long or  f a t arrow". However, I r e a l i z e d t h a t I c o u l d not g i v e  has  for a  not f i n d out,  i n the  " i m p l i c a t i o n " , f o r he  c r i t i c i z e s and  r e j e c t s the  "horseshoe" o p e r a t o r as an  first i s the  i n t e r p r e t a t i o n of  "implication".  indeed an o f f i c i a l d e f i n i t i o n of " s t r i c t this:  i t s truth rule a  Lewis  implication".  It  1  C.I. Lewis' d e f i n i t i o n of s t r i c t i m p l i c a t i o n . According to Lewis, P s t r i c t l y i m p l i e s Q i f f " i t i s f a l s e t h a t i t i s poss i b l e t h a t P should be true and Q f a l s e . " Does the strict  " f a t arrow" ( v i z . , =4 ) s a t i s f y Lewis' d e f i n i t i o n of  implication?  I t t u r n s out  i s : l e t us have the f o l l o w i n g two HI1  A l l b r o t h e r s are male,  H2:  A l l s i s t e r s are  Then, we  have t h a t  of ~H1=±~H2  ~H1  One  counter-example  sentences,  female. strictly  i m p l i e s ~H2,  f a t arrow i s not a s t r i c t  However, l e t us w r i t e c l e a r t h a t i f we  " |(P=>Q)  H  but  the  truth-value  implication.  as "P |f=»Q".  have t h a t P|f=* Q then we  although not  1. Lewis and  to.  i s n.  Hence, the  p l i e s Q,  not  Then, i t seems  have t h a t P s t r i c t l y  conversely.  Langford: Symbolic L o g i c ,  p.124  and  p.244.  im-  12k  So, ||=* i s s t r o n g e r than  ||-.  We  w i l l come back to the problem  of t h e i r s t r e n g t h s i n a moment. Quine a l s o has a d e f i n i t i o n of " i m p l i c a t i o n " , which he i s a l o g i c a l r e l a t i o n between two 2 implication i s thist  sentences.  thinks  H i s d e f i n i t i o n of  Quine's d e f i n i t i o n of i m p l i c a t i o n . Sentence SI i m p l i e s sentence S2 i f f no i n t e r p r e t a t i o n makes SI t r u e and S2 f a l s e , hence i f and o n l y i f no i n t e r p r e t a t i o n f a l s i f i e s the m a t e r i a l c o n d i t i o n a l whose antecedent i s SI and whose consequence i s S2. In a word, i m p l i c a t i o n i s v a l i d i t y of the c o n d i t i o n a l . Hence, both the semantical  entailments  of c l a s s i c a l l o g i c  of c o n f i r m a t i o n l o g i c are " i m p l i c a t i o n " a c c o r d i n g to Q u i n e .  and  J  Thus, i n c o n f i r m a t i o n l o g i c \[ i s the i m p l i c a t i o n as w e l l as the semantical  entailment.  Since ||=> i s s t r o n g e r than |(-, the  can be c a l l e d  "the s t r o n g i m p l i c a t i o n " or "the s t r o n g  entailment".  But what does t h a t mean i n l o g i c  In l o g i c  " |[-" as a semantical  entailment  p r e s e r v i n g i n any l o g i c a l i n f e r e n c e . The  answer i s no.  Is  former  (semantical)  though?  relation i s truth-  ||=* a l s o  truth-preserving?  Thus, f o r i n s t a n c e , the f o l l o w i n g i n f e r e n c e , P  .  P->Q  |f- Q  i s a v a l i d argument; but, c o r r e s p o n d i n g l y , P  ,  P^Q  we  do not havej  Q  However, i f a l l premises of an argument are n e c e s s a r i l y t r u e , and  the argument i s v a l i d , then ||=> w i l l preserve  the c o n c l u s i o n w i l l be a l s o a necessary  the p r o p e r t y , i . e .  truth.  Thus " ||=»" i s n e c e s s a r y - t r u t h - p r e s e r v i n g (or v a l i d i t y - p r e s e r v i n g ) . Since t r u t h c o n s i s t s of necessary necessary-truth  t r u t h and c o n t i n g e n t t r u t h ,  p r e s e r v i n g i s , of course,  s t r o n g e r than t r u t h pre-  2. Quine» Methods of L o g i c , r e v i s e d ed., p.3^. 3 . There may be a problem of "use" and "mention" n e g l e c t e d  here.  125  serving. Thus " s t r o n g i m p l i c a t i o n " c l e a r l y means t h a t i t s necessaryt r u t h - p r e s e r v i n g i s stronger than the t r u t h - p r e s e r v i n g o f \\~ , i . e . the s e m a n t i c a l  entailment  Similarly,  r e l a t i o n as " i m p l i c a t i o n " .  i s s t r o n g e r than the c o n d i t i o n a l —* i n C3« So,  i t can be c a l l e d  "the s t r o n g c o n d i t i o n a l " .  But what does t h a t  r e a l l y mean? The P-»Q  t r u t h - v a l u e o f P => Q can be d e s c r i b e d as i s e q u i v a l e n t t o  provided  t h a t i n the f i r s t p l a c e P i s presupposed t o be t r u e .  Then, what i s the s t r o n g e r p a r t i s , o f course,  the c l a u s e t h a t P i s  f i r s t l y presupposed t o be t r u e . Hence, " s t r o n g c o n d i t i o n a l " can a l s o be c a l l e d " p r e s u p p o s i t i o n a l c o n d i t i o n a l " and "P=*Q" can be read as " i f P then Q, where P i s p r e supposed t o be t r u e " or, more b r i e f l y , Below we l i s t Theorem 8.  "P o n l y i f presupposed by Q".  some i n t e r e s t i n g p r o p e r t i e s about ||=> » II"  * ~* *  anc  I n L^,  C3 and QC3 |(=> i s s t r o n g e r than  \\- , which  i s i n t u r n s t r o n g e r than —> . Theorem 9«  I f P ||=>Q then P||-i3, but not c o n v e r s e l y ;  other words, i f |f-(P=»Q) then The  or i n  ||-(P-*Q). but not c o n v e r s e l y .  f o l l o w i n g example shows t h a t the converse p a r t o f the p r e -  c e d i n g theorem does not h o l d i We have t h a t P||-P, but n o t t h a t P |f=>P. To see t h i s ,  just l e t V(P)=n  (or f ) .  (Note t h a t f o r l a c k o f a s h o r t e r term, "P=»Q" w i l l be s t i l l read as "P i m p l i e s Q", while we r e s e r v e "entailment" "strong entailment"  for  \[ and  o r " s t r o n g i m p l i c a t i o n " f o r ||=» , although i n  E n g l i s h the best r e a d i n g t h a t I can t h i n k o f f o r "P=*Q" i s i P i s f o l l o w e d by Q ( f a c t u a l l y o r l o g i c a l l y ) . ) 4. T h i s has been much f u l l y d e s c r i b e d i n next s e c t i o n on p.128.  126 13. A LONG WAY TO REACH THE TRUTH RULE OF THE IMPLICATION In the p r e v i o u s f o u r s e c t i o n s I have t r i e d to d e s c r i b e d  how  I reached the t r u t h r u l e f o r i m p l i c a t i o n as w e l l as how i t s t r u t h rule i s j u s t i f i e d .  I t seems somewhat complicated;  the way I reached the complicated.  truth  rule  but, a c t u a l l y ,  f o r i m p l i c a t i o n i s even more  Since i t a l s o p r o v i d e s a way t o j u s t i f y the t r u t h  r u l e f o r i m p l i c a t i o n , I would l i k e t o d e s c r i b e i t below. (I) F i r s t , my i n i t i a l  i n s i g h t i n c o n s t r u c t i n g a three-valued  c o n f i r m a t i o n l o g i c was t o i d e n t i f y ,  o r a t l e a s t a s s o c i a t e , the  three t r u t h - v a l u e s t ( t r u t h ) , f ( f a l s i t y ) and n ( n e i t h e r t r u t h nor f a l s i t y ) w i t h the three c o n f i r m a t i o n - s t a t e s c ( c o n f i r m a t i o n ) , d ( d i s c o n f i r m a t i o n ) and n  (neutrality).  I b e l i e v e d t h a t the i d e n t i f i c a t i o n s are j u s t i f i a b l e . At l e a s t , we should t r y t o pursue the matter i n t h i s  light.  ( I I ) Second, I b e l i e v e t h a t the paradoxes of c o n f i r m a t i o n are c o u n t e r - i n t u i t i v e , i . e . genuinely  puzzling.  F u r t h e r , I thought  t h a t the o n l y way out of these paradoxes i s to "extend" and/or "modify" the c l a s s i c a l l o g i c , f i r m a t i o n theory proper  s i n c e i t seems t h a t Hempel's con-  i s flawless.  This c o n v i c t i o n re-enforces  point ( I ) . ( I I I ) I, then, semantical)approach, lying logic  constructed,  i n a syntactical  (and f o r m a l l y  a minimal c o n f i r m a t i o n theory, whose under-  i s a three-valued  q u a n t i f i c a t i o n a l l o g i c , which  avoids  the paradoxes o f c o n f i r m a t i o n . Unfortunately,  a f t e r r e f l e c t i o n the avoidance of the para-  doxes of c o n f i r m a t i o n i s only apparent! they are avoided  by "sup-  p r e s i n g " , i . e . , n e g l e c t i n g t h e i r t o t a l meaning and complete i n t e r p r e t a t i o n from a f u l l y developed semantical Some important  l e s s o n s have been l e a r n e d i  perspective.  12? Lesson 1.  A purely s y n t a c t i c a l  (and f o r m a l l y  semantical)  approach to c o n f i r m a t i o n theory i s i n s u f f i c i e n t . Lesson 2.  Any  u n d e r l y i n g l o g i c of a c o n f i r m a t i o n  must have a f u l l y - d e v e l o p e d Lesson 3»  The  semantics.  s e n t e n t i a l l o g i c of the u n d e r l y i n g l o g i c must  be t r u t h - f u n c t i o n a l l y complete, and Lesson 4. by s u p p r e s s i n g Lesson 5«  The  so i s i t s q u a n t i f i c a t i o n a l one.  paradoxes of c o n f i r m a t i o n should not be  The  paradoxes of c o n f i r m a t i o n are n a t u r a l conse-  " s o l v e d " by s u p p r e s s i o n  (IV) So f a r , i t s t i l l  otherwise,they would  but they should not be the consequ-  ence of the I m p l i c a t i o n a l Raven  Hypothesis.  remained u n c l e a r what i m p l i c a t i o n i s .  (V) However, b e f o r e we  can answer what " i m p l i c a t i o n " i s , we  would b e t t e r c o n s t r u c t a t h r e e - v a l u e d i s so c l o s e to the c l a s s i c a l l o g i c  " q u a s i - c l a s s i c a l " l o g i c which  t h a t the paradoxes of  t i o n are n a t u r a l consequence of the " q u a s i - c l a s s i c a l t h eo ry " with such an u n d e r l y i n g (VI) So, we  solved  e x p r e s s i b i l i t y , meaning or t o t a l i n t e r p r e t a t i o n .  quences of the C o n d i t i o n a l Raven Hypothesis be  theory  have L  and  2  confirma-  confirmation  logic.  QL . 2  (VII) Since L^ i s not t r u t h - f u n c t i o n a l l y complete, a new con2 cept of " i m p l i c a t i o n " can be added to L^ i n our present approach. ( V I I I ) I a l s o found 2 to Ly  t h a t i f we add the i n t e r n a l n e g a t i o n  1 then the f o l l o w i n g concept of " p r e s u p p o s i t i o n " i s d e f i n a b l e i  D e f i n i t i o n of P r e s u p p o s i t i o n , ing c o n d i t i o n holdst where "&" L  2  and  (-)  p presupposes Q i f the f o l l o w -  (P ||-Q) & (-P l|-Q)  f  i s the mata-language analogue of the " c o n j u n c t i o n "  " fr" i s the semantical  entailment  of  relation.  1. C f . Haack, Susans Deviant L o g i c , p . l 4 l . The i d e a i s to capture Frege's d e f i n i t i o n or p r e s u p p o s i t i o n , which i s : P presupposes Q * df P i s n e i t h e r t r u e nor f a l s e u n l e s s Q i s t r u e .  128 (IX)  Implication,  denoted as •=», can have the f o l l o w i n g  r u l e i n order t o capture von Wright's i d e a of " n a t u r a l  truth  range of r e -  levance" i ( i ) I n the f i r s t p l a c e , V(P)  t then V(P  P=*Q presupposes P.  Hence, i f  Q) = n; and  ( i i ) i f V(P) = t , then V(P =^Q) = V(P->Q), where "-•" i s the 2 c o n d i t i o n a l o f L^. It follows has  the t r u t h  from the above t r u t h r u l e t h a t  the new  implication  table:  t f n  t f n n n n n n n  I t t u r n s out t h a t the above t r u t h t a b l e  i s i d e n t i c a l t o the  one reached a t s e c t i o n 10 although they are reached by q u i t e ferent routes.  The above route seems to g i v e us a  dif-  presuppositional  j u s t i f i c a t i o n o f the t r u t h r u l e o f the i m p l i c a t i o n . Note t h a t  I d i d not i n f a c t r e a c h the above t r u t h r u l e f o r the  i m p l i c a t i o n as smoothly as I d e s c r i b e d  i t here.  I n f a c t , before I  reached i t I s t i l l made many mistakes and t r i e d many o t h e r r o u t e s . The f a i l u r e s have been p a r t i a l l y recorded as C o n d i t i o n 1 t o Condit i o n 5 o f s e c t i o n 9 and, e s p e c i a l l y , as C o n d i t i o n 6 of s e c t i o n 1 0 .  129  14. A BRIEF VIEW OF C3 AND  QC3  B e f o r e the c o n s t r u c t i n g a new c o n f i r m a t i o n t h e o r y to s o l v e the paradoxes of c o n f i r m a t i o n , we'd b e t t e r stop a t t h i s p o i n t t o have a b r i e f view of i t s u n d e r l y i n g l o g i c C3 i s .  QC3.  p l u s the i n t e r n a l n e g a t i o n - and the i m p l i c a t i o n  Thus C3 c o n t a i n s the f o l l o w i n g s e n t e n t i a l  connectivess  ~ (the e x t e r n a l n e g a t i o n ) , v  (disjunction),  & (conjunction), —» ( c o n d i t i o n a l ) , <-> ( b i - c o n d i t i o n a l ) , — (the i n t e r n a l  negation),  =4. (the i m p l i c a t i o n ) . Then, the f i r s t r e s u l t about C3 i s i Theorem 10. Proof.  C3 i s t r u t h - f u n c t i o n a l l y complete.  We know from E.L.Post's work t h a t any t h r e e - v a l u e d  s e n t e n t i a l l o g i c i s t r u t h - f u n c t i o n a l l y complete i f i t c o n t a i n s the d i s j u n c t i o n o f C3 p l u s the f o l l o w i n g unary o p e r a t i o n "o" called truth  " ( P o s t i a n ) c y c l i c o p e r a t i o n " , which has the f o l l o w i n g tablet  1  p  OP  t f n  f n t  S i n c e oP can be d e f i n e d i n G3 as f o l l o w s : D e f i n i t i o n of oP.  oP =df (~P & ((P=4P)&(~P=»~P)))v~(P v -P)  as shown by the f o l l o w i n g t r u t h t a b l e s p  oP (~P&  t f n  f n t  f tf tfn t nn  ((P=»P)&(~P =*~p)))v' ~_ (p t t t n f t n f t f nf n tf nt f n n n n tn t tn  -P) f? f t t f t n f f t tf i i i t n n nn  1. C f . Rescher*N» Many-valued L o g i c , p.53.  V  (I969)  130 we have t h a t C3 i s t r u t h f u n c t i o n a l l y complete. Since we have i m p l i c a t i o n , define  QED.  i t would a l s o he n a t u r a l  f o r us to  a " b i - i m p l i c a t i o n " i n C3 as f o l l o w s :  D e f i n i t i o n of b i - i m p l i c a t i o n .  P<^ Q =df (P=*Q) & (Q=^P).  Then, the b i - i m p l i c a t i o n w i l l have the t r u t h  table:  1  Q p P<s=» t f n t f n f n n n n n  t f n  Note t h a t our b i - i m p l i c a t i o n t u r n s out t o be d i f f e r e n t from the b i - c o n d i t i o n a l t h a t we have had. Note a l s o t h a t the b i - c o n d i t i o n a l and the b i - i m p l i c a t i o n are i n an important sense not an "equivalence**, which l i t e r a l l y means "of equal ( t r u t h ) v a l u e " . so the f o l l o w i n g  In view of t h i s we i n t r o d u c e i n t o C3 a l -  " e q u i v a l e n c e " which has the t r u t h  table:  P == Q t f n t f n  t f f f t f f f t  Since C3 i s t r u t h - f u n c t i o n a l l y complete, i t i s b e t t e r f o r us to i n t r o d u c e i n t o i t the f o l l o w i n g as  and  read as "a c o n t r a d i c t i o n  d i c t i o n " , whose t r u t h t a b l e  t f n  " i n t r i n s i c n e g a t i o n " , denoted o f " and i n t e r p r e t e d  "contra-  i s g i v e n as f o l l o w s :  f t f  Thus i n C3 "P& 'P" i s a c o n t r a d i c t i o n ; "P&-P" are.  as  but n e i t h e r  "P&~P" nor  (On the other hand, the law o f excluded-middle i n C3  131 can o n l y be expressed and  the law  i n terms of the e x t e r n a l n e g a t i o n  of double-negation  which w i l l be our c o n f i r m a t i o n l o g i c f o r the  c o n f i r m a t i o n theory t h a t w i l l  (3 )  and  (\/ )  new  solve the paradoxes of c o n f i r m a t i o n ,  i s C3 p l u s both the e x i s t e n t i a l and denoted as  only i n terms of  as: ||-P=—P. f  the i n t e r n a l n e g a t i o n Then, QC3,  can be expressed  as: ||-Pv~P:  the u n i v e r s a l q u a n t i f i c a t i o n s  which are r e s p e c t i v e l y the g e n e r a l i z a -  t i o n s of the d i s j u n c t i o n and  the c o n j u n c t i o n of  G3.  A f t e r adding the c o n f i r m a t i o n a l v a l u a t i o n d e s c r i b e d i n s e c t i o n 12 of P a r t II to QG3  (with or without  i d e n t i t y ) , we  QC3  (with or without  i d e n t i t y ) i s a "decent l o g i c " ,  Theorem 11.  claim:  meaning: ( i ) i t has a f u l l y - d e v e l o p e d or complete semantics of i t s own,  and  ( i i ) i t i s " s e l f - s u f f i c i e n t " i n the sense t h a t the under-  lying logic etc.,  (and  can be QC3  Thus from now  semantics) of i t s meta-language, meta-meta-language, itself. on the l o g i c a l words and  appear i n our d i s c u s s i o n s (except stood as those  of QC3  reason  can use  t h a t we  not c o n v e r s e l y  ( i n s t e a d of those  in  concepts t h a t  s t a t e d ) should be  of QC2),  the v o c a b u l a r i e s of QC3  f o r example, we  " i m p l i c a t i o n " of QC3  otherwise  semantical  under-  f o r the obvious  to d i s c u s s QC2  do not have the c o u n t e r p a r t  but of  QC2.  2. See the t r u t h t a b l e s f o r the e x t e r n a l n e g a t i o n ~ and n a l n e g a t i o n - g i v e n r e s p e c t i v e l y on p.100 and p.11?  the i n t e r f o r checking.  132  15.  ELEMENTARY INTERNAL CONFIRMATION  THEORY  We i n t r o d u c e the f o l l o w i n g two d e f i n i t i o n s about tion  confirma-  theoriesi D e f i n i t i o n of an elementary c o n f i r m a t i o n theory.  A con-  f i r m a t i o n theory i s an "elementary c o n f i r m a t i o n theory", of i t s hypotheses are atomic o r molecular  i f all  sentences.  D e f i n i t i o n of a g e n e r a l c o n f i r m a t i o n theory.  A confirmation  theory i s a "general c o n f i r m a t i o n theory", i f i t has g e n e r a l  senten-  ces as w e l l as atomic e.nd molecular sentences among i t s hypotheses. Now we w i l l have a non-Hempelean i n t e r n a l elementary c o n f i r mation theory,  i f we add to QC3 the seven b a s i c concepts borrowed  from Hempel's theory of c o n f i r m a t i o n , v i z . , " o b s e r v a t i o n r e p o r t " , "hypothesis"  (which i s c o n f i n e d only to atomic or molecular  t e n c e s ) , "development of a h y p o t h e s i s " , "confirmation",  "direct confirmation" ,  " d i s c o n f i r m a t i o n " (which i s d e f i n e d i n terms o f  the i n t e r n a l negation, "neutrality",  sen-  thus "B d i s c o n f irms H" i s "B|(--H " ) and d  and a l s o the f o l l o w i n g "equivalence  The E q u i v a l e n c e  Condition.  condition":  Whatever confirms  or i s n e u t r a l with r e s p e c t t o ) one of two l o g i c a l l y sentences a l s o confirms  1  (disconfirms, equivalent  ( d i s c o n f i r m s , or i s n e u t r a l with  respect  to) the o t h e r . The  first  important  r e s u l t of t h i s elementary i n t e r n a l  con-  f i r m a t i o n theory i s t h i s : Theorem 1 2 . or a molecular  Suppose t h a t a h y p o t h e s i s  i s e i t h e r an atomic  sentence. Then, ( i ) i n Hempel's c o n f i r m a t i o n  theory,  1. Note t h a t "H " i s "the development of H w.r.t. the s e t of i n d i v i d u a l s e s s e n t i a l l y mentioned i n B". d  133  g i v e n t h a t the only sentence l e t t e r s t h a t occur i n an o b s e r v a t i o n r e p o r t B are those following  t h a t occur i n the  hypothesis  H, we have the  result:  (1.1) H i s confirmed  by B i f f H i s T (which i s "True" i n the  sense of QC2); (1.2) H i s d i s c o n f i r m e d by B i f f H i s F (which i s " F a l s e " i n the sense of QC2); ( 1 . 3 ) H i s never n e u t r a l w i t h r e s p e c t to B. On the other hand, internal  ( I I ) i n the newly c o n s t r u c t e d  elementary  c o n f i r m a t i o n theory, g i v e n t h a t the o n l y sentence l e t t e r s  t h a t occur  i n an o b s e r v a t i o n r e p o r t B are those  hypothesis  H, we have the f o l l o w i n g neat  (11.1) H i s confirmed  that  occurinthe  result:  by B i f f H i s t r u e (which i s t of QC3);  (11.2) H i s d i s c o n f i r m e d by B i f f H i s f a l s e  (which i s f of  QC3); (11.3) H i s n e u t r a l w i t h r e s p e c t to B i f f H i s n e i t h e r true nor f a l s e  ( i . e . , H i s n of QC3).  Examples of (II.3) of Theorem 12 a r e : HI: T h i s centaur  i s black;  H2: That u n i c o r n i s b l u e ; H3: Satan i s c l e v e r ; H4: The ether wind i s c o o l i n g . (And,  perhaps, a l s o H5« Pegasus = Pegasus, i f we add  the i d e n t i t y , v i z . "=",  to the v o c a b u l a r y  Note t h a t Theorem 12 has j u s t i f i e d  our intended  to C3.) identifica-  t i o n t h a t the three t r u t h - v a l u e s t , f , and n are r e s p e c t i v e l y i d e n t i f i a b l e to the three c o n f i r m a t i o n - s t a t u s e s c ( c o n f i r m a t i o n ) ,  134 d ( d i s c o n f i r m a t i o n ) and n  (neutrality).  The r e s u l t of Theorem 12 a l s o j u s t i f i e s  our c a l l i n g  the  elementary i n t e r n a l c o n f i r m a t i o n theory "non-Hempelean", f o r the r e s u l t s of (I) and ( I I ) are q u i t e d i f f e r e n t . Note a l s o t h a t only the equivalence nor the b i - i m p l i c a t i o n ) of QC3 observes given i n t h i s section. Theorem 13. responding  (not the b i - c o n d i t i o n a l , the E q u i v a l e n c e  Condition  In other words, we have*  I f ||-(Hl«H2)  t  then we have the f o l l o w i n g c o r -  p r o p e r t y : HI i s confirmed  ( d i s c o n f i r m e d , or n e u t r a l  with r e s p e c t to an o b s e r v a t i o n r e p o r t ) i f f H2 i s confirmed  (dis-  conf irmed, or n e u t r a l with r e s p e c t to the same o b s e r v a t i o n report)? on the other hand we cannot have the above c o r r e s p o n d i n g  property  even i f we have: ||-(HI <-* H2 ), or we have: |f- (HI 4=*H2 ). T h i s Theorem thus  j u s t i f i e s the i n t r o d u c t i o n of the e q u i v a -  l e n c e , v i z . , " = " , i n t o the v o c a b u l a r y  of C3 and  QC3.  135 16.  THE GENERAL INTERNAL CONFIRMATION THEORY  The  elementary i n t e r n a l c o n f i r m a t i o n theory can be e a s i l y ex-  tended now t o a g e n e r a l i n t e r n a l c o n f i r m a t i o n theory, which i s a l s o non-Hempelean, by making the f o l l o w i n g two major t h i n g s t (I) Add any g e n e r a l sentences,  b e s i d e s atomic and molecular  ones, as p o s s i b l e hypotheses. (II)  On top o f the elementary i n t e r n a l c o n f i r m a t i o n theory we  a l s o add the f o l l o w i n g few concepts! D e f i n i t i o n s of p o s i t i v e , n e g a t i v e pose t h a t G  V//c  and n e u t r a l i n s t a n c e s .  , which i s the development of the hypothesis  with r e s p e c t to the r e f e r e n c e c l a s s {cJ, i s confirmed firmed by, or n e u t r a l t o ) some o b s e r v a t i o n r e p o r t .  Sup-  ()/v)Gv  by ( d i s c o n -  Then, G  a p o s i t i v e (negative, or n e u t r a l ) i n s t a n c e o f hypothesis  V / /  °is  (Vv)Gv.  D e f i n i t i o n of " s t r o n g c o n f i r m a t i o n " . A u n i v e r s a l hypothesis i s s t r o n g l y confirmed i f a l l o f i t s i n s t a n c e s are p o s i t i v e ones. D e f i n i t i o n o f "pure n e u t r a l i t y " . A u n i v e r s a l hypothesis i s p u r e l y n e u t r a l i f a l l of i t s i n s t a n c e s a r e n e u t r a l ones. D e f i n i t i o n of "weak c o n f i r m a t i o n " (or "mixed n e u t r a l i t y " ) . A u n i v e r s a l h y p o t h e s i s i s weakly confirmed (or mixed n e u t r a l ) i f ( i ) some o f i t s i n s t a n c e s a r e p o s i t i v e and ( i i ) some other i n s t a n c e s are n e u t r a l ones, w h i l e ( i i i ) none o f i t s i n s t a n c e s are negative. The  last  Hempelean. to  three concepts are something new; and they are nonThey w i l l p l a y some important  r o l e l a t e r when we come  the s o l u t i o n o f the paradoxes o f c o n f i r m a t i o n . Thus, a t l a s t ,  we have a non-Hempelean g e n e r a l i n t e r n a l con-  f i r m a t i o n theory, which w i l l be c a l l e d  "the i n t e r n a l  t h e or y" from now on f o r the sake o f s i m p l i c i t y .  confirmation  136 17- A QUASI-SOLUTION OF THE PARADOXES OF CONFIRMATION I f we frame the Raven Hypothesis t h a t a l l ravens a r e b l a c k i n terms of the i m p l i c a t i o n of QC3 as the f o l l o w i n g  Implicational  Raven Hypothesis,  (Vx)(Rx =^Bx)  IRH:  where R J 0 B« ® to be read s t i l l  i s a raven, i s black,  as " a l l ravens a r e b l a c k " , then we c l a i m the  following resulti  Theorem 14. plicational  I f we re-frame the Raven Hypothesis as an im-  law as (Vx) (Rx =^ Bx), then the paradoxes of c o n f i r -  mation a r e not d e r i v a b l e i n the i n t e r n a l c o n f i r m a t i o n t h e o r y . Proof. (1) Suppose t h a t a i s a b l a c k raven. Ra & Ba  Then, we have:  \\-(Ra =*3a),  i . e . , any b l a c k r a v e n confirms the I m p l i c a t i o n a l Raven H y p o t h e s i s . (2) Suppose t h a t b i s a n o t - b l a c k raven.  Then, we have:  Rb & -Bb ||--(Rb =*3b), i . e . , any n o t - b l a c k raven d i s c o n f i r m s the I m p l i c a t i o n a l Raven Hypothesis. (3.1) Suppose t h a t c i s a b l a c k peony.  Since i t i s not  a raven, we have t h a t -Rc; and s i n c e i t i s b l a c k , we have t h a t Be.  Together we have: (-Rc&Bc).  So, together we have:  -Rc 4Bc^Jf(Rc «r>Bc) and -Rc & Be Jjf- (Rc =*Bc ),  13?  i . e . , any b l a c k not-raven c a t i o n a l Raven (3.2)  i s n e u t r a l w i t h r e s p e c t t o the I m p l i -  Hypothesis.  Suppose t h a t d i s a white swan. Since i t i s not a raven  and i t i s not b l a c k , we have: -Rd&-Bd.  So, t o g e t h e r we have:  -Rd& -BdJ^(Rd =*Bd) and -Rd & -Bd Jjf- (Rd =»Bd), i . e . , any n o t - b l a c k not-raven Raven  i s n e u t r a l to the I m p l i c a t i o n a l  Hypothesis.  (3.3)  Suppose t h a t "e" i s "a b l u e u n i c o r n " .  Then, a c c o r d i n g  to the c o n f i r m a t i o n a l v a l u a t i o n ( i n the sense o f a b s o l u t e semant i c s ) we have: V(Re)= n = V ( B e ) , f o r t h e r e are no u n i c o r n s actual world).  Hence, we have: ~-Re &~BeJ~ ~ R e & ~ B e J p ( R e =*Be)  ( i n the  So, t o g e t h e r we have:  and  i . e . , any non-black non-raven i s n e u t r a l t o the I m p l i c a t i o n a l Raven  QED.  Hypothesis.  To sum up, we have: Theorem 1 5 .  I n the i n t e r n a l c o n f i r m a t i o n theory we have the  f o l l o w i n g r e s u l t s about the I m p l i c a t i o n a l Raven (1)  Only b l a c k ravens c o n f i r m i t :  (2)  Only a n o t - b l a c k raven d i s c o n f i r m s i t ;  Hypothesis:  and  (3) A l l o t h e r cases are n e u t r a l to i t .  1. Note t h a t we do not have: -Re &-Be  i n the present  case.  138 18. SOME CRITICISMS OF THE  QUASI-SOLUTION OF THE  PARADOXES  OF CONFIRMATION At t h i s p o i n t when  I was  i n c l i n e d to c l a i m  c o n f i r m a t i o n had been s o l v e d , we  the paradoxes of  f i n d the f o l l o w i n g two  criticisms  about the " s o l u t i o n " of the paradoxes of c o n f i r m a t i o n o f f e r e d i n the p r e v i o u s  s e c t i o n i n p a r t i c u l a r and  the i n t e r n a l  confirmation  theory i n g e n e r a l t 1) The  t r u t h - v a l u e s of the I m p l i c a t i o n a l Raven Hypothesis i s  n e i t h e r t r u e nor f a l s e i n the a c t u a l world;so l o n g as there e x i s t s one  i n d i v i d u a l other than a raven  or an i n d i v i d u a l which i s not  b l a c k , the t r u t h - v a l u e of the I m p l i c a t i o n a l Raven Hypothesis cannot be true i n the a c t u a l world.  T h i s i s odd.  In c o n t r a s t the  t r u t h of the Raven Hypothesis i n Hempel's t h e o r y of c o n f i r m a t i o n i s not excluded  by such c o n s i d e r a t i o n s .  Here i t seems t h a t there i s an exchange: On the one paradoxes of c o n f i r m a t i o n are avoided theory, when we Raven Hypothesis?  hand the  inthe internal confirmation  frame the Raven Hypothesis as the I m p l i c a t i o n a l on the other hand the t r u t h - v a l u e of the  c a t i o n a l Raven Hypothesis i s n e i t h e r t r u t h nor f a l s i t y ,  Impli-  instead  of t r u t h . In c o n t r a s t i n Hempel's c o n f i r m a t i o n theory,  although  we  have  the paradoxes of c o n f i r m a t i o n , the t r u t h - v a l u e of the Raven Hypot h e s i s i s , simply, T r u t h 2) value  (we may  suppose).  In g e n e r a l , i n the i n t e r n a l c o n f i r m a t i o n theory the t r u t h of any  u n i v e r s a l i m p l i c a t i o n a l hypothesis  so l o n g there e x i s t s one  cannot be t r u t h  i n d i v i d u a l other than those  mentioned  i n the antecedent of the i m p l i c a t i o n . T h i s i s more than odd.  Any  such  theory of c o n f i r m a t i o n i s  139  s e r i o u s l y inadequate,  f o r the theory cannot r e s u l t i n a n y t h i n g  t r u e , i f an e m p i r i c a l h y p o t h e s i s i s framed i n terms of the imp l i c a t i o n o f QG3, though i t may r e s u l t i n some hypotheses "being confirmed.  T h i s c r i t e r i o n o f inadequacy i s due to the nature o f  s c i e n c e t h a t s c i e n c e i s o b v i o u s l y i n search of " t r u t h " , not, or not  just,  " n e i t h e r t r u t h nor f a l s i t y " .  I s u f f e r e d a shocking experience when I d i s c o v e r e d how v u l n e r a b l e I was t o the above v i g o r o u s c r i t i c i s m s , 2).  F o r a l o n g time I consoled myself  e s p e c i a l l y point  t h a t , perhaps, s c i e n c e i s  a l s o i n search of " n o n - f a l s i t y " and, hence, t h a t the i n t e r n a l c o n f i r m a t i o n theory i s not t o t a l l y inadequate own l i m i t e d v a l u e .  and s t i l l has i t s  140 19. A PLAUSIBLE SOLUTION OF THE PARADOXES OF CONFIRMATION The  above two v i g o r o u s c r i t i c i s m s o f the " s o l u t i o n " o f the pa-  radoxes o f c o n f i r m a t i o n i n p a r t i c u l a r and the inadequacy o f the i n t e r n a l c o n f i r m a t i o n theory i n g e n e r a l , as s t a t e d i n the l a s t t i o n , posed a s e r i o u s problem f o r me f o r a l o n g time.  sec-  However,  they can be overcome. The  key t o a p l a u s i b l e s o l u t i o n which answers the two above-  mentioned c r i t i c i s m s i s t h i s i p l i c a t i o n a l Raven IRHi  Although the t r u t h - v a l u e o f the Im-  Hypothesis,  (VX)(RX-*BX),  i s n e i t h e r t r u e nor f a l s e ,  i t s c o n f i r m a t i o n - s t a t u s i s "weakly con-  firmed", which i s s t r o n g e r and b e t t e r than " p u r e l y n e u t r a l " .  1  Thus  i f we can exclude  those n e u t r a l i n s t a n c e s from the IRH, the r e f i n e d  IRH,  "the d e - n e u t r a l i z e d I m p l i c a t i o n a l Raven Hypothe-  sis" ly  t o be c a l l e d  (IRH^), w i l l be t r u e , and i t s c o n f i r m a t i o n - s t a t u s i s now s t r o n g 2 confirmed. In order t o achieve  t h i s we w i l l r e s t r i c t  p o t h e s i s t o i t s " n a t u r a l range o f r e l e v a n c e " , and  the domain o f any hyf o l l o w i n g von Wrightj  i n order t o see how t h i s i s done we w i l l c o n s i d e r the s i m p l e s t  form o f a u n i v e r s a l i m p l i c a t i o n a l h y p o t h e s i s i Hi  (VX) (FX  =*GX),  and  denote i t s " d e - n e u t r a l i z e d h y p o t h e s i s " as "H^". The " n a t u r a l u range o f r e l e v a n c e " o f W i s then d e f i n e d as f o l l o w s i f x i F x & (Gx v -Gx)> . 1. The d e f i n i t i o n s o f "weak c o n f i r m a t i o n " and "pure n e u t r a l i t y " a r e g i v e n i n s e c t i o n 16 o f t h i s P a r t , p.135. 2. See i t s d e f i n i t i o n on p.135-  141 In g e n e r a l , Hs  suppose a h y p o t h e s i s H i s of the f o l l o w i n g forms  H(F1,F2  F i , G1,G2  Gj ),  where F l , F 2 , . . . F i are antecedents of any f  Gl,G2,...,Gj are consequents of any Then, the  {x:  i m p l i c a t i o n i n H,  " n a t u r a l range of r e l e v a n c e "  thesis H i s defined  as  i m p l i c a t i o n i n H while  of }r  a s s o c i a t e d w i t h hypo-  (F1&F2&...&Fi)&(Glv -G1)&(G2v -G2)&...&(Gjv -Gj)}  p l i c a t i o n a l Raven Hypothesis IRH {x: Hence, IRH^ i t s reference Thus we  Rx&  i s not  (Bx v -Bx)}  t h e s i s IRH  c l a s s of i n d i v i d u a l s ) but  can be  2) In g e n e r a l , vide truths  IRH i s :  a l s o now  (with r e s p e c t  to  true.  criticisms raised  earlier:  I m p l i c a t i o n a l Raven Hypo-  true. the  i n t e r n a l confirmation  theory  can a l s o p r o -  (besides n e i t h e r t r u t h nor f a l s i t y ) i f any  h y p o t h e s i s i s framed i n terms of the i m p l i c a t i o n of i t follows that  ( i ) we  the i n t e r n a l c o n f i r m a t i o n rous c r i t i c i s m s once we  empirical  QG3.  have solved the paradoxes of c o n f i r -  mation i f the Raven Hypothesis i s understood as the IRH  A few  Im-  .  only s t r o n g l y confirmed  have answered the two  .  of the d e - n e u t r a l i z e d  a s s o c i a t e d w i t h the  1) In p a r t i c u l a r , the d e - n e u t r a l i z e d  pothesis  i,j^.l.  follows:  Thus the n a t u r a l range of relevance  So,  and  theory  and ( i i )  remains adequate under the  have i n t r o d u c e d  the  vigo-  # - r e s t r i c t i o n of a  into i t . important s i d e r e s u l t s can a l s o be mentioned here:  hy-  142 ( I ) Our s t r a t e g y f o r a p l a u s i b l e s o l u t i o n of the paradoxes of c o n f i r m a t i o n v i a the concept of "the d e - n e u t r a l i z e d a s s o c i a t e d with a hypothesis  hypothesis"  (say, e.g., the Raven Hypothesis) can-  not be a p p l i e d to Hempel's theory of c o n f i r m a t i o n .  T h i s i s mainly  because no n e u t r a l i n s t a n c e about an o b j e c t whose complete mation r e l a t i v e to the Raven Hypothesis (or to any other  infor-  hypothe-  s i s ) can be knocked out i n Hempel's theory of c o n f i r m a t i o n .  So,  any such move to a v o i d the paradoxes of c o n f i r m a t i o n i s d e s t i n e d to be a f a t a l f a i l u r e by committing the f a l l a c y of " t a k i n g (some) r e l e v a n t i n s t a n c e s as i r r e l e v a n t  ones".  ( I I ) In the i n t e r n a l c o n f i r m a t i o n theory there i s always a chanic way  to t e l l whether  range of r e l e v a n c e  me-  or not an o b j e c t c i s i n the n a t u r a l  of a h y p o t h e s i s  Suppose t h a t we have a hypothesis  H.  The mechanic  way  i s this.  H and suppose t h a t we a l s o have  a complete i n f o r m a t i o n of an o b j e c t c r e l e a t i v e to a l l p r e d i c a t e s appearing  i n H.  Then, we  have:  c i s i n the n a t u r a l range of r e l e v a n c e r e p o r t B(c) i s not n e u t r a l to H.  of H i f f o b s e r v a t i o n  Since B(c) i s never n e u t r a l to H i n Hempel's theory of c o n f i r mation due to i t s p r i n c i p l e of u n i v e r s a l r e l e v a n c e , we can never 1  have the above r e s u l t i n Hempel's t h e o r y of c o n f i r m a t i o n . there i s no way  t o l a y down the n a t u r a l range of r e l e v a n c e  p o t h e s i s i n Hempel's theory.  Hence, of a hy-  T h i s poses a g r e a t d i f f i c u l t y f o r  von Wright i n h i s s o l u t i o n of the paradoxes of c o n f i r m a t i o n v i a the concept o f " n a t u r a l range of r e l e v a n c e " . ( I I I ) Another c r u c i a l t h i n g to note i s t h a t i n von Wright's s o l u t i o n of the paradoxes of c o n f i r m a t i o n v i a the concept of "na1. Cf. s e c t i o n 7, P a r t I, p . 2 0 f f .  143 t u r n a l range of r e l e v a n c e " two i n g e n e r a l have two  different  t h i s i s not the case  (x)(Rxr>Bx)  H2:  (x) (-Bx =>-Rx)  w i l l o b v i o u s l y have two But  " n a t u r a l ranges of r e l e v a n c e " ;  i n our approach.  approach the f o l l o w i n g two Hit  l o g i c a l l y e q u i v a l e n t hypotheses w i l l  Thus, e.g.,  " n a t u r a l ranges of r e l e v a n c e " .  i n our approach the f o l l o w i n g two  ((Vx)(Rx=>Bx)) ,  H4:  ((l/x)(-Bx=V-Rx)) ,  i n von Wright's  l o g i c a l l y e q u i v a l e n t hypotheses:  different  H3«  but  counterpart-hypotheses,  #  #  are not l o g i c a l l y e q u i v a l e n t nor have they the same " n a t u r a l range of r e l e v a n c e " .  In f a c t , what i s l o g i c a l l y e q u i v a l e n t to H3  i s the  hypothesis:  ((Vx)((Pxv -Px) & (Rx=>Bx))) ,  H5:  #  where "P" But,  i s any p r e d i c a t e .  then, H3 and H5 have the same " n a t u r a l range of r e l e v a n c e " .  Thus the E q u i v a l e n c e  C o n d i t i o n i s s t i l l preserved  i n the  inter-  n a l c o n f i r m a t i o n theory even under the # - r e s t r i c t i o n ; and t h i s i s why  we  a l s o f e e l t h a t von Wright's s i m i l a r s o l u t i o n of the  doxes of c o n f i r m a t i o n i s not r e a l l y s a t i s f a c t o r y , f o r the  paraequiva-  l e n c e c o n d i t i o n i s not preserved under the t r a n s f o r m a t i o n of n a t u r a l range of r e l e v a n c e " of a h y p o t h e s i s  i n h i s approach.  (IV) F i n a l l y , a l s o note the f o l l o w i n g important Theorem 17.  The  "the  property:  " c o n f i r m a t i o n " of a d e - n e u t r a l i z e d i m p l i c a -  t i o n a l hypothesis  ((Vx)(Fx=»Gx))^ i s a u t o m a t i c a l l y a " s e l e c t i v e  c o n f i r m a t i o n " i n the i n t e r n a l c o n f i r m a t i o n theory i f we d e f i n e the " c o n t r a r y " o f the g i v e n h y p o t h e s i s  as " ((Vx) (Fx=*—Gx) ) ^ . M  Proof. T h i s i s due t o the f a c t t h a t the c o n f i r m a t i o n of a d e - n e u t r a l i z e d i m p l i c a t i o n a l hypothesis  H« #  such as,  ((\/x)(Fx=*Gx)) , #  i s confirmed  i f and o n l y i f ( i )  i s confirmed  and ( i i ) the "con-  t r a r y " CH# of CH t #  ((Vx)(Fx=*-Gx)) , #  i s d i s c o n f i r m e d i n the i n t e r n a l c o n f i r m a t i o n t h e o r y .  QED.  145  PART IV.  ADEQUACY CONDITIONS FOR CONFIRMATION AND THE GOODMAN  PARADOX  146  1.  ADEQUACY CONDITIONS FOR CONFIRMATION LAID DOWN BY HEMPEL  A c c o r d i n g t o Hempel any c o n f i r m a t i o n theory must observe the f o l l o w i n g c o n d i t i o n s i n order t o be adequate; (8.1)  Entailment  Condition.  Any sentence which i s e n t a i l e d  by an o b s e r v a t i o n r e p o r t i s confirmed (8.2)  1  Consequence C o n d i t i o n .  by i t .  I f an o b s e r v a t i o n r e p o r t con-  f i r m s every one of a c l a s s K o f sentences,  then i t a l s o  confirms  any sentence which i s a l o g i c a l consequence o f K. (The f o l l o w i n g two more c o n d i t i o n s are consequences o f the above c o n d i t i o n . ) (8.21) S p e c i a l Consequence C o n d i t i o n .  I f an o b s e r v a t i o n r e -  p o r t confirms a hypothesis H, then i t a l s o confirms  every conse-  quence of H. (8.22) Equivalence  Condition.  I f an o b s e r v a t i o n r e p o r t con-  f i r m s a h y p o t h e s i s H, then i t a l s o confirms  every hypothesis which  i s l o g i c a l l y e q u i v a l e n t with H. (8.3)  Consistency  Condition.  Every l o g i c a l l y c o n s i s t e n t ob-  s e r v a t i o n r e p o r t i s l o g i c a l l y compatible hypotheses which i t c o n f i r m s .  with the c l a s s of a l l the  (This i m p l i e s t h a t  (8.31) Unless an o b s e r v a t i o n r e p o r t i s s e l f - c o n t r a d i c t o r y , i t does n o t c o n f i r m any hypotheses which c o n t r a d i c t each other, and that (8.32) Unless an o b s e r v a t i o n r e p o r t i s s e l f - c o n t r a d i c t o r y , i t does not c o n f i r m any hypotheses which c o n t r a d i c t each (8.33) C o n j u n c t i o n 1.  Hempelt Aspects  Condition.  other.)  I f an o b s e r v a t i o n r e p o r t con-  of S c i e n t i f i c E x p l a n a t i o n ,  pp.31-3 . k  147  f i r m s each o f two hypotheses, then i t a l s o confirms  t h e i r conjunc-  tion. These c o n d i t i o n s must be met f o r any adequate theory firmation according  of con-  t o Hempel; i n other words, they are necessary  c o n d i t i o n s f o r any adequate theory In t h i s P a r t we w i l l  of confirmation.  t r y t o examine t h e i r n e c e s s i t y , and come  to a c o n c l u s i o n which i s much weaker than Hempel's view when he laid  down the c o n d i t i o n s . However, b e f o r e we proceed t o examine them, l e t us examine a  rejected  "Converse Consequence C o n d i t i o n " , which i s of some i n t e r -  e s t and has been proposed by some  authors.  148  2. THE CONVERSE CONSEQUENCE CONDITION AND ITS REJECTION The Converse Consequence C o n d i t i o n can be s t a t e d as f o l l o w s : (8.4)  Converse Consequence C o n d i t i o n .  p o r t confirms  a hypothesis  1  I f an o b s e r v a t i o n r e -  H, then i t a l s o confirms  any sentence  K which has H as i t s consequence. Hempel r e j e c t s the above Converse Consequence C o n d i t i o n , f o r the f o l l o w i n g two reasons: (I) The Converse Consequence C o n d i t i o n i s incompatible the Consequence C o n d i t i o n .  with  To show t h i s , l e t us assume t h a t Hem-  p e l ' s theory o f c o n f i r m a t i o n  ( o r any theory  adequate and, hence, c o n s i s t e n t .  of c o n f i r m a t i o n ) i s  Then, l e t us make the f o l l o w i n g  two n o t a t i o n a l a b b r e v i a t i o n s * 1. B* r i s a raven. 2.  H: Hook's law (or any p h y s i c a l law o r  I t f o l l o w s by Hempel's  statement).  definition:  3 . B c o n f i r m s B. Since  i n classical logic  the f o l l o w i n g statement i s always  true, 4. ( B » H ) O B , by the Converse Consequence C o n d i t i o n we have: 5. B confirms Likewise,  (B-H).  i t i s always t r u e i n c l a s s i c a l l o g i c  6. (B-H)  "D H .  Hence, by the Consequence C o n d i t i o n we have* 1. Hempel*  Aspects of S c i e n t i f i c E x p l a n a t i o n , p . 3 2 .  t h a t we have*  149  ?.  B confirms  H.  The above r e s u l t means t h a t any o b s e r v a t i o n r e p o r t B c o u l d c o n f i r m any h y p o t h e s i s H, t o r y statement.  even i f H i s a f a l s e or a c o n t r a d i c -  T h i s i s absurd.  Thus the Consequence C o n d i t i o n and the Converse Consequence C o n d i t i o n are i n c o m p a t i b l e . Hence, they cannot both be v a l i d .  Hempel t h i n k s t h a t here  the Converse Consequence C o n d i t i o n has to go f o r the f o l l o w i n g reasont ( I I ) A h y p o t h e s i s i s confirmed by an o b s e r v a t i o n r e p o r t should not imply t h a t a s t r o n g e r h y p o t h e s i s i s a l s o confirmed by observation report.  the same  Thus the h y p o t h e s i s t h a t r i s a raven i s con-  f i r m e d by the o b s e r v a t i o n r e p o r t t h a t r i s a raven; but the  obser-  v a t i o n r e p o r t should not without any other good reason a l s o  con-  f i r m the s t r o n g e r h y p o t h e s i s t h a t r i s a raven and The above reason seems to be sound.  So, we  to r e j e c t the Converse Consequence C o n d i t i o n .  Hook's law.  agree w i t h Hempel  150 3.  CARNAP AND  THE  CONSEQUENCE CONDITION  Carnap t h i n k s t h a t the consequence c o n d i t i o n i s i n v a l i d . He  g i v e s the f o l l o w i n g c o u n t e r - e x a m p l e t  1  The i n i t i a l evidence i s t h a t t e n chess p l a y e r s w i l l p a r t i c i p a t e i n a tournament. Some are men (M) and others women (W). Some are from New York and others are not. Some are j u n i o r p l a y e r s and others are s e n i o r p l a y e r s . T h e i r d i s t r i b u t i o n i s as f o l l o w s t New Juniors Seniors  M, M,  Yorkers  Strangers  W, W  M, M W, W,  M  W  Moreover, we know t h a t only one w i l l win and t h a t each has an equal chance of winning. L e t H = 'a woman wins', and l e t K = 'a s t r a n g e r wins'. Then the degree of c o n f i r mation of H and the degree of c o n f i r m a t i o n of K on the i n i t i a l evidence i s . 5 . Now l e t us suppose t h a t we are g i v e n the a d d i t i o n a l i n f o r m a t i o n t h a t a s e n i o r p l a y e r has won. Then the degree of c o n f i r m a t i o n of H g i v e n t h i s new evidence i s .6; so t h i s new evidence confirms H. S i m i l a r l y , the degree of c o n f i r m a t i o n of K g i v e n t h i s new evidence i s .6; so t h i s new evidence confirms K. The degree of c o n f i r m a t i o n of Hv K on the i n i t i a l evidence was, however, . 7 ; but, g i v e n t h i s new i n f o r m a t i o n , i t i s only .6. T h i s new evidence, t h e r e f o r e , d i s c o n f irms H v K even though i t confirms H and confirms K, and e i t h e r of the l a t t e r e n t a i l s the former. The  above "counter-example" shows t h a t the consequence con-  d i t i o n i s indeed  not v a l i d  i n Carnap's theory of c o n f i r m a t i o n ,  where "degree of c o n f i r m a t i o n " ( a s , i n a sense,  probability) i s  c l e a r l y a q u a n t i t a t i v e concept of c o n f i r m a t i o n : and t h i s i s q u i t e p  d i f f e r e n t from Hempel's q u a l i t a t i v e concept of c o n f i r m a t i o n . So,  i t c o u l d happen t h a t the consequence c o n d i t i o n holds i n Hem-  p e l ' s theory of c o n f i r m a t i o n , although nap's t h e o r y of c o n f i r m a t i o n . rise: Ql: 1.  2.  i t does not h o l d i n Car-  Thus the  following  question  a-  Do. Carnap and Hempel t a l k about the same t h i n g when they  C f . B.A. Brody: " C o n f i r m a t i o n and e x p l a n a t i o n " , the J o u r n a l of Philosophy, LXV, 10; p.288; or Carnap's o r i g i n a l v e r s i o n i n h i s L o g i c a l Foundations of P r o b a b i l i t y . 2nd ed., p.383. Carnap: L o g i c a l Foundations of P r o b a b i l i t y , p . 2 5 f f .  151 t a l k about " c o n f i r m a t i o n " ? Q2» Is i t p o s s i b l e  t h a t the  consequence  c o n d i t i o n holds  i n the c o n t e x t of Hempel's q u a l i t a t i v e theory of c o n f i r m a t i o n but not i n the context of Carnap*s q u a n t i t a t i v e theory of con' firmation? We w i l l t r y t o answer the above two q u e s t i o n s and r e l a t e d ones i n the next few s e c t i o n s .  152  4. EXPLICATION OF THE CONCEPT OF CONFIRMATION Carnap t h i n k s t h a t there are two e x p l i c a t i o n s of the concept of  confirmation! (I)  1  C o n f i r m a t i o n as firmness o f h y p o t h e s i s .  Here " f i r m -  ness" means "of h i g h p r o b a b i l i t y " . (II)  C o n f i r m a t i o n as i n c r e a s e i n firmness o f h y p o t h e s i s .  As an i n t e r p r e t a t i o n o f Hempel's n o t i o n of c o n f i r m a t i o n the first  e x p l i c a t i o n i s e a s i l y seen to be untenable  or i n a p p l i c a b l e .  To see t h i s , l e t us make the f o l l o w i n g two assumptions! HIs  A l l f l o w e r s are p u r p l e .  Bis  a^ i s a purple f l o w e r .  Then i t i s obvious t h a t the o b s e r v a t i o n r e p o r t B l confirms the h y p o t h e s i s HI i n Hempel's t h e o r y .  Thus were the f i r s t  explica-  t i o n o f c o n f i r m a t i o n a p p l i c a b l e to Hempel's theory o f c o n f i r m a t i o n , the h y p o t h e s i s HI would be f i r m . in  fact, false.  But hypothesis HI i s ,  Hence, i t cannot be f i r m , nor can i t have h i g h  p r o b a b i l i t y f o r most f l o w e r s are not p u r p l e . Carnap's second e x p l i c a t i o n of c o n f i r m a t i o n as i n c r e a s e i n f i r m n e s s o f hypothesis  seems to be a p p l i c a b l e or compatible to  Hempel's t h e o r y p r o v i d e d t h a t we make the f o l l o w i n g two c l a r i f i cations s i)  Some hypotheses,  increase i n firmness. H2s  cannot  F o r example, the h y p o t h e s i s  There are f l o w e r s  i s w e l l confirmed 1.  when they are w e l l confirmed,  by the o b s e r v a t i o n r e p o r t ,  Carnaps L o g i c a l Foundations  of P r o b a b i l i t y , p.xv f f .  153  B2:  a  2  i s a flower.  How c o u l d such a hypothesis  as H2, which i s confirmed  now by  the o b s e r v a t i o n r e p o r t B2 and hence has p r o b a b i l i t y one, be made " f i r m e r " present  one  i . e . , to have a h i g h e r p r o b a b i l i t y than the by any new  evidence?  However, t h i s d i f f i c u l t y can be overcome e a s i l y by unders t a n d i n g what Carnap means by " i n c r e a s e i n firmness o f a hypot h e s i s " as " i n c r e a s e of the p r o b a b i l i t y of the hypothesis up to p r o b a b i l i t y one", not as " i n c r e a s e of the p r o b a b i l i t y of the hypothesis without  any upper  limit".  i i ) In Hempel's theory of q u a l i t a t i v e c o n f i r m a t i o n he does not have a concept  of a p r i o r i p r o b a b i l i t y .  we say that Hempel's concept  In consequence when  of c o n f i r m a t i o n can be i n t e r p r e t e d  as " i n c r e a s e i n firmness of a h y p o t h e s i s " , we encouter  the f o l -  lowing d i f f i c u l t y : How can we t e l l whether an o b s e r v a t i o n r e p o r t i n c r e a s e s or decreases  the firmness o f a hypothesis  theory of q u a l i t a t i v e c o n f i r m a t i o n ?  i n Hempel's  Here what we can say i s  t h a t we can see no d i f f i c u l t y f o r Hempel to i n t r o d u c e a concept of a p r i o r i  p r o b a b i l i t y i n t o h i s theory of q u a l i t a t i v e c o n f i r -  mation i n such a way t h a t Hempel's concept be i n t e r p r e t e d as, o r i s compatible  of c o n f i r m a t i o n can  with, Carnap's n o t i o n o f  " i n c r e a s e i n firmness o f a h y p o t h e s i s " .  This suggestion i s i n 2 accord with Hempel's l a t e s t view when he says« Perhaps the problem of f o r m u l a t i n g adequate c r i t e r i a of q u a l i t a t i v e c o n f i r m a t i o n had best be t a c k l e d , a f t e r a l l , by means of the q u a n t i t a t i v e concept of confirmation. T h i s has been suggested e s p e c i a l l y by Carnap, who holds t h a t "any adequate explicatum f o r the c l a s s i f i c a t o r y concept o f c o n f i r m a t i o n must be i n accord with a t 2. Hempeli Aspects  of S c i e n t i f i c  E x p l a n a t i o n , p.50.  154 l e a s t one adequate explicatum f o r the q u a n t i t a t i v e concept of c o n f i r m a t i o n . ... In other words: on some s u i t a b l e d e f i n i t i o n of l o g i c a l p r o b a b i l i t y , the p r o b a b i l i t y of H on B should exceed the a p r i o r i probab i l i t y of H whenever B q u a l i t a t i v e l y c o n f i r m s H. With the above two c l a r i f i c a t i o n s , we second  say t h a t Carnap*s  e x p l i c a t i o n of the concept of c o n f i r m a t i o n as i n c r e a s e  i n f i r m n e s s of hypothesis i s a p p l i c a b l e or, or a t l e a s t compit a b l e with, Hempel's t h e o r y of c o n f i r m a t i o n .  155  5.  CARNAP AND THE CONJUNCTION CONDITION  Carnap t r i e s t o r e j e c t not only the Consequence C o n d i t i o n but a l s o the C o n j u n c t i o n confirmation.  C o n d i t i o n of Hempel's theory of q u a l i t a t i v e  H i s "counter-example" f o r the l a t t e r i s t h i s *  1  L e t the p r i o r evidence e c o n t a i n the f o l l o w i n g i n f o r mation. Ten chess p l a y e r s p a r t i c i p a t e i n a chess tournament i n New York C i t y . Some o f them a r e l o c a l people, some from out of town: some are j u n i o r p l a y e r s , some are s e n i o r s ; some a r e men (M) and some are women (W). The d i s t r i b u t i o n i s known t o be as f o l l o w s : Local players  Out-of-towners  Juniors  M, W, W  M, M  Seniors  M, M  W, W, W  Furthermore, the evidence e i s assumed t o be such t h a t on i t s b a s i s each of the t e n p l a y e r s has an equal chance o f becoming the winner, hence the chance f o r any p l a y e r to win i s l / l O . I t i s a l s o assumed t h a t i n each case o f e v i dence t h a t c e r t a i n p l a y e r s have been e l i m i n a t e d , the r e maining p l a y e r s have equal chance o f winning. L e t h be the h y p o t h e s i s t h a t a man wins. L e t i, be the evidence t h a t a l o c a l p l a y e r wins. L e t j be the evidence t h a t a j u n i o r wins. ~~ U s i n g the background i n f o r m a t i o n g i v e n above i n the t a b l e , we can o b t a i n the f o l l o w i n g v a l u e s : c(h,e) = 1 / 2 ;  Thus, while other firms So,  c(h, e.i) = 3/5; c(h, e.j) = 3/5; c(h, e . i . j ) = 1/3.  i, and j a r e each p o s i t i v e l y r e l e v a n t t o h y p o t h e s i s h, the c o n j u n c t i o n i . j i s n e g a t i v e l y r e l e v a n t t o h. I n words, i. c o n f i r m s h and j confirms h b u t i . j d i s c o n h. —  the C o n j u n c t i o n C o n d i t i o n i s not v a l i d a c c o r d i n g t o Car-  nap. I t seems t h a t we c a n t r y t o save Hempel's C o n j u n c t i o n  Condi-  1. Carnap: L o g i c a l Foundations o f P r o b a b i l i t y , p p . 3 8 2 - 3 8 3 . Also Marsha Hanen: " C o n f i r m a t i o n , e x p l a n a t i o n and acceptance",p.126. In f a c t the v e r s i o n t h a t we have here i s adopted from Hanen's, g i v e n i n the a r t i c l e . Note t h a t " c ( h , e ) " i n the counter-example means "the degree of c o n f i r m a t i o n c o f the h y p o t h e s i s h, g i v e n evidence e".  156  t i o n f o r the same reason t h a t we have t r i e d to save the Consequence C o n d i t i o n , i . e . , Carnap's concept of q u a n t i t a t i v e c o n f i r m a t i o n i n h i s theory of c o n f i r m a t i o n i s q u i t e d i f f e r e n t from p e l ' s concept of q u a l i t a t i v e c o n f i r m a t i o n .  Hempel's concept of  " q u a l i t a t i v e c o n f i r m a t i o n " i s the r e l a t i o n between an t i o n r e p o r t and a h y p o t h e s i s ,  i s the r e l a t i o n of t o t a l evidence  So,  Obviously  observa-  while Carnap's " q u a n t i t a t i v e con-  f i r m a t i o n " employing "degree of c o n f i r m a t i o n " as  hypothesis.  "probability"  plus p r i o r p r o b a b i l i t y with a  they are two  quite different  relations.  i t c o u l d t u r n out to be the case t h a t i n Carnap's theory  p r o b a b i l i t y as q u a n t i t a t i v e c o n f i r m a t i o n the C o n j u n c t i o n tion  Hem-  Condi-  (as w e l l as the Consequence C o n d i t i o n ) does not h o l d ,  i n Hempel's theory of c o n f i r m a t i o n the C o n j u n c t i o n w e l l as the Consequence C o n d i t i o n ) We  of  while  Condition  (as  holds.  w i l l come back to the problem of the v a l i d i t y of the Con-  j u n c t i o n C o n d i t i o n (as w e l l as t h a t of the Consequence  Condition)  l a t e r i n s e c t i o n 11, where we w i l l have more d e c i s i v e t h i n g to say.  15?  6. GOODMAN'S PARADOX AND THE CONJUNCTION CONDITION Besides Carnap's  "counter-example"  to r e j e c t the c o n j u n c t i o n  c o n d i t i o n , Goodman a l s o o f f e r s another reason t o show the i n c r e d i b i l i t y of the c o n j u n c t i o n c o n d i t i o n . Goodman's counter-example, which i s adapted from Goodman's t o Hempel*s o r n i t h o l o g i c a l paradigm here, i s t h i s one.  C o n s i d e r the f o l l o w i n g two hypotheses!  HI:  A l l ravens are black?  H2:  A l l ravens are b l i t e , where " b l i t e "  i s d e f i n e d as f o l l o w s :  An o b j e c t i s s a i d to be b l i t e  i f i t has been ex-  amined b e f o r e time t ^ (which i s any f i x e d  future  time, say, the year A.D. 2000) and i s b l a c k or has not been examined b e f o r e time t ^ and i s white. A c c o r d i n g to Hempel's theory of c o n f i r m a t i o n both HI and H2 are  w e l l confirmed, i . e . , they are both confirmed by each and  every a c t u a l o b s e r v a t i o n r e p o r t a v a i l a b l e to us i n the w o r l d . Now Goodman asks: What should we p r e d i c t from the two w e l l confirmed hypotheses (1)  HI and H2?  Should we s a y :  Any raven examined a f t e r t ^ i s b l a c k ?  Or should we say* (2)  Any raven examined a f t e r t ^ i s b l i t e ? I f the p r e d i c t i o n i s the a f f i r m a t i o n o f (1), then we w i l l have:  (3) 1.  Any raven examined a f t e r  i s black.  C f . Goodman: F a c t . F i c t i o n , and F o r e c a s t . p p . 7 3 - 7 ; Hempel: Aspects of S c i e n t i f i c E x p l a n a t i o n , p . 5 ° . k  and a l s o  158 On the o t h e r hand i f the p r e d i c t i o n i s the a f f i r m a t i o n of ( 2 ) , then we w i l l ( )  have:  Any raven examined a f t e r ^  k  f o r a f t e r time ± (3)  1  i s white,  any b l i t e raven i s , by d e f i n i t i o n , white.  and ( ) are two incompatible p r e d i c t i o n s . k  s i m i l a r examples) w i l l be c a l l e d  This  "Goodman's paradox"  But  (and other or "the Good-  2 man  paradox". One immediate consequence,  some people^ w i l l  say, i s t h a t the  c o n j u n c t i o n c o n d i t i o n should be r e j e c t e d , otherwise we would have two  i n c o m p a t i b l e p r e d i c t i o n s from the above example of Goodman's  paradox. To t h i s , t h e r e are a few d i f f e r e n t  responses:  ( i ) One response i s t o agree t o r e j e c t the c o n j u n c t i o n c o n d i tion. ( i i ) Another d i f f e r e n t response i s t o s o l v e Goodman's Were Goodman's paradox be r e t a i n e d .  paradox.  s o l v a b l e , the c o n j u n c t i o n c o n d i t i o n c o u l d  We are going to review some proposed  s o l u t i o n s i n the  next few s e c t i o n s . ( i i i ) One i n t e r e s t i n g response not r e a l l y connected with our present d i s c u s s i o n o f the c o n j u n c t i o n c o n d i t i o n , by the way, i s Hempel's response.  F a c i n g Goodman's paradox Hempel has t h i s to say:  ... c o n f i r m a t i o n whether i n i t s q u a l i t a t i v e or i n i t s q u a n t i t a t i v e form cannot be adequately d e f i n e d by s y n t a c t i c a l means a l o n e . That has been made c l e a r e s p e c i a l l y by Goodman, who has shown t h a t some hypothe2. C f . , s a y , Skyrms: Choice & Chance, p . 6 l f f . 3. C f . Marsha Hanen: An Examination of Adequacy C o n d i t i o n s f o r Conf i r m a t i o n . Ph.D. d i s s e r t a t i o n , B r a n d e i s Univ., 19?0. . Hempel: A s p e c t s of S c i e n t i f i c E x p l a n a t i o n , p.50.  k  159  ses of the form * ( x ) ( P x o Q x ) ' can o b t a i n no c o n f i r m a t i o n a t a l l even from evidence sentence of the form • Pa.Qa'. Thus Hempel admits t h a t h i s s y n t a c t i c a l  approach to c o n f i r -  mation t h e o r y i s a f a i l u r e i n the l i g h t o f Goodman's paradox.  160  7. CARNAP'S SOLUTION OF THE GOODMAN PARADOX Carnap's tible  s o l u t i o n of the Goodman paradox i s t h a t any p r o j e c -  p r e d i c a t e must be n o n - t e m p o r a l T h u s  "blite"  i s not pro-  j e c t i b l e , f o r i t o b v i o u s l y i n v o l v e s some time element i n i t s definition. Unfortunately, out, can be d e f i n e d  both "black" and "white", as Goodman p o i n t e d i n terms of " b l i t e " and "whack" (which i s t o  be d e f i n e d below) and, hence, both of them a l s o i n v o l v e time e l e ment as can be shown by the f o l l o w i n g d e f i n i t i o n s : Df 1. before  An o b j e c t i s s a i d to be b l a c k  time t ^ (which i s any f i x e d f u t u r e time, say, the year  2000) and i s b l i t e Df 2. before  i f i t has been examined A.D.  or has not been so examined and i s whack.  An o b j e c t i s s a i d to be white i f i t has been examined  time t ^ and i s whack or not so examined and i s b l i t e .  where " b l i t e " and "whack" are understood as f o l l o w s : Df 3. before  i f i t has been examined  time t ^ and i s b l a c k or i s not so examined and i s white.  Df 4. before  An o b j e c t i s s a i d to be b l i t e  An o b j e c t i s s a i d to be whack i f i t has been examined  time t ^ and i s white or not so examined and i s b l a c k .  Thus from the view p o i n t of " b l i t e " and "whack" the p r e d i c a t e s " b l a c k " and "white" a l s o i n v o l v e time element i n a symmetric Hence, Carnap's p r o p o s a l  way.  of the s o l u t i o n of the Goodman p a r a -  dox does not work. 1. Carnap: "On the a p p l i c a t i o n of i n d u c t i v e l o g i c " , P h i l o s o p h y and Phenomenological Research, 19^7, pp.133-147.  161  8.  SALMON'S SOLUTION OF THE  Wesley Salmon's proposed this:  Any  GOODMAN PARADOX  s o l u t i o n of the Goodman paraodx i s  p r o j e c t i b l e predicate  Salmon t h i n k s t h a t " b l i t e " and f i n a b l e w h i l e " b l a c k " and  must be  "whack" are not  "white" a r e .  "whack" are not p r o j e c t i b l e , but the Goodman paradox i s  ostensively  ostensively  Accordingly  "black"  and  definable.  1  de-  "blite"  and  "white" a r e .  Hence,  s o l u t i o n f o r two  main  solved.  2  Some people  reject  t h i s as an ad hoc  reasons: ( i ) The who  are  i d e a of " o s t e n s i v e  d e f i n i t i o n " i s ambiguous.  To  us  "black"-and-"white" language speakers,a " b l a c k " ( o r "white")  o b j e c t can have the a "black"  "ostensive  d e f i n i t i o n " simply by p o i n t i n g  (or, r e s p e c t i v e l y , "white") t h i n g .  To us who  are  accus-  tomed to such a "black"-and-"white" language, i t seems t h a t i s no way  f o r " b l i t e " and  "whack" to have such an  to  there  "ostensive  de-  finition" . But  "blite"-and-"whack" language speakers c o u l d  f o l l o w i n g ways " b l i t e "  (or "whack") c o u l d  d e f i n i t i o n " i n the f o l l o w i n g ways F i r s t and  say t h a t an o b j e c t  examined, say,  b e f o r e the year A.D.  to a white t h i n g and  a l s o have i t s  i f i t looks l i k e 2 0 0 0 ; and,  " b l i t e " object  i n the  "ostensive  they p o i n t to a blaek. t h i n g  say t h a t a " b l i t e " o b j e c t  t h a t white t h i n g i f the the year A.D.  is "blite"  object  that thing  and  then, they p o i n t should  i s examined  look  like  i n or a f t e r  2000.  Thus "blite"-and-"whack" language speakers c o u l d a l s o have "ostensive 1. 2.  d e f i n i t i o n s " f o r the p r e d i c a t e s  "blite","whack", etc.  Salmons "On v i n d i c a t i n g i n d u c t i o n " , P h i l o s o p h y of Science, 1963. Cf. Hanens An Examination of Adequacy C o n d i t i o n s f o r Confirmat i o n , Ph.D. d i s s e r t a t i o n , 1970, p p . 9 6 - 1 1 0 .  162  ( i i ) " B l i t e " and "whack" a r e a l s o " o s t e n s i v e l y d e f i n a b l e " i n another sense so l o n g as we a l l o w a certain a r t i f i c i a l  observation  t a i n d i s c r e t e time element. observation on a b l a c k and  d i s c r e t e elements o r we  allow  instrument which i n v o l v e s a c e r -  So, f o r i n s t a n c e ,  a "blite"  colour  instrument w i l l be a c e r t a i n meter whose p o i n t e r r e s t s t h i n g when an o b j e c t ha& been examined b e f o r e time t ^  whose p o i n t e r r e s t s on a white t h i n g when the o b j e c t has not  been so examined b e f o r e time t ^ .  3  Thus i t appears t h a t Salmon's proposed s o l u t i o n may s u f f e r from a c e r t a i n ambiguity i n " o s t e n s i v e  3.  definition".  Goodman: " P o s i t i o n a l i t y and p i c t u r e " , P h i l o s o p h i c a l Review, I960, p p . 5 2 3 - 5 2 5 .  163  9.  AN ATTEMPTED SOLUTION OF THE GOODMAN PARADOX  In t h i s s e c t i o n I w i l l attempt a semantical  (and s y n t a c t i c a l )  F i r s t , l e t us g e n e r a l i z e the two  s o l u t i o n of the Goodman paradox.  p r e d i c a t e s " b l i t e " and "whack" i n t o two f a m i l i e s of p r e d i c a t e s as follows: t  Df 5.  i  An o b j e c t i s b l i t e  i f i t has been examined  before  time t ^ and i s white or not so examined and i s b l a c k . t. Df 6.  An o b j e c t i s whack  i f i t has been examined  before  time t ^ and i s white or not so examined and i s b l a c k . Now l e t t ^ = t ^ , tg» ...» t , .... Then, we have the f o l l o w i n g two f a m i l i e s of i n f i n i t e l y many p r e d i c a t e s * n  blite whack  t,  , blite  t  2  ,  blite  t  ,  "^1 ^2 "^n , whack , ..., whack  Then, we ask: Can the p r e d i c a t e s " b l a c k " and "white" be d e f i n e d tt. i n terms o f any p a i r of the p r e d i c a t e s " b l i t e " and "whack "? And the answer i s : Yes, they can be so d e f i n e d by any p a i r of t t. "blite " and "whack ", where t- i s any f i x e d time. Thus the p r e i  h d i c a t e s "black" and "white" can be d e f i n e d by the p a i r b l i t e  and  whack , or by the p a i r b l i t e and whack , and so on and so f o r t h . Now l e t time t . be any f i x e d time l a t e r than now, and l e t us J  c o n s i d e r the f o l l o w i n g two hypotheses: HI:  A l l ravens are b l a c k . t  H2:  A l l ravens are b l i t e  i J  .  I t appears t o be obvious t h a t both hypotheses are w e l l confirmed.  164 But they have i n c o m p a t i b l e p r e d i c t i o n s : HI p r e d i c t s t h a t any examined l a t e r  raven  w i l l be b l a c k , w h i l e H2 p r e d i c t s t h a t t. than t . w i l l be b l i t e , v i z . , w h i t e .  than t .  raven examined l a t e r  any  J  J  T h i s i s the Goodman  paradox.  A q u e s t i o n i s r a i s e d now: To answer t h i s f a m i l y of H2.1:  How  q u e s t i o n , l e t us  first  c o n s i d e r the  H2?  following  hypotheses: A l l ravens are b l i t e  , 2  t  H2.2:  t o make c h o i c e between HI and  A l l ravens are b l i t e  ,  • •t  H2.n-1:  A l l ravens are b l i t e  ,  t H2.n:  A l l ravens are b l i t e where t < t 1  time now)  and  " <"  Then, l e t us  2  ,  <... < t _ ^ < t n  i s "the e a r l i e r assume t h a t we  to be black. Then, we ( i ) Hypotheses  n  n  (which i s supposed  than"  to be the  relation.  j u s t examined a raven and found i t  have:  H 2 . 1 , H 2 . 2 , . . . , H 2 . n - l , H2.n are  a l l disconfirmed?  ( i i ) Both HI and H2 are conf irmed (with r e s p e c t t o the l a s t examined b l a c k raven.) In the above sense of ( i ) and H2 "not a h i s t o r i c a l l y "historically p a i r of b l i t e  ( i i ) we w i l l c a l l  the h y p o t h e s i s  w e l l - e s t a b l i s h e d hypothesis", while  w e l l - e s t a b l i s h e d h y p o t h e s i s " , f o r no matter t. t. and whack  H is a  which  i s employed t o d e f i n e " b l a c k " HI i s  always w e l l confirmed. Thus i t appears t h a t HI says -more t h a n £ 2 a l t h o u g h both of them are  well confirmed.  165 So, we propose the f o l l o w i n g t h e s i s i Preference Thesis.  Between two hypotheses  i f one of them  i s h i s t o r i c a l l y w e l l - e s t a b l i s h e d while the other i s not, we should p r e f e r the f i r s t h y p o t h e s i s t o the second  i n c o n f i r m a t i o n theory  f o r the simple reason t h a t the l a t t e r says l e s s h i s t o r i c a l l y the  than  former. Once we have the above T h e s i s , we should p r e f e r HI t o H2, and  the Goodman paradox w i l l have a s e m a n t i c a l ( s y n t a c t i c a l ) s o l u t i o n becuase the above T h e s i s can be d e s c r i b e d i n terms of  semantics  (and s y n t a x ) . Thus we have a s e m a n t i c a l (and s y n t a c t i c a l ) s o l u t i o n of the Goodman paradox.  166 10. THE GOODMAN PARADOX REGAINED By a symmetric argument to the p r e v i o u s of the Goodman paradox we f i n d regained  attempted  solution  t h a t the Goodman paradox can be  as f o l l o w s i  First,  g e n e r a l i z e the two p r e d i c a t e s  "black" and "white" i n t o  two f a m i l i e s o f p r e d i c a t e s as f o l l o w s ! t  Df 7.  An o b j e c t i s b l a c k  time t ^ and i s b l i t e and  i i f i t has been examined  or not so examined and i s whackt (where " b l i t e "  "whack" are understood, to repeat, Df 8.  before  An o b j e c t i s b l i t e  as f o l l o w s :  i f i t has been examined  before  the year A.D. 2000 and i s black o r not so examined and i s white; Df 9.  An o b j e c t i s whack i f i t has been examined  before  the year A.D. 2000 and i s white or i s not so examined and i s b l a c k . ) . * i Df 10.  An o b j e c t i s white  i f i t has been examined  before  time t ^ and i s whack or not so examined and i s b l i t e . Now l e t t ^ = t ^ , tg t , .... Then, we have the f o l l o w i n g two f a m i l i e s of i n f i n i t e l y many p r e d i c a t e s : black  t,  t  white  , black  l  t  , white  t  2  2  t  ,  black  , ...;  ,  *n white , ....  Then, we ask: Can the two p r e d i c a t e s  " b l i t e " and "whack" be t. t.  d e f i n e d i n terms o f any p a i r o f the p r e d i c a t e s "black  " and "white  "?  And the answer i s : Yes, they can be d e f i n e d i n the f o l l o w i n g waysi Df 11.  An o b j e c t i s b l i t e t  t. and i s b l a c k  i  i f i t has been examined before  * i or i s not so examined and i s white  time  167  Df 1 2 .  An o b j e c t i s whack i f i t has been examined b e f o r e t.  time t ^ and i s white Now  1  t. or i s not so examined and i s b l a c k . 1  l e t time t . be any f i x e d time l a t e r than t h e y e a r A. D. 2000  and l e t us c o n s i d e r the f o l l o w i n g two  hypotheses:  t. H3«  A l l ravens are b l a c k  H4:  A l l ravens are b l i t e .  J  .  I t appears to be obvious t h a t both hypotheses are w e l l conf i r m e d . But they have i n c o m p a t i b l e p r e d i c t i o n s : H3 p r e d i c t s t h a t 2000 w i l l be b l a c k , w h i l e H4  any raven examined l a t e r than A.D.  p r e d i c t s t h a t any raven examined l a t e r than A.D. 2000 w i l l be w h i t e . A c c o r d i n g t o the attempted s e c t i o n , we  s o l u t i o n d e s c r i b e d i n the p r e v i o u s  should form the f o l l o w i n g f a m i l y of hypotheses:  H3.1:  A l l ravens are b l a c k  H3.2:  A l l ravens are b l a c k  . *2  t  H3.n-1:  A l l ravens are b l a c k  H3.n:  A l l ravens are b l a c k  n-l  t  where t < t < . . . 1  be the time now)  2  and " <"  Then, we  ( i ) Hypotheses firmed;  ' _i< * t  n  n  (which i s supposed  i s "the e a r l i e r than"  Then, we assume t h a t we to be b l a c k .  , to  relation.  j u s t examined a raven and found i t  have:  H3.1,  H3.2,  H3.n-1, H3.n  are a l l d i s c o n -  168  ( i i ) Both H3 and H  k  are confirmed  (with r e s p e c t to the  last  examined raven.) In the above sense of ( i ) and well-established  ( i i ) H3  i s not  "historically  h y p o t h e s i s " , while H4 i s .  So, by the Preference T h e s i s g i v e n i n the p r e v i o u s we  should p r e f e r H  k  to H3.  And  section  t h i s c o n c l u s i o n i s e x a c t l y oppo-  s i t e to the p r e f e r e n c e we made i n the p r e v i o u s s e c t i o n by same evidence  of a j u s t examined b l a c k raven.  s o l u t i o n of the l a s t have a c o n t r a d i c t i o n .  the  So, the attempted  s e c t i o n cannot be r i g h t , otherwise we  would  169  11.  A TENTATIVE CONCLUSION OF THE VALIDITY OF THE ADEQUACY  CONDITIONS FOR CONFIRMATION IN LIGHT OF MARSHA HANEN'S STUDY Hanen has a v e r y thorough study of the adequacy c o n d i t i o n s for confirmation.  1  Her c o n c l u s i o n is«  2  I t thus seems t h a t none of the most obvious appearing adequacy c o n d i t i o n s f o r c o n f i r m a t i o n i s v a l i d f o r the i n c r e a s e of firmness n o t i o n of c o n f i r m a t i o n with a possible  exception  3  an e x c e p t i o n may be the E q u i v a l e n c e i s v a l i d f o r Carnap's explicatum.  C o n d i t i o n , which  F i r s t note t h a t "the most obvious appearing  adequacy c o n d i -  t i o n s f o r c o n f i r m a t i o n " mentioned i n the above quote i n c l u d e a l l adequacy c o n d i t i o n s l a i d down by Hempel g i v e n i n s e c t i o n one of t h i s P a r t , and t h a t "Carnap*s explicatum",  j u s t mentioned,is ex-  a c t l y *the i n c r e a s e o f firmness n o t i o n o f c o n f i r m a t i o n " to i n the p r e c e d i n g  quote.  I agree w i t h Hanen's c o n c l u s i o n w i t h one p o s s i b l e and  the e x c e p t i o n  referred  i s the E n t a i l m e n t  exception,  C o n d i t i o n which says t h a t any  sentence which i s e n t a i l e d by an o b s e r v a t i o n r e p o r t i s confirmed by i t .  I n f a c t , my view about the E n t a i l m e n t  C o n d i t i o n does not  d i f f e r too much from what Hanen says about i t provided possible confusions  t h a t two  are c l a r i f i e d , f o r Hanen sayst  ... c o n s i d e r Hempel's E n t a i l m e n t  Condition.... I t i s  1. C f . Haneni An Examination of Adequacy C o n d i t i o n s f o r c o n f i r mation, Ph.D. d i s s e r t a t i o n , Brandeis Univ., 1 9 7 0 , a l s o "Conf i r m a t i o n and adequacy c o n d i t i o n s " , Philosophy of Science. 1971, pp.361-368, and " C o n f i r m a t i o n , e x p l a n a t i o n and acceptance" i n A n a l y s i s and Metaphysics, ed. by K e i t h Lehrer, 1975 pp.93-128. 2. Haneni " C o n f i r m a t i o n , e x p l a n a t i o n and acceptance", p.102. 3. 4.  I b i d , p.126, f o o t n o t e 2 9 . I b i d , p.102.  170 u s u a l to regard t h i s c o n d i t i o n as v a l i d f o r the f i r m n e s s concept of c o n f i r m a t i o n . But i f we say t h a t , f o r evidence to count as c o n f i r m i n g a g i v e n hypothesis there must be some as yet undetermined cases, then t h i s c o n d i t i o n seems to f a i l . I f p_ e n t a i l s q_, p_ can s c a r c e l y count as evidence f o r g_ , s i n c e , i f £ i s t r u e , £ w i l l be t r u e as w e l l , and there w i l l be n o t h i n g l e f t to determine. In t h i s s i t u a t i o n , g. f u n c t i o n s as a h y p o t h e s i s t h a t i s exhausted in the sense t h a t a l l of i t s i n s t a n c e s have, as i t were, a l ready been examined. I t l o o k s as though we can say t h a t p_ v e r i f i e s i n which case i t does not merely c o n f i r m i t .  Since i n Hempel's theory of c o n f i r m a t i o n , as w e l l as i n our i n t e r n a l c o n f i r m a t i o n theory, the Goodman condition-* on a u n i v e r s a l c o n d i t i o n a l hypothesis  t h a t i s s a i d to be  "confirmed"  i s not  imposed, Hanen's p o i n t about "undetermined cases" mentioned i n the above quote does not a r i s e .  T h i s i s the f i r s t p o s s i b l e  con-  c o n f u s i o n t h a t we need to c l a r i f y . Hanen a l s o s a y s i ^ But i f we take an i n c r e a s e of firmness n o t i o n of conf i r m a t i o n , the s i t u a t i o n i s e n t i r e l y d i f f e r e n t . Here the entailment c o n d i t i o n holds o n l y i n a r e s t r i c t e d form. The h y p o t h e s i s must not be l o g i c a l l y true f o r then, even though i t i s e n t a i l e d by any o b s e r v a t i o n statement, i t i s i n c a p a b l e of r e c e i v i n g support i n the i n c r e a s e of firmness sense, i t s degree of c o n f i r m a t i o n a l r e a d y b e i n g 1. Similarly, i f the degree of c o n f i r m a t i o n of the h y p o t h e s i s (say, (3x)Fx) i s a l r e a d y 1 on the b a s i s of p r i o r evidence (say, F a ) , then new evidence (say, Fb) cannot i n c r e a s e t h a t degree of conf i r m a t i o n and so cannot be s a i d to be conf i r m i n g i n the i n crease of firmness sense. Note t h a t we «  t h a t may 5.  have a l r e a d y r e a l i z e d the above awkward s i t u a t i o n  a r i s e i n Carnap's " i n c r e a s e i n f i r m n e s s " n o t i o n of c o n f i r -  The Goodman c o n d i t i o n of a u n i v e r s a l c o n d i t i o n a l h y p o t h e s i s to be confirmed i s t h a t ( i ) i t be "supported" ( i . e . "there be some p o s i t i v e i n s t a n c e s " ) , ( i i ) " u n v i o l a t e d " ( i . e . "there be no neg a t i v e i n s t a n c e s " ) and ( i i i ) "unexhausted" ( i . e . "there be some as y e t undetermined i n s t a n c e s " ) . C f . Goodmani F a c t , F i c t i o n , and F o r e c a s t . Chapter IV. 6. Hanem "Confirmation, e x p l a n a t i o n and acceptance", p.102.  171 mation i n s e c t i o n f o u r of t h i s P a r t , and there we  have  suggested  a small m o d i f i c a t i o n of Carnap's explicatum from " i n c r e a s e i n firmness of a h y p o t h e s i s " to " i n c r e a s e i n firmness of a hypothes i s up to p r o b a b i l i t y one"  i n order to a v o i d the above awkward  s i t u a t i o n and to save Carnap's e x p l i c a t u m .  The m o d i f i c a t i o n of  Carnap*s explicatum a l s o avoids Hanen's c r i t i c i s m above.  This  i s the second c l a r i f i c a t i o n of another p o s s i b l e c o n f u s i o n t h a t may  a r i s e from the remark t h a t my view about the E n t a i l m e n t Con-  d i t i o n does not d i f f e r too much from what Hanen's; although  I  t r y to save the Entailment C o n d i t i o n while Hanen t r i e s to r e j e c t it.  In f a c t , Hanen has t h i s to say too:' On the other hand, no p a r t i c u l a r harm seems to r e s u l t from viewing entailment as a degenerate case of confirmation. Thus, the disagreement about the v a l i d i t y of the  Entailment  C o n d i t i o n between Hanen and myself c o u l d be only apparent,  for  she uses Carnap's explicatum of " i n c r e a s e i n f i r m n e s s " of conf i r m a t i o n while I t r y to save both Carnap's explicatum and  the  Entailment C o n d i t i o n by m o d i f y i n g Carnap's explicatum i n order to avoid the awkward s i t u a t i o n p o i n t e d out by Hanen and a l s o d i s cussed by us i n s e c t i o n f o u r of t h i s P a r t . In the f o l l o w i n g I w i l l t r y to g i v e reasons why  we  reject  the  C o n j u n c t i o n C o n d i t i o n , the Consequence C o n d i t i o n , the S p e c i a l Consequence C o n d i t i o n , and the C o n s i s t e n c y C o n d i t i o n . (I) The r e j e c t i o n of the C o n j u n c t i o n C o n d i t i o n .  In s e c t i o n  s i x of t h i s P a r t we have a l r e a d y d e s c r i b e d a "counter-example" made by Carnap to r e j e c t the C o n j u n c t i o n C o n d i t i o n .  However, we  tried  to defend Hempel's C o n j u n c t i o n C o n d i t i o n by s a y i n g t h a t Carnap's 7. I b i d , p.102.  172  counter-example i s g i v e n i n terms of " d e g r e e of conf i r m a t i o n " which i s a concept of " q u a n t i t a t i v e conf irmation", while Hempel's Conjunct i o n C o n d i t i o n may confirmation".  he only an adequacy c o n d i t i o n f o r " q u a l i t a t i v e  Since the q u a l i t a t i v e c o n c e p t i o n of c o n f i r m a t i o n  and the q u a n t i t a t i v e c o n c e p t i o n of c o n f i r m a t i o n are q u i t e d i f f e r ent, i t may  t u r n out t h a t the C o n j u n c t i o n C o n d i t i o n h o l d s i n  Hem-  p e l ' s theory of c o n f i r m a t i o n while i t f a i l s to h o l d i n Carnap's t h e o r y of q u a n t i t a t i v e c o n f i r m a t i o n . Once we  have Goodman's paradox, the C o n j u n c t i o n C o n d i t i o n i s  not d e f e n s i b l e i f the p r e d i c a t e s " b l i t e " ,  "whack", e t c . , are i n -  g  troduced  i n t o Hempel's t h e o r y .  To see t h i s , l e t us repeat  the  argument g i v e n i n s e c t i o n s i x of t h i s P a r t . First,  c o n s i d e r the two  hypotheses:  HI:  A l l ravens are b l a c k .  H2:  A l l ravens are  blite  Then, suppose t h a t we B:  r i s a black So,  2 0 0 0  .  have the f o l l o w i n g o b s e r v a t i o n r e p o r t :  raven.  i n Hempel*s theory of c o n f i r m a t i o n we  (1)  B confirms  HI:  (2)  B confirms  H2.  Hence, by the C o n j u n c t i o n C o n d i t i o n we (3) 8.  B confirms  have:  have:  H1&H2.  The i n t r o d u c t i o n of these new p r e d i c a t e s , by the way, f o r c e s Hempel to say: I b e l i e v e Carnap i s r i g h t i n h i s estimate t h a t the concept of c o n f i r m a t i o n d e f i n e d i n my e s s a y " . . . i s c l e a r l y too narrow." See Aspects of S c i e n t i f c E x p l a n a t i o n , p.50.  173 But  H1&H2  i s a hypothesis w i t h a c o n t r a d i c t o r y p r e d i c t i o n  which says t h a t any raven w i l l be both b l a c k and white a f t e r the year A.D. 2000.  Since we do not want any adequate theory of con-  f i r m a t i o n t o have any c o n t r a d i c t o r y p r e d i c t i o n , the C o n j u n c t i o n C o n d i t i o n must be r e j e c t e d . ( I I ) The r e j e c t i o n o f the Consequence C o n d i t i o n .  Carnap has  a counter-example t o r e j e c t the Consequence C o n d i t i o n as has been d e s c r i b e d i n s e c t i o n three of t h i s P a r t .  We t r i e d t o defend Hem-  p e l ' s Consequence C o n d i t i o n by s a y i n g t h a t i t would h o l d i n Hempel's theory of q u a l i t a t i v e c o n f i r m a t i o n , while i t f a i l s t o h o l d i n Carnap's q u a n t i t a t i v e c o n f i r m a t i o n t h e o r y .  Once we have the Goodman  paradox, the Consequence C o n d i t i o n i s not d e f e n s i b l e , f o r c o n s i der the two hypotheses HI and H2 and the o b s e r v a t i o n r e p o r t B desc r i b e d above.  We have the f o l l o w i n g r e s u l t d i r e c t l y from the de-  f i n i t i o n o f c o n f i r m a t i o n (without employing the C o n j u n c t i o n Condition) : (4)  B confirms H1&H2. Since HI & H2 has some c o n t r a d i c t o r y p r e d i c t i o n a f t e r the year  A.D. 2000, we always have the f o l l o w i n g r e s u l t i (5)  ( H l & H 2 ) o P i , where P i i s any p r e d i c t i o n o f HI & H2 a f t e r the year A.D. 2000. So, by the Consequence C o n d i t i o n we havei  (6)  B confirms P i . Since the r e s u l t of (6) i s n o t a c c e p t a b l e i n any adequate the-  ory of c o n f i r m a t i o n , we have t o r e j e c t the Consequence C o n d i t i o n (and, f o r the same reason,  the S p e c i a l Consequence C o n d i t i o n . )  174  ( I I I ) The r e j e c t i o n o f the C o n s i s t e n c y C o n d i t i o n .  Consis-  tency C o n d i t i o n s a y s i Every l o g i c a l l y c o n s i s t e n t o b s e r v a t i o n r e p o r t i s l o g i c a l l y compatible w i t h the c l a s s o f a l l the hypotheses which i t c o n f i r m s .  I n the above example o f r e j e c t i n g the Conse-  quence C o n d i t i o n once we have the r e s u l t o f (4), the C o n s i s t e n c y C o n d i t i o n i s a t the same time r e j e c t e d f o r the reasons  ( i ) the  o b s e r v a t i o n r e p o r t B i s l o g i c a l l y c o n s i s t e n t , and ( i i ) HI and H2 are l o g i c a l l y i n c o m p a t i b l e - - - s i n c e t h e i r p r e d i c t i o n s about the c o l o r s o f ravens a f t e r the year A.D. 2000 are incompatible and, hence, ( i i i ) B and HI &H2 a r e l o g i c a l l y incompatible  at  l e a s t B, HI and H2 a l t o g e t h e r are incompatible about the c o l o r of ravens a f t e r the year A.D. 2000.  Hence, the C o n s i s t e n c y Con-  d i t i o n must be r e j e c t e d . Thus we have r e j e c t e d Hempel*s C o n j u n c t i o n C o n d i t i o n , Consequence C o n d i t i o n ( i n c l u d i n g the S p e c i a l Consequence C o n d i t i o n ) and the C o n s i s t e n c y C o n d i t i o n as we c l a i m e d .  I t seems t h a t i n  the end Hempel h i m s e l f has come t o r e a l i z e t h e i r as w e l l , f o r he says i n h i s l a t e s t P o s t s c r i p t mation:  implausibility  (1964) on C o n f i r -  Q 7  One and the same observable phenomenon may w e l l be accounted f o r by each o f two incompatible hypotheses, and the o b s e r v a t i o n r e p o r t d e s c r i b i n g i t s occurrence would then normally be regarded as c o n f i r m a t o r y f o r e i t h e r h y p o t h e s i s . T h i s p o i n t does seem t o me t o c a r r y c o n s i d e r a b l e weighttbut i f i t i s granted, then the consequence c o n d i t i o n has t o be g i v e n up a l o n g w i t h the cons i s t e n c y c o n d i t i o n . Otherwise, a r e p o r t c o n f i r m i n g each of two incompatible hypotheses would count as c o n f i r m i n g any consequence of the two, and thus any h y p o t h e s i s whatever, (my i t a l i c s ) And,  hence, Hempel concludes i n h i s P o s t s c r i p t  (1964) on C o n f i r -  9. See Hempel: Aspects o f S c i e n t i f i c E x p l a n a t i o n , p.50.  175  mation: I b e l i e v e Carnap i s r i g h t i n h i s estimate t h a t the concept of c o n f i r m a t i o n d e f i n e d i n my essay " i s not c l e a r l y too wide but i s c l e a r l y too narrow." A c c o r d i n g l y , I t h i n k t h a t the c r i t e r i a s p e c i f i e d i n my d e f i n i t i o n may be s u f f i c i e n t , but are not necessary f o r the c o n f i r m a t i o n of hyp o t h e s i s H by an o b s e r v a t i o n r e p o r t B. (my i t a l i c s ) Thus we  can agree with Hempel t h a t those  l a i d down by him sufficient. Condition  are not r e a l l y necessary,  In f a c t , the C o n j u n c t i o n  adequacy c o n d i t i o n s  although  they may  C o n d i t i o n , the Consequence  ( i n c l u d i n g the S p e c i a l Consequence C o n d i t i o n ) and  Consistency  be  C o n d i t i o n are a l l unnecessary, i f what we  the  have argued  above i s r i g h t . (IV) The sympathetic  problem of the E q u i v a l e n c e w i t h the E q u i v a l e n c e  Goodman's and  Hanen i s v e r y  C o n d i t i o n , although  she knows t h a t  S c h e f f l e r ' s theory of s e l e c t i v e c o n f i r m a t i o n does not  observe the E q u i v a l e n c e  C o n d i t i o n and,  t i o n must be r e j e c t e d i n Goodman's and tive  Condition.  hence, the E q u i v a l e n c e  Condi  S c h e f f l e r ' s theory of s e l e c -  confirmation. In her Ph.D.  d i s s e r t a t i o n she  c o n c l u s i o n : On the primary  seems to come to the f o l l o w i n g  l e v e l of c o n f i r m a t i o n i n the sense of  Hempel's S a t i s f a c t i o n C r i t e r i o n of C o n f i r m a t i o n we valence  need the E q u i -  C o n d i t i o n ; but on the secondary l e v e l of conf i r m a t i o n which  i s d e r i v e d from the primary Equivalence  Condition.  c o n f i r m a t i o n , we  have to r e l i n q u i s h the  1 1  12  In her l a t e r a r t i c l e  she t r i e s to r e l i n q u i s h the  Equivalence  C o n d i t i o n as an adequacy c o n d i t i o n f o r c o n f i r m a t i o n on the one hand, 10. I b i d . . p.50. 1 1 . Hanen: An Examination of the Adequacy C o n d i t i o n s f o r Confirmat i o n , pp.221-222. 1 2 . Hanen, op. c i t . , p . H 8 f f .  176 but r e t a i n i t as a good c o n d i t i o n f o r the theory of acceptance  on  the o t h e r . Thus, to sum  up,  a c c o r d i n g to Hanen we  do not need any of the  adequacy c o n d i t i o n s laldjiown by Hempel f o r conf irmation;, although need the E q u i v a l e n c e C o n d i t i o n i n t h e o r y of But why  acceptance.  should the E q u i v a l e n c e C o n d i t i o n be r e j e c t e d i n the the-  ory of c o n f i r m a t i o n while r e t a i n e d i n the theory of Hanen's reasons First,  we  acceptance?  are:  i n any adequate theory of acceptance  we  need the f o l l o w -  i n g r u l e or c o n d i t i o n : E q u i v a l e n c e Rule  ( f o r acceptance).  two l o g i c a l l y e q u i v a l e n t hypotheses. is  Suppose t h a t KI and K2  are  Then, KI i s accepted i f f K2  accepted. T h i s i s c l e a r and,  I suppose, nobody would argue a g a i n s t i t .  Second, "questions about c o n f i r m a t i o n are d i f f e r e n t from quest i o n s about acceptance".  13 Or more f u l l y i n Hanen's words:  I am s u g g e s t i n g ... t h a t c o n f i r m a t i o n and acceptance can be viewed as o c c u r r i n g on two l e v e l s .... The u s u a l c o n d i t i o n s of adequacy should not be r i g i d l y a p p l i e d to c o n f i r mation r e l a t i o n s . But they may be a p p l i e d once we ask which among confirmed hypotheses should be accepted ... . But one does not want to make too much of the i d e a of two l e v e l s , f o r I do not wish to c l a i m t h a t they are separate l o g i c a l l e v e l s or a n y t h i n g of the s o r t . Rather, the important c o n s i d e r a t i o n i s t h a t q u e s t i o n s about c o n f i r m a t i o n are d i f f e r e n t from q u e s t i o n s about acceptance .,~. (my i t a l i c s ) And,  i n p a r t i c u l a r , Hanen has t h i s to say about the E q u i v a l e n c e Con-  d i t i o n as an adequacy c o n d i t i o n f o r c o n f i r m a t i o n and as an adequacy c o n d i t i o n (or r u l e ) f o r  acceptance:  lk  Hempel argues on b e h a l f of the E q u i v a l e n c e C o n d i t i o n t h a t hypotheses f u n c t i o n i n d e d u c t i v e e x p l a n a t i o n s and p r e d i c t i o n s and i t would be v e r y odd to suppose t h a t a g i v e n hy13. Hanen, op. c i t . . 14. Hanen, op. c i t . ,  p.121. p.121.  177  p o t h e s i s c o u l d serve as a premise i n a deductive argument where i t s l o g i c a l e q u i v a l e n t c o u l d not. And indeed that would be odd. But there i s an e x t r a step needed the move from s a y i n g t h a t H i s confirmed to s a y i n g t h a t i t i s a c c e p t a b l e as a premise i n a p a r t i c u l a r deductive argument. I f i t is_ so a c c e p t a b l e , then c e r t a i n l y so i s any hypothesis l o g i c a l l y e q u i v a l e n t t o i t but i t s u r e l y does not f o l l o w t h a t i f i t i s confirmed, then so i s any e q u i v a l e n t hypothesis. Whether the l a t t e r c l a i m i s t r u e depends upon the p a r t i c u l a r theory of c o n f i r m a t i o n i n use and the d i f f i c u l t i e s i n t o which attempts t o adhere s t r i c t l y t o the u s u a l adequacy c o n d i t i o n s seem t o l e a d us i s good reason t o a v o i d imposing them a p r i o r i . Now t o sum up without  making any c r i t i c a l review o f what Hanen  says above, Hanen t h i n k s t h a t we do not need any o f the adequacy c o n d i t i o n s l a i d down by Hempel. One s e r i o u s q u e s t i o n i s immediately r a i s e d : " I f we no l o n g e r r e q u i r e o f a theory o f c o n f i r m a t i o n t h a t i t meets the c o n d i t i o n of adequacy, i s not everyone f r e e to propose h i s own t h e o r y ? " if  i t i s the case,  then we would be l e f t without  (For  any means o f choos-  i n g among them.) -' 1  To b r i n g out the p o i n t , c o n s i d e r the f o l l o w i n g new type o f selective  confirmation: ^ 1  A u n i v e r s a l c o n d i t i o n a l hypothesis ( x ) ( F x a G x ) i s s e l e c t i v e l y confirmed, i f the hypothesis i s confirmed and i t s "antecedent d e n i a l " , v i z . (x)(-Fx o G x ) , i s d i s c o n f i r m e d (where both "conf i r m e d " and " d i s c o n f i r m e d " a r e i n the Hempelean sense o f conf i r m a t i o n and d i s c o n f i r m a t i o n . ) Under t h i s new type o f s e l e c t i v e c o n f i r m a t i o n we havet a nonb l a c k non-raven s e l e c t i v e l y c o n f i r m s b l a c k raven  i s neutral to i t  the raven h y p o t h e s i s w h i l e a  a complete r e v e r s a l o f the s i t u a -  t i o n o f Goodman*s and S c h e f f I e r • s theory of s e l e c t i v e  confirmation.  So, which i d e a of s e l e c t i v e c o n f i r m a t i o n i s l e g i t i m a t e ? I f both a r e l e g i t i m a t e , then the paradoxes of c o n f i r m a t i o n 15. 16.  Stewart, J . P i "Comments on Marsha Hanen's "Confirmation, n a t i o n and acceptance"", p . l . Ibid.. p.3.  will expla-  178 o b v i o u s l y come back. and  However, Hanen t h i n k s t h a t o n l y Goodman's  S c h e f f l e r ' s i d e a of s e l e c t i v e c o n f i r m a t i o n i s the l e g i t i m a t e 17  one,  f o r she  says:  I f one i s n ' t wedded to the equivalence c o n d i t i o n , and i f one i s prepared to view hypotheses as s p e c i f i c i n s c r i p t i o n s or u t t e r a n c e s , then i t seems n a t u r a l to say t h a t , a necessary c o n d i t i o n f o r t r e a t i n g hypotheses as i n comp e t i t i o n i s t h a t t h e i r antecedent c l a s s e s be the same,... In the case of a hypothesis and i t s antecedent d e n i a l , t h i s condition f a i l s . (my i t a l i c s ) I am q u i t e sympathetic w i t h the necessary called  c o n d i t i o n , to be  "Hanen's c o n d i t i o n " , g i v e n i n the above quote, ( f o r t h a t  i s , i n a way, the l i n e of thought c a r r i e d out i n the i n t e r n a l 1o firmation But,  con-  theory.) then, a second q u e s t i o n a r i s e s : Is Hanen's c o n d i t i o n a  necessary  condition f o r confirmation?  I f i t i s a necessary  or f o r acceptance? or what?  c o n d i t i o n f o r c o n f i r m a t i o n , does Hanen con-  t r a d i c t h e r s e l f to r e j e c t any adequacy c o n d i t i o n s l a i d down by Hempel? Since Hanen says n o t h i n g about the above q u e s t i o n s , venture  the f o l l o w i n g answers, which are  I would  the most reasonable  ones,  t h a t I can t h i n k o f : ( i ) Hanen's c o n d i t i o n i s an adequacy c o n d i t i o n f o r comparative (not q u a l i t a t i v e ) c o n f i r m a t i o n , f o r as she  says: I t i s "a  c o n d i t i o n f o r t r e a t i n g hypotheses as i n c o m p e t i t i o n " ( m y  necessary italics)  ( i i ) Since Hanen's c o n d i t i o n i s not among Hempel's adequacy c o n d i t i o n s f o r q u a l i t a t i v e c o n f i r m a t i o n , she i s not i n c o n s i s t e n t i n rejecting  a l l the adequacy c o n d i t i o n s l a i d down by Hempel.  17. Hanen, op. c i t . , p.104. 18. The concept of " c o n f i r m a t i o n " of the i n t e r n a l c o n f i r m a t i o n theory i s a u t o m a t i c a l l y a " s e l e c t i v e c o n f i r m a t i o n " i n the sense of Goodman and S c h e f f l e r . (Cf., p.144) Hence, i t s a t i s f i e s Hanen's condition automatically. 19. See Note 17.  179 I t h i n k t h a t Hanen's r e j e c t i o n of a l l adequacy c o n d i t i o n l a i d down by Hempel i s a l s o a p l a u s i b l e p o s i t i o n with two g r e a t m e r i t s * 1) I t f r e e s us from imposing any "adequacy" c o n d i t i o n a p r i o r i in  c o n s t r u c t i n g a theory of q u a l i t a t i v e 2) That, o f course,  is  confirmation.  does n o t mean any theory o f c o n f i r m a t i o n  as good as any o t h e r .  Still,  Hanen's p o s i t i o n has the g r e a t  m e r i t o f s e p a r a t i n g the problem o f the c o n s t r u c t i o n of a c o n f i r m a t i o n theory and i t s a p p l i c a b i l i t y .  Hence, an odd o r absurd  theory  of c o n f i r m a t i o n w i l l be r e j e c t e d when i t s a b s u r d i t y i s found l a t e r in its  applications.  Thus we should not impose a p r i o r i any "adequacy"  c o n d i t i o n t o any c o n f i r m a t i o n theory, tion period.  e s p e c i a l l y i n i t s construc-  T h i s i s e s p e c i a l l y t r u e when we know v e r y  about the p r a c t i c a l i t y of many c o m p e t i t i v e tion.  little  t h e o r i e s of confirma-  By p r a c t i c e , we w i l l e v e n t u a l l y s e l e c t the l e a s t odd one.  However, I do not h e s i t a t e to add t h i s to Hanen's p o s i t i o n * Any is  c o n f i r m a t i o n theory which v i o l a t e s the E q u i v a l e n c e  Condition  l i k e l y t o have the f o l l o w i n g two demerits i n i t s a p p l i c a b i l i t y * 1) Since d i f f e r e n t f o r m u l a t i o n s  of an e q u i v a l e n t hypothesis  have d i f f e r e n t c o n f i r m i n g i n s t a n c e s , t h i s r e s u l t would c r e a t e inconvenience  may  great  i n i t s a p p l i c a t i o n as p o i n t e d out by Hempel.  2) The theory o f q u a l i t a t i v e c o n f i r m a t i o n may become i n c o h e r e n t or u n n e c e s s a r i l y complicated t i t a t i v e confirmation.  when i t extends i n t o a theory o f quan-  F o r i n the standard  theory o f p r o b a b i l i t y  we need the f o l l o w i n g p r i n c i p l e e i t h e r p o s t u l a t e d as an axiom or 20 d e r i v e d as a theorem* Equivalence P r i n c i p l e f o r P r o b a b i l i t y . I f two statements are l o g i c a l l y e q u i v a l e n t , then they have the same p r o b a b i l i t y . Thus any theory o f q u a l i t a t i v e c o n f i r m a t i o n must observe the 20.  Cf., E l l e r y E e l l s *  R a t i o n a l D e c i s i o n and C a u s a l i t y , p.222.  180 above E q u i v a l e n c e P r i n c i p l e i n o r d e r to be extendable  to a theory  of q u a n t i t a t i v e c o n f i r m a t i o n t h a t i s i d e n t i c a l to or i n accord w i t h the standard theory of p r o b a b i l i t y , f o r the E q u i v a l e n c e C o n d i t i o n i n c o n f i r m a t i o n theory i s the c o u n t e r p a r t c o n d i t i o n or p r i n c i p l e to the E q u i v a l e n c e P r i n c i p l e i n the standard t h e o r y of p r o b a b i l i t y . I n defence  f o r a t h e o r y of q u a l i t a t i v e c o n f i r m a t i o n which v i o -  l a t e s the E q u i v a l e n c e C o n d i t i o n , "... one c o u l d argue t h a t t h e r e need be no c o n n e c t i o n between the p r o b a b i l i t y c a l c u l u s which i s f o r m a l , mathem a t i c a l , and q u a n t i t a t i v e i n nature, and a q u a l i t a t i v e confirmation f u n c t i o n . " 2 1  22 But Hanen s a y s i "... t h i s i s a r a t h e r f a c i l e and s u p e r f i c i a l way out of our d i f f i c u l t i e s . There are important s i m i l a r i t i e s i n g o a l between the two approaches to i n d u c t i o n , and i t would be odd i f the r e s u l t s of one approach bore no f r u i t a t a l l f o r the o t h e r . One wants to say t h a t the statements of the p r o b a b i l i t y c a l c u l u s accord w i t h our i n t u i t i o n s about the nature of evidence,..." (my i t a l i c s ) Or someone who  defends f o r the q u a l i t a t i v e c o n f i r m a t i o n theory 23  which v i o l a t e s the E q u i v a l e n c e C o n d i t i o n may  say«  J  "... some p r o b a b i l i t y t h e o r i s t s f o r the s u b j e c t i v i s t or p e r s o n a l i s t s p e r s u a s i o n d e l i b e r a t e l y v i o l a t e s the prob a b i l i t y c a l c u l u s i n the sense t h a t they do not r e q u i r e of an i n t e r p r e t a t i o n of ' p r o b a b i l i t y * t h a t i t render a l l the axioms of the c a l c u l u s t r u e . " To t h i s Hanen quotes Shimony's remarkt "... The remarkable r e s u l t obtained by Ramsey and De F i n e t t i i s t h i s * the c o n f i r m a t i o n e v a l u a t i o n s made by an i n d i v i d u a l must s a t i s f y the axioms of fthe p r o b a b i l i t y c a l c u l u s ) , i f these e v a l u a t i o n s are to c o n s t i t u t e a coherent s e t of b e l i e f s . Thus the axiom of the q u a n t i t a t i v e concept of c o n f i r m a t i o n are j u s t i f i e d , i n t h a t they are necessary c o n d i t i o n s f o r the coherence, and hence f o r the r a t i o n a l i t y , of b e l i e f s . " (my i t a l i c s ) 21. Hanent An Examination 22. I b i d . , p.195. 23. I b i d . , p.195. 24. I b i d . . p.58.  of Adequacy C o n d i t i o n s f o r C o n f i r m a t i o n ,  p.195  181 Since the r e s u l t obtained by Ramsey and s u b j e c t i v i s t ' s or p e r s o n a l i s t ' s approach to  De F i n e t t i about probability  mentioned i n the above quote i s c o n c l u s i v e and t h a t Goodman's and  the  theory  s i n c e Hanen t h i n k s  S c h e f f l e r ' s theory of s e l e c t i v e  confirmation  can be extended to a theory of q u a n t i t a t i v e c o n f i r m a t i o n , I suspect t h a t i t s e x t e n s i o n would be incoherent  at some p o i n t s i n c e  2 5  the Equivalence  Condition i s v i o l a t e d .  However, the p o i n t i s s t i l l t h i s t  J  Not  to k i l l  a theory of qua-  l i t a t i v e c o n f i r m a t i o n j u s t because i t does not observe the  Equiva-  l e n c e C o n d i t i o n , although  Condi-  the v i o l a t i o n of the Equivalence  t i o n i s l i k e l y to make the theory of q u a l i t a t i v e c o n f i r m a t i o n i n coherent  l a t e r when the theory  firmation.  i s extended i n t o a q u a n t i t a t i v e con-  Anyway, i t s l a t e r "incoherence"  even i f i t i s h i g h l y p l a u s i b l e .  Still,  we  i s only a s p e c u l a t i o n , should k i l l  q u a l i t a t i v e or q u a n t i t a t i v e c o n f i r m a t i o n only when we  a theory  of  have i n f a c t  found t h a t i t i s i n c o h e r e n t . Note t h a t my and my  appeal  agreement with Hanen's p o s i t i o n on the one  the a d m i s s i b i l i t y and w o r k a b i l i t y of the  C o n d i t i o n as the c o u n t e r p a r t the context Hempel's and his  25.  26.  of the standard  of the equivalence  hand  Equivalence  Principle within  p r o b a b i l i t y theory i s i n accord  Carnap's l a t e s t view, f o r Hempel has  with  t h i s to say i n  " P o s t s c r i p t (1964) on C o n f i r m a t i o n " ! Perhaps the problem of f o r m u l a t i n g adequate c r i t e r i a o f q u a l i t a t i v e c o n f i r m a t i o n had best be t a c k l e d , a f t e r a l l , by means of the q u a n t i t a t i v e concept of c o n f i r m a 2 6  Note t h a t the theory of s e l e c t i v e c o n f i r m a t i o n t u r n s out to be " i n c o h e r e n t " i n i t s c l a i m t h a t i t can s o l v e the paradoxes of c o n f i r m a t i o n , f o r I f i n d t h a t new paradoxes of c o n f i r m a t i o n can be e a s i l y d e r i v e d i n Goodman's and S c h e f f l e r ' s theory of s e l e c t i v e c o n f i r m a t i o n to be d i s c u s s e d i n s e c t i o n 3, P a r t V. Hempel, op. c i t . , p.50.  182 tion. T h i s has been suggested e s p e c i a l l y by Carnap, who holds t h a t "any adequate explicatum f o r the c l a s s i f i c a t o r y concept o f c o n f i r m a t i o n must be i n accord w i t h a t l e a s t one adequate explicatum f o r the q u a n t i t a t i v e concept of c o n f i r m a t i o n " .... (my i t a l i c s ) Thus, i n c o n c l u s i o n , we agree w i t h Hanen t o r e j e c t a l l adequacy c o n d i t i o n s l a i d down by Hempel f o r q u a l i t a t i v e c o n f i r m a t i o n w i t h the understanding  t h a t there i s l i k e l y a problem of p o t e n t i a l i n -  coherence when the q u a l i t a t i v e c o n f i r m a t i o n theory i s extended t o a q u a n t i t a t i v e one i f the E q u i v a l e n c e the q u a l i t a t i v e c o n f i r m a t i o n  C o n d i t i o n i s not observed i n  theory.  However, f o r i n t r i n s i c reasons  (say, f o r the sake o f c o n s i s -  tence, coherence and g r e a t e r e x p r e s s i b i l i t y ) the i n t e r n a l mation theory needs the E q u i v a l e n c e  confir-  C o n d i t i o n ( s i n c e any l o g i c a l l y  e q u i v a l e n t hypotheses of a g i v e n h y p o t h e s i s  are confirmed,  or d i s -  conf irmed, by the same o b s e r v a t i o n r e p o r t s i n the i n t e r n a l  confir-  mation theory) and, perhaps, a l s o the E n t a i l m e n t  C o n d i t i o n ( i f peo-  p e l would accept my m o d i f i c a t i o n of Carnap*s explicatum mation d i s c u s s e d  earlier.)  of c o n f i r -  PART V. TENTATIVE CONCLUSIONS  184 1. INTRODUCTION In t h i s l a s t P a r t we w i l l  sum up and a t the same time compare  the d i f f e r e n c e s of Hempel's theory of c o n f i r m a t i o n , Goodman's and S c h e f f l e r ' s theory o f s e l e c t i v e c o n f i r m a t i o n and the i n t e r n a l conf i r m a t i o n theory as they r e p r e s e n t , t a n t approaches to c o n f i r m a t i o n  I t h i n k , the three most impor-  theory.  With what we have s t u d i e d up t o now and, e s p e c i a l l y , with the h e l p of some newly d i s c o v e r e d paradoxes o f c o n f i r m a t i o n d e r i v e d i n the theory o f s e l e c t i v e c o n f i r m a t i o n as w e l l as i n the i n t e r n a l c o n f i r m a t i o n theory,  I come to the f o l l o w i n g most reasonable  con-  c l u s i o n t h a t I can t h i n k o f t the o n l y p l a u s i b l e s o l u t i o n of the paradoxes o f c o n f i r m a t i o n i s t h e i r  dissolution.  185 2. A SUMMARY OF HEMPEL'S THEORY OF CONFIRMATION Hempel's theory o f c o n f i r m a t i o n has the f o l l o w i n g  outstanding  propertiesi 1. I t i s the f i r s t comprehensive theory o f q u a l i t a t i v e  confir-  mation. 2. I t s u n d e r l y i n g l o g i c cational  i s the c l a s s i c a l two-valued  quantifi-  logic.  3. H i s t h e o r y o f c o n f i r m a t i o n o f f e r s a S a t i s f a c t i o n  Criterion  of C o n f i r m a t i o n ; i . e . , i n h i s t h e o r y o f c o n f i r m a t i o n we have* Theorem 18.  An o b s e r v a t i o n r e p o r t B ( c , c , ...,c^) confirms a 1  hypothesis H i f f f c ^ . C g  c^  are the i n d i v i d u a l c o n s t a n t s  2  s a t i s f i e s H, (where c ^ . C g , • • . » c ^  e s s e n t i a l l y mentioned i n B.)  In p a r t i c u l a r , we have« Theorem 18.1.  The o b s e r v a t i o n r e p o r t Rc.Bc confirms the raven  hypothesis i f f c s a t i s f i e s  (x)(Rx»Bx).  (This i s why sometimes people say l o o s e l y t h a t c " c o n f i r m s " the raven h y p o t h e s i s , although what they mean i s t h a t c s a t i s f i e s the raven h y p o t h e s i s and, hence, the t o t a l i n f o r m a t i o n about c, v i z . Rc.Bc, c o n f i r m s the raven  hypothesis.)  4. A l l adequacy c o n d i t i o n s o f c o n f i r m a t i o n theory d i s c u s s e d i n the l a s t P a r t hold i n h i s theory of c o n f i r m a t i o n .  Above  all,  the E q u i v a l e n c e C o n d i t i o n holds i n h i s theory of c o n f i r m a t i o n . 5. The paradoxes o f c o n f i r m a t i o n are d i s s o l v e d i n h i s theory of c o n f i r m a t i o n , f o r a c c o r d i n g t o Hempelt ( i ) they are not genuine, ( i i ) they have no l o g i c a l and o b j e c t i v e grounds and, hence, ( i i i ) they are p s y c h o l o g i c a l i l l u s i o n s .  186 Or i n Hempel's words: "The impression of a p a r a d o x i c a l s i t u a t i o n i s not object i v e l y founded: i t i s a p s y c h o l o g i c a l i l l u s i o n . " (my i t a l i c s ) "... our f a c t u a l knowledge t h a t not a l l o b j e c t s are b l a c k tends t o c r e a t e an impression o f p a r a d o x i c a l i t y which i s not j u s t i f i e d on l o g i c a l grounds .... Thus i t t u r n s out t h a t the paradoxes o f c o n f i r m a t i o n . . . a r e due t o a misguided i n t u i t i o n i n the matter r a t h e r than t o a l o g i c a l flaw i n t h e . . . s t i p u l a t i o n s from which they are d e r i v e d . " (my i t a l i c s ) 6. What i s c o n t r o v e r s i a l i s t h a t many people t h i n k t h a t the paradoxes of c o n f i r m a t i o n are genuine and they can be l o g i c a l l y and o b j e c t i v e l y formulated  and, hence, they are not p s y c h o l o g i c a l  illu-  s i o n s as Hempel t h i n k s and persuades us to b e l i e v e . 7. One s e r i o u s t r o u b l e of Hempel's theory of c o n f i r m a t i o n i s t h a t Hempel's theory of c o n f i r m a t i o n allows the p r a c t i c e o f "indoor p ornithology".  Thus the f o l l o w i n g hypotheses are a l l confirmed i n  Hempel's theory  (simply because I have no o f f i c e a t UBC):  HI:  A l l ravens i n my o f f i c e a t UBC are b l a c k .  H2:  A l l ravens i n my o f f i c e a t UBC are white.  H3«  A l l ravens i n my o f f i c e a t UBC are both b l a c k and white. Perhaps the above type o f hypotheses i s too t r i v i a l f o r o r n i -  t h o l o g i s t s , f o r who c a r e s  about the ravens i n my o f f i c e a t UBC. So,  c o n s i d e r the f o l l o w i n g b i g g e r  hypotheses:  H4:  A l l pandas i n Canada a r e b l a c k .  H5*  A l l pandas i n Canada are w h i t e .  H6:  A l l pandas i n Canada a r e both b l a c k and w h i t e .  H7«  Each panda i n the moon has two eyes.  H8:  Each panda i n the moon has three  H*  Each panda i n the moon has one thousand and one eyes.  Q  eyes.  1. Hempel: Aspects of S c i e n t i f i c E x p l a n a t i o n , p.18 and p.20. 2. Goodman: F a c t . F i c t i o n , and F o r e c a s t , pp.70-71.  18? Now  each of the above b i g g e r hypotheses H4-  H9  i s confirmed  (and,  hence, presumably t r u e ) simply because there i s no panda i n Canada nor i n the moon and,  hence, the t o t a l evidence  about a l l f i n i t e  ex-  i s t i n g o b j e c t s i n Canada or i n the moon confirms each of hypotheses  H9.  H4-  Here i t seems q u i t e c l e a r now  as the above examples shown t h a t  the p r a c t i c e of "indoor o r n i t h o l o g y " may l o g i c a l or f i c t i o n a l o r n i t h o l o g y .  produce a k i n d of mytho-  That Hempel's theory of c o n f i r -  mation a l l o w s such a m y t h o l o g i c a l or f i c t i o n a l p r a c t i c e  indicates  t h a t there i s something wrong w i t h Hempel's theory of q u a l i t a t i v e c o n f i r m a t i o n or t h a t the u n d e r l y i n g l o g i c  (i.e. classical  logic)  of Hempel's t h e o r y of c o n f i r m a t i o n cannot  d e a l s a t i s f a c t o r i l y with  some problems a r i s i n g from the empty domain, or both. 8. However, I t h i n k t h a t there i s an easy way  out of the above  d i f f i c u l t y f o r Hempel, when he comes to the t o p i c s of (theory o f ) comparative following H^ : !  H : 1 2  P  a n c  *  a s  a  r  e  black  and,  3.  eyes  the hypotheses H2,  e l i m i n a t e d as l e s s good and 1 2  of the  are b l a c k  Each panda has two  as t r u t h s or laws,  H  For, then, w i t h the e s t a b l i s h m e n t  hypotheses:  A l l ravens  Q  ^11  confirmation.  H3,  H5,  H6,  H8  and H9 can  i m p r a c t i c a l , s i m p l y because H ,  are t r u e and more g e n e r a l on the one hand and  1Q  H  1 1  be and  simpler-^ i n form  hence, more u s e f u l on the other hand.  Cf. Goodman: " s a f e t y , s t r e n g t h , s i m p l i c i t y " , ence, 1961, p p . 1 5 0 - 1 5 1 .  P h i l o s o p h y of S c i -  188 3.  A SUMMARY OF GOODMAN'S AND SCHEFFLER'S THEORY O F SELECTIVE  CONFIRMATION Goodman's and S c h e f f l e r ' s theory o f s e l e c t i v e c o n f i r m a t i o n has the f o l l o w i n g i n t e r e s t i n g , important  as w e l l as p e c u l i a r proper-  ties » 1. I t i s based on top o f Hempel's theory o f c o n f i r m a t i o n . 2. The theory o f s e l e c t i v e c o n f i r m a t i o n appears not to allow the p r a c t i c e o f indoor 3.  ornithology.  The theory o f s e l e c t i v e c o n f i r m a t i o n does n o t have a S a t i s -  f a c t i o n C r i t e r i o n of Confirmation. 4. The paradoxes o f c o n f i r m a t i o n appear not; t o be d e r i v a b l e i n the theory o f s e l e c t i v e c o n f i r m a t i o n .  I f so, i t would f o l l o w :  ( i ) the paradoxes of c o n f i r m a t i o n are genuine, and ( i i ) they are l o g i c a l l y and o b j e c t i v e l y grounded and,  hence,  ( i i i ) they are n o t p s y c h o l o g i c a l i l l u s i o n s . 5. I n the theory o f s e l e c t i v e c o n f i r m a t i o n the E q u i v a l e n c e d i t i o n does n o t h o l d  (as f a r as s e l e c t i v e c o n f i r m a t i o n i s  Con-  concerned).  Hence, i n g e n e r a l , 5.1.  the f o r m u l a t i o n o f a h y p o t h e s i s  i s c r u c i a l to i t s selec-  t i v e c o n f i r m a t i o n , and 5.2.  there a r e hypotheses, e.g.  (x)Gx, which cannot have s e -  l e c t i v e c o n f i r m a t i o n simply because they do n o t have c o n t r a r i e s . 6. By d e f i n i t i o n o f s e l e c t i v e c o n f i r m a t i o n the c o n t r a r y o f a hypothesis  must be d i s c o n f i r m e d  l y confirmed.  t o be s e l e c t i v e -  I t follows that a u n i v e r s a l c o n d i t i o n a l hypothesis  such as the f o l l o w i n g one, H:  f o r the hypothesis  ( X ) ( F X D G X ) ,  189  i n order to be s e l e c t i v e l y confirmed must have the f o l l o w i n g hypothesis, -(x)(Px3-Gx),  CHi  confirmed.  That amounts t o i we must have both H and 1  the f o l l o w -  ing hypothesis, CH*:  (3x)(Fx.Gx),  confirmed  i n order to s e l e c t i v e l y c o n f i r m the u n i v e r s a l c o n d i t i o n a l  h y p o t h e s i s H. sumption.  T h i s seems to be a v e r y s t r o n g and u n d e s i r a b l e as-  At l e a s t , i t i s a s t r o n g e r assumption than the assump-  t i o n of adding an e x i s t e n t i a l c l a u s e to a u n i v e r s a l c o n d i t i o n a l h y p o t h e s i s made by the A r i s t o t a l i a n s o l u t i o n of the paradoxes of c o n f i r m a t i o n c r i t i c i z e d by Hempel. A l s o some c r i t i c s f e e l t h a t p r o p e r t y 5*1 above are u n d e s i r a b l e , i n c o n v e n i e n t and, theory or, a t l e a s t , p e r t y 5.2  seems a l s o to suggest  firmation i s Above a l l , one  in scientific  and p r o p e r t y 5.2  made  perhaps, a l s o wrong i n  p r a c t i c e ; and,  furthermore,  t h a t the theory of s e l e c t i v e  pro-  con-  inadequate. the p e c u l i a r p r o p e r t y 5 seems to be the most  to the t h e o r y of s e l e c t i v e c o n f i r m a t i o n , f o r without  harmful  observing  the e q u i v a l e n c e c o n d i t i o n the t h e o r y of s e l e c t i v e c o n f i r m a t i o n i s l i k e l y to be i n c o n f l i c t w i t h the standard p r o b a b i l i t y theory when it  i s extended i n t o a theory of q u a n t i t a t i v e c o n f i r m a t i o n as we  have  t r i e d to i n d i c a t e i n s e c t i o n 11 of the p r e v i o u s P a r t . 7. 1.  2.  F i n a l l y , my  l a t e s t d i s c o v e r y i s t h a t Goodman's and  Scheffler's  However, from the t r a n s f o r m a t i o n of CH t o CH* the equivalence c o n d i t i o n has been employed, and we know t h a t the equivalence c o n d i t i o n i s r e j e c t e d i n the t h e o r y s e l e c t i v e c o n f i r m a t i o n . So, the c r i t i c i s m f o l l o w e d maybe not r e a l l y t r u e . Hempeli Aspects of S c i e n t i f i c E x p l a n a t i o n , p p . 1 6 - 1 7 .  190  theory of s e l e c t i v e  c o n f i r m a t i o n does not r e a l l y solve the p a r a -  doxes of c o n f i r m a t i o n . (1)  Thus, c o n s i d e r the o b s e r v a t i o n r e p o r t :  Ra.Ba,  which s e l e c t i v e l y c o n f i r m s the raven h y p o t h e s i s : H:  (x)(Rxr>Bx);  w h i l e each of the o b s e r v a t i o n r e p o r t s , (2.1)  -Rb.-Bb,  (2.2)  -Rc.Bc,  i s " s e l e c t i v e l y n e u t r a l " to ( i . e . n e i t h e r " s e l e c t i v e l y c o n f i r m s " nor  " d i s c o n f i r m s " ) the raven h y p o t h e s i s H.  But each of the f o l -  lowing o b s e r v a t i o n r e p o r t s , (3.1)  (Ra.Ba).(-Rb.-Bb)  (3-2)  (Ra.Ba).(-Rc.Bc),  (3.3)  (Ra.Ba).(-Rb.-Bb).(-Rc.Bc),  turns out to s e l e c t i v e l y c o n f i r m the raven h y p o t h e s i s H, f o r we (4.1)  (Ra.Ba). (-Rb.-Bb) \[ (Ra a B a ) . (Rb t>Bb),  (4.2)  (Ra.Ba). (-Rc.Bc) ||-(RaoBa). (Rc o B c ) ,  (4.3)  (Ra.Ba). (-Rb.-Bb). (-Rc.Bc) ||- ( R a o B a ) . (Rb=>Bb). (Rc o B c ),  i.e.,  (3.1),  and we a l s o  (3-2)  and  (3.3)  have:  each c o n f i r m s the raven h y p o t h e s i s H;  have:  (5-1)  (Ra.Ba). (-Rb.-Bb) ||- -((Ra=> -Ba). (Rb => -Bb)),  (5.2)  (Ra.Ba). (-Rc.Bc)fl-- ( ( R a o -Ba). (Rc=> -Be ) ) ,  (5-3)  (Ra.Ba). (-Rb.-Bb).(-Rc .Be ) ||- -((Ra=?-Ba). ( R b o - B b ) . ( R c P - B c ) ) ,  i.e.,  (3.1).  contrary.  (3-2)  and  (3.3)  each d i s c o n f i r m s , r e s p e c t i v e l y , i t s  Hence, (3.1), (3*2) raven h y p o t h e s i s  H,  and (3.3)  each s e l e c t i v e l y confirms the  these are the newly d i s c o v e r e d  of c o n f i r m a t i o n d e r i v e d i n the theory of s e l e c t i v e  paradoxes  confirmation.  So, the theory of s e l e c t i v e c o n f i r m a t i o n does not r e a l l y a l l paradoxes of c o n f i r m a t i o n .  solve  In other words i t i s an u n j u s t i f i e d  c l a i m t o say t h a t Goodman's and S c h e f f l e r ' s theory of s e l e c t i v e c o n f i r m a t i o n has s o l v e d the paradoxes of c o n f i r m a t i o n .  192  4. A TENTATIVE CONCLUSION FOR THE INTERNAL CONFIRMATION THEORY In the development  o f the i n t e r n a l  c o n f i r m a t i o n theory when  we understood the Raven Hypothesis (which says t h a t a l l ravens are b l a c k ) as the h y p o t h e s i s : RH:  (tyx) (Rx =»Bx),  we almost had a s o l u t i o n of the paradoxes of c o n f i r m a t i o n , f o r we can have the f o l l o w i n g r e s u l t s about the RH i n the i n t e r n a l mation  confir-  theory: ( i ) A raven t h a t i s b l a c k and only a raven  t h a t i s b l a c k can  c o n f i r m the RH; ( i i ) A raven t h a t i s non-black w i l l d i s c o n f i r m the RH; and ( i i i ) A l l other i n s t a n c e s are n e u t r a l to the RH. So f a r so good. The  only t r o u b l e with the above " s o l u t i o n " of the paradoxes o f  c o n f i r m a t i o n i s t h a t the RH can never be true i n the a c t u a l world where, as i t t u r n s out t o be, there are non-black o b j e c t s which w i l l make the RH never be t r u e i n the i n t e r n a l c o n f i r m a t i o n when we come t o a c t u a l a p p l i c a t i o n .  The non-truth,  theory  albeit also  n o n - f a l s i t y , of the RH i s t h e o r e t i c a l l y both odd and u n d e s i r a b l e . So, we do not have a genuine s o l u t i o n of the paradoxes of conf i r m a t i o n thus f a r . Then, I t h i n k t h a t I can perhaps r e f i n e and improve the above " s o l u t i o n " o f the paradoxes of c o n f i r m a t i o n by i n t r o d u c i n g the concept of " # - r e s t r i c t i o n " o f a h y p o t h e s i s t i o n theory  i n t o the i n t e r n a l  i n order t o get r i d of the n e u t r a l cases  t h a t the r e f i n e d RH can have the t r u t h - v a l u e " t r u t h " . Thus i f we understand the Raven Hypothesis a s :  confirma-  of the RH so  193  RH : #  ((Vx)(Rx  =*Bx)) , #  we can s t i l l preserve  the above three p r o p e r t i e s ( i ) , ( i i ) and  ( i i i ) on the one hand and have RJr had the t r u t h - v a l u e " t r u t h " on the other hand. Thus i t seems t h a t f i n a l l y we have a s o l u t i o n of the paradoxes of c o n f i r m a t i o n . At t h i s p o i n t some important  and c r u c i a l c r i t i c i s m s have been  made about the above " s o l u t i o n " of the paradoxes of c o n f i r m a t i o n s (I) The f i r s t c r i t i c i s m i s t h i s .  Suppose t h a t we have the Con-  sequence C o n d i t i o n i n t h e i n t e r n a l c o n f i r m a t i o n t h e o r y .  Then, note  t h a t we have the f o l l o w i n g s e m a n t i c a l entailment r e l a t i o n i n the i n t e r n a l confirmation theory: (1)  (3x)Rx& (3x)-Bx |K(Vx)(Rx=*Bx)) s #  ((Vx) (-Bx*»-Rx) )  #  .  Since i t i s t r u e t h a t i n the a c t u a l world we have the observ a t i o n reports (2)  (Ra& Ba) & (-Rb  &-Bb)  where "a" i s "a b l a c k raven" and "b" i s "a non-black non-raven", it  f o l l o w s w i t h the h e l p of the Consequence C o n d i t i o n t h a t the hy-  pothesis, (3)  ((VX)(RX=*BX)) SE #  ((\/x)(-Bx=>-Rx)) , #  i s confirmed  by (2) f o r (2) c o n f i r m s  (3x)Rx& (3x)-Bx, which i s the  "antecedent"  of the semantical entailment r e l a t i o n ( l ) .  From (3) we can d e r i v e the paradoxes of c o n f i r m a t i o n i n the i n t e r n a l c o n f i r m a t i o n theory  (with the h e l p of the E q u i v a l e n c e  Condi-  t i o n ) a c c o r d i n g to the c r i t i c i s m . One response to the above c r i t i c i s m i s t h a t the Consequence Con-  194  d i t i o n does not hold i n the i n t e r n a l c o n f i r m a t i o n t h e o r y ,  1  for i t  has been r e j e c t e d as an adequacy c o n d i t i o n f o r c o n f i r m a t i o n i n section 11,  P a r t IV.  hence, we  (3)  Moreover,  i s not a l o g i c a l equivalence  cannot employ the E q u i v a l e n c e  and,  C o n d i t i o n to i t i n order  to d e r i v e the paradoxes of c o n f i r m a t i o n from i t . ( I I ) The  second c r i t i c i s m  t i o n theory the two (4)  is this.  In the i n t e r n a l  confirma-  hypothesesi  (3x)Rx& (3x)-Bx&((yx)(Rx=*Bx))  #  and (5)  (3x)Rx& (3x)-Bx&  are l o g i c a l l y  ((t/x)(-Bx=*-Rx))  equivalent. (4)  L e t us c a l l  "the s t r o n g Raven H y p o t h e s i s " .  l o g i c a l l y e q u i v a l e n t to  (5),  t h e s i s " , e s p e c i a l l y when we internal confirmation  (5)  f i r m s both ( 4 )  and  is  C o n d i t i o n i n the  theory. (or d i s c o n f i r m s )  (or, r e s p e c t i v e l y , d i s c o n f i r m ) In p a r t i c u l a r , we  (4)  Since  i s a l s o "the s t r o n g Raven Hypo-  have the E q u i v a l e n c e  Consequently, what confirms firm  #  have t h i s i  (5),  and  (5)»  and v i c e  (4)  w i l l a l s o con-  versa.  the o b s e r v a t i o n r e p o r t  (2)  con-  t h i s i s one v e r s i o n of the paradoxes  of c o n f i r m a t i o n . T h i s new blow to my  v e r s i o n of the paradoxes of c o n f i r m a t i o n i s a f a t e f u l  c l a i m t h a t the paradoxes of c o n f i r m a t i o n can be  i n the i n t e r n a l c o n f i r m a t i o n ( I I I ) Worse s t i l l , 1.  without  solved  theory. employing the E q u i v a l e n c e  Condition  Note t h a t the f i r s t c r i t i c i s m was made b e f o r e I came to the r e j e c t i o n of the Consequence C o n d i t i o n as an adequacy c o n d i t i o n f o r the i n t e r n a l c o n f i r m a t i o n t h e o r y . So, the c r i t i c i s m was then much more f o r c e f u l than i t appears now.  195 (or any other adequacy c o n d i t i o n s ) some new forms of the paradoxes of c o n f i r m a t i o n c a n be d e r i v e d as f o l l o w s : Although i n the i n t e r n a l c o n f i r m a t i o n theory the o b s e r v a t i o n report, BI:  Ra&Ba  confirms  the d e - n e u t r a l i z e d I m p l i c a t i o n a l Raven Hypothesis RH ,and  each o f the f o l l o w i n g two o b s e r v a t i o n r e p o r t s , B2:  -Rb&-Bb,  B3:  -Rc&Bc  i s n e u t r a l t o RH^, we have t h a t each of the f o l l o w i n g three  obser-  vation reports, B4:  (Ra& Ba) & (-Rb &-Bb),  B5:  (Ra & Ba) & (-Rc & Be ),  B6:  ( R a & B a ) & (-Rb&-Bb)& (-Rc & Be)  confirms RH^.  And t h i s i s because B4, B5 and B6 each e n t a i l s the  statement, (6)  Ra=*Ba  which i s the development o f RH  w i t h r e s p e c t to the c l a s s of i n d i -  v i d u a l s e s s e n t i a l l y mentioned i n each o f the o b s e r v a t i o n r e p o r t s B4,  B5 and B6 i n the i n t e r n a l c o n f i r m a t i o n  theory.  In other words, i n the i n t e r n a l c o n f i r m a t i o n theory we have the f o l l o w i n g two important  results:  ( I I I . l ) I n d i v i d u a l l y , the o b s e r v a t i o n r e p o r t t h a t a raven i s b l a c k confirms RH , while the o b s e r v a t i o n r e p o r t t h a t a non-raven i s non-black or the o b s e r v a t i o n r e p o r t t h a t a non-raven i s b l a c k u  (or both) i s n e u t r a l t o RW .  196 ( I I I . 2 ) C o l l e c t i v e l y , as a statement o f t o t a l evidence, ment t h a t a raven i s black c o n j o i n e d  t o a statement t h a t a non-raven  i s non-black o r t o a statement t h a t a non-raven i s black confirms The  (or t o both)  RH^. result  ( I I I . l ) i s f i n e ; but the r e s u l t  ( I I I . 2 ) i s some new  v e r s i o n of the paradoxes of c o n f i r m a t i o n gained firmation The  a state-  i n the i n t e r n a l  con-  theory.  result  ( I I I . l ) g i v e s us the impression  t h a t the paradoxes  of c o n f i r m a t i o n are genuine and s o l v a b l e ; but i n f a c t t h a t i s o n l y a psychological i l l u s i o n  f o r the paradoxes o f c o n f i r m a t i o n a r e not  r e a l l y s o l v a b l e as the r e s u l t The  above s u r p r i s i n g  ( I I I . 2 ) has shown.  results  mation can be d e r i v e d from RH  t h a t some new paradoxes of c o n f i r -  as w e l l as from the s t r o n g raven hy-  p o t h e s i s seem to suggest the f o l l o w i n g c o n c l u s i o n s ! 1) Even i f we change the u n d e r l y i n g l o g i c of c o n f i r m a t i o n from c l a s s i c a l l o g i c t o QC3 of a l l , r a d i c a l and q u i t e u n d e s i r a b l e  of Hempel's  theory  t h i s change i s , f i r s t  i n the sense t h a t l o g i c i s  always s a i d t o be the l a s t t h i n g t o go f o r change  new  difficul-  t i e s as w e l l as new and o l d paradoxes o f c o n f i r m a t i o n w i l l  occur  or r e c u r i n some unexpected ways. 2) Thus, f i r s t , when we understand the Raven Hypothesis as RH, although we c a n a v o i d the paradoxes of c o n f i r m a t i o n , the t r u t h - v a l u e of RH unexpectedly t u r n s out t o be n e u t r a l ( i . e . n e i t h e r t r u e nor false). 3) Then, we d e - n e u t r a l i z e RH t o have RH  u  i n order t o get r i d of  u  a l l n e u t r a l i n s t a n c e s of RH, although  RH ^ t u r n s out t o be t r u e , some 77  new form of the paradoxes of c o n f i r m a t i o n such as the above r e s u l t ( I I I . 2 ) now unexpectedly  occurs.  197 4) A l l these,  i n t u r n , seem t o suggest t h a t the paradoxes of  c o n f i r m a t i o n are much more deeply rooted  i n confirmation theories  than I had thought and expected, and the i n t e r n a l c o n f i r m a t i o n ory i s no e x c e p t i o n , raven hypothesis plausible  the-  even i f one t r i e s t o read and understand the  i n other p l a u s i b l e ways, as w e l l as to make other  refinements.  5) So, Hempel i s perhaps r i g h t a f t e r a l l t o conclude t h a t we can o n l y have a " d i s s o l u t i o n " of the paradoxes of c o n f i r m a t i o n . At l e a s t t h i s seems t o be the most reasonable  c o n c l u s i o n t h a t I can  draw from what I have done here i n t h i s d i s s e r t a t i o n . 6) However, note t h a t there are some s l i g h t d i f f e r e n c e s  between  the r e s u l t s of the d i s s o l u t i o n i n Hempel's theory of c o n f i r m a t i o n and  the r e s u l t s of the d i s s o l u t i o n i n the i n t e r n a l c o n f i r m a t i o n  the-  ory i (A) In Hempel's theory of c o n f i r m a t i o n from the d i s s o l u t i o n of the paradoxes of c o n f i r m a t i o n we have: ( i ) Ra.Ba, -Rb.-Bb o r -Rc.Bc (or any number of t h e i r conj u n c t i o n ) confirms ( i i ) Rd.-Bd  the raven h y p o t h e s i s  H;  (or i t s c o n j u n c t i o n with any other  r e p o r t ) d i s c o n f i r m s the raven h y p o t h e s i s  observation  H.  (B) In the i n t e r n a l c o n f i r m a t i o n theory from the d i s s o l u t i o n of the paradoxes of c o n f i r m a t i o n what we have a r e : ( i ) R a & B a as the t o t a l evidence sis  confirms  the raven  hypothe-  RH j #  ( i i ) -Rb&-Bb o r -Rc & Be (or both) as the t o t a l evidence i s n e u t r a l t o the raven hypothesis  RH ;  ( i i i ) Rd& -Bd (or i t s c o n j u n c t i o n with any other r e p o r t ) d i s c o n f i r m s the raven hypothesis  RH ;  observation  198  ( i v ) R a & B a and -Rc & Be  i t s c o n j u n c t i o n w i t h e i t h e r -Rb&  (or both) as the t o t a l evidence  a l s o confirms  -Bb  the  or  raven  it hypothesis The  RH  .  s i g n i f i c a n c e of ( A ) ( i ) means t h a t the paradoxes of c o n f i r -  mation can o n l y be c o n s i d e r e d theory;  as p s y c h o l o g i c a l i l l u s i o n s i n Hempel's  however, the s i g n i f i c a n c e of  ( B ) ( i i ) seems to suggest t h a t  i n a s e n s e t h a t the paradoxes of c o n f i r m a t i o n are not p s y c h o l o g i c a l i l l u s i o n s i n the i n t e r n a l c o n f i r m a t i o n theory, out to be e v e n t u a l l y i n the sense of  although  they  turn  ( B ) ( i v ) when we a l s o come a c r o s s  a raven t h a t i s b l a c k . However, the s i g n i f i c a n t d i f f e r e n c e s j u s t noted above c o u l d o n l y apparent once we  r e a l i z e t h a t to c o n f i r m  the raven  RH ^ i n the i n t e r n a l c o n f i r m a t i o n theory a t l e a s t one must be  e s s e n t i a l l y mentioned i n the t o t a l evidence  t r u e there must e x i s t a t l e a s t one  i n the world.  there i s no  the raven h y p o t h e s i s  raven  or, i n other  words, to make RH But  hypothesis  black  77  black  raven  such " r e s t r i c t i o n " b e i n g b u i l t  H i n the c o n f i r m a t i o n of H i n Hempel's  of c o n f i r m a t i o n , nor i n the making of the t r u t h of the raven thesis H i n c l a s s i c a l  be  into  theory hypo-  logic.  7) F i n a l l y , note t h a t the s t r i k i n g s i m i l a r i t y of the d e r i v a t i o n s of the new  paradoxes of c o n f i r m a t i o n t h a t the t o t a l Ra and  conjoined  Ba  to e i t h e r -Rb  and  -Bb  -Rc  and  Be  or  evidence,  199 (or both) confirms  or s e l e c t i v e l y confirms  the raven  hypothesis  t h a t a l l ravens are b l a c k , r e s p e c t i v e l y , i n the i n t e r n a l mation theory and  confir-  i n the theory of s e l e c t i v e c o n f i r m a t i o n .  s t r i k i n g s i m i l a r i t y of the new i n the above mentioned two  The  paradoxes of c o n f i r m a t i o n d e r i v e d  t h e o r i e s i s i n f a c t due  to the same r o o t :  both t h e o r i e s employ ( i ) Hempel's concept of t o t a l evidence ( i . e . t o t a l o b s e r v a t i o n r e p o r t ) and development of a hypothesis t i o n " and  "selective  ( i i ) the concept of entailment  i n d e f i n i n g the concepts of  of the  "confirma-  confirmation".  I t i s t r u e t h a t Hempel a l s o employs the concept of t o t a l dence and  the concept of entailment  evi-  of the development of a hypo-  t h e s i s i n d e f i n i n g h i s concept of " c o n f i r m a t i o n " .  But he a l s o ac-  c e p t s g l a d l y the paradoxes of c o n f i r m a t i o n , once he d e r i v e s them. Thus c o n d i t i o n s ( i ) and  ( i i ) s t a t e d above seem to impose on  i n t e r n a l c o n f i r m a t i o n theory and t i o n the new  the theory of s e l e c t i v e  confirma-  paradoxes of c o n f i r m a t i o n .  Since there i s no obvious way  to overcome the new  (and i n view of the above a n a l y s i s we p e c t i o n so l o n g we and  the  difficulties  should not have such an  employ Hempel's concepts to d e f i n e  ex-  "confirmation"  " s e l e c t i v e c o n f i r m a t i o n " i n the i n t e r n a l c o n f i r m a t i o n theory  and  i n the theory of s e l e c t i v e c o n f i r m a t i o n ) , I come to t h i s more r e a sonable we  c o n c l u s i o n i under the present  s i t u a t i o n the best t h i n g t h a t  can do i s to agree with Hempel t o d i s s o l v e ( i . e . to l i v e  a l l paradoxes of c o n f i r m a t i o n .  T h i s means t h a t Hempel may  with) be  right  a f t e r a l l to say t h a t the paradoxes of c o n f i r m a t i o n are not genuine and  to t h i n k otherwise  s i o n s as Hempel  says..  i s to have some k i n d of p s y c h o l o g i c a l  illu-  200 BIBLIOGRAPHY A c h i n s t e i n , P. and S.P. 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