Arts, Faculty of
Philosophy, Department of
DSpace
UBCV
Lin, Chao-tien
Doctor of Philosophy - PhD
University of British Columbia
The title of my dissertation is "confirmation theory & confirmation logic", and it consists of five Parts. The motivation of the dissertation was to construct an adequate confirmation theory that could solve "the paradoxes of confirmation" discovered by Carl G. Hempel. In Part One I try mainly to do the three things, (i) introduce the fundamentals of Hempel's theory of qualitative confirmation as the common background for subsequent discussions, (ii) review the major views of the paradoxes of confirmation, (iii) present a new view, which is more radical than other known views, and argue that a solution to the paradoxes of confirmation may require a change of logic. In Part Two I construct a number of promising three-valued logics. I employ these "quasi confirmation logics" as the underlying logics of some new confirmation theories which, I had hoped, would solve the paradoxes of confirmation. I consider three-valued logics instead of any other many-valued logics as the underlying logic for any promising confirmation theory, because I believe that there is some intimate relationship or, even, a one-to-one correspondence between the (controversial) three truth-values of "truth", "falsity" and "neither truth nor falsity" and, respectively, the (non-controversial) three confirmation-statuses of "confirmation", "disconfirmation" and "neutrality". Unfortunately, these theories were found to be semantically inadequate. This became clear after a complete semantics for them had been developed. Thus, one negative result of Part Two is that our syntactical approach to confirmation theory is wrong from the very beginning. However, from this negative result we learn a positive lesson: a semantical approach is more fundamental and decisive than a syntactical one, at least this is so for constructing an adequate theory of confirmation. It is rewarding to note that the three-valued semantics worked out in Part Two is simple, complete and the first of its kind. In fact, the new three-valued semantics is in the spirit of Frege, although the line of thought is much neglected (even by Frege himself). In Part Three I shift the search for a confirmation logic and an adequate theory of confirmation from a syntactical to a semantical approach because of the lesson learned in Part Two. After a systematic search through several promising three-valued logics I come, at last, to a plausible confirmation logic and to a confirmation theory that could solve all known paradoxes of confirmation. The promising three-valued confirmation theory is called "the internal confirmation theory". In Part Four I review and appraise the adequacy conditions laid down by Hempel as the necessary conditions for any adequate confirmation theory. Under the criticisms of Carnap, Goodman and, especially, with the help of Hanen's thorough studies, I come to almost an identical conclusion to Hanen's we should not impose a priori in a theory of qualitative confirmation any adequacy conditions laid down by Hempel except perhaps the Entailment Condition, although the internal confirmation theory also adopts the Equivalence Condition for some intrinsic reasons. In the last Part Five I try to appraise the three most important confirmation theories discussed and/or constructed in this dissertation. They are Hempel's theory of confirmation, Goodman's and Scheffler's theory of selective confirmation and the internal confirmation theory. After some more vigorous criticisms are made and some new paradoxes of confirmation are unexpectedly derived in both the theory of selective confirmation and the internal confirmation theory, I arrive at, perhaps reluctantly, this more reasonable conclusion under the present situation when there is no obvious way to overcome the new difficulties the best thing that we can do is to dissolve (i.e. to live with) all new and old paradoxes of confirmation, for Hempel may be after all right to say that the paradoxes of confirmation are not genuine and to think otherwise is to have psychological illusions as Hempel says.
CONFIRMATION THEORY & CONFIRMATION LOGIC By LIN CHAO-TIEN B .A . , National Taiwan Un ivers i t y , 1962 M.A., National Taiwan Un ivers i t y , I966 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF PHILOSOPHY) We accept t h i s thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1987 © L i n Chao - t ien , 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Philosophy The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date March 23, 1988 DE-6G/81) ABSTRACT The t i t l e of my d i s s e r t a t i o n i s " c o n f i r m a t i o n t h e o r y & c o n -f i r m a t i o n l o g i c " , and i t c o n s i s t s of f i v e P a r t s . The m o t i v a t i o n o f the d i s s e r t a t i o n was to c o n s t r u c t an adequate c o n f i r m a t i o n t h e -o r y tha t c o u l d s o l ve " the paradoxes o f c o n f i r m a t i o n " d i s c o v e r e d by C a r l G. Hempel. In Pa r t One I t r y ma in ly to do the t h r e e t h i n g s , ( i ) i n t r o d u c e the fundamenta ls of Hempel ' s t heo ry 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 as the common background f o r subsequent d i s c u s s i o n s , ( i i ) rev iew the major v iews of the paradoxes of c o n f i r m a t i o n , ( i i i ) p re sen t a new v iew, which i s more r a d i c a l than o the r known v iews, and argue tha t a s o l u t i o n to the paradoxes of c o n f i r m a t i o n may r e q u i r e a change of l o g i c . In Pa r t Two I c o n s t r u c t a number of p rom i s i n g t h r e e - v a l u e d l o -g i c s . I employ these " q u a s i c o n f i r m a t i o n l o g i c s " as the under -l y i n g l o g i c s of some new c o n f i r m a t i o n t h e o r i e s which, I had hoped, would s o l ve the paradoxes o f c o n f i r m a t i o n . I c o n s i d e r t h r e e - v a l u e d l o g i c s i n s t e a d o f any o the r many-valued l o g i c s as the u n d e r l y i n g l o g i c f o r any p romi s i ng c o n f i r m a t i o n t h e o r y , because I b e l i e v e t h a t t he re i s some i n t i m a t e r e l a t i o n s h i p o r , even, a one - to -one correspondence between the ( c o n t r o v e r s i a l ) t h r e e t r u t h - v a l u e s o f " t r u t h " , " f a l s i t y " and " n e i t h e r t r u t h nor f a l s i t y " and, r e s p e c t i v e l y , the ( n o n - c o n t r o v e r s i a l ) th ree c o n f i r m a t i o n - s t a t u s e s of " c o n f i r m a t i o n " , " d i s c o n f i r m a t i o n " and " n e u t r a l i t y " . U n f o r t u n a t e l y , these t h e o r i e s were found to be s e m a n t i c a l l y i nadequa te . T h i s became c l e a r a f t e r a complete semant ics f o r them had been d e v e l o p e d . i i i Thus, one negative r e s u l t of Part Two i s that our s y n t a c t i c a l approach to conf i rmation theory i s wrong from the very beginning. However, from t h i s negative r e s u l t we lea rn a pos i t i ve l e s s o n : a semantical approach i s more fundamental and dec is i ve than a syn-t a c t i c a l one, at l e a s t th i s i s so fo r const ruct ing an adequate the-ory of conf i rmat ion . It i s rewarding to note that the three-valued semantics worked out i n Part Two i s simple, complete and the f i r s t of i t s k i n d . In f a c t , the new three-valued semantics i s i n the s p i r i t of Frege, a l -though the l i n e of thought i s much neglected (even by Frege h imse l f ) . In Part Three I s h i f t the search fo r a conf i rmat ion log i c and an adequate theory of conf i rmat ion from a s y n t a c t i c a l to a seman-t i c a l approach because of the lesson learned i n Part Two. Af ter a systematic search through several promising three-valued log i cs I come, at l a s t , to a p laus ib le conf i rmat ion log ic and to a conf i rmat ion theory that could solve a l l known paradoxes of con-f i r m a t i o n . The promising three-va lued conf i rmat ion theory i s c a l l e d "the i n t e r n a l conf i rmat ion theory" . In Part Four I review and appraise the adequacy condi t ions l a i d down by Hempel as the necessary condi t ions fo r any adequate c o n f i r -mation theory . Under the c r i t i c i s m s of Carnap, Goodman and, espe-c i a l l y , with the help of Hanen's thorough s tud ies , I come to a l -most an i d e n t i c a l conclus ion to Hanen's J we should not impose a p r i - o r i i n a theory of q u a l i t a t i v e conf i rmat ion any adequacy condit ions l a i d down by Hempel except perhaps the Entailment Condi t ion, a l -though the i n t e r n a l conf i rmat ion theory a lso adopts the Equivalence Condit ion fo r some i n t r i n s i c reasons. In the l a s t Part F ive I t ry to appraise the three most impor-t a n t c o n f i r m a t i o n t h e o r i e s d i s c u s s e d and/or c o n s t r u c t e d i n t h i s d i s s e r t a t i o n . They are Hempel ' s t h e o r y 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 t heo ry of 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 c o n f i r m a t i o n t h e o r y . A f t e r some more v i g o r o u s c r i t i c i s m s are made and some new p a -radoxes o f c o n f i r m a t i o n are unexpec ted l y d e r i v e d i n both the theo ry 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 c o n f i r m a t i o n t h e o r y , I a r r i v e a t , perhaps r e l u c t a n t l y , t h i s more rea sonab le c o n c l u s i o n t u n -der the p re sen t s i t u a t i o n when t h e r e i s no obv ious way to overcome the new d i f f i c u l t i e s the be s t t h i n g t h a t we can do i s to d i s s o l v e ( i . e . to l i v e w i th ) a l l new and o l d paradoxes of c o n f i r m a t i o n , f o r Hempel may be a f t e r a l l r i g h t 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 o therwi se i s to have p sycho -l o g i c a l i l l u s i o n s as Hempel says . 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 cal led the attention of the phi -losophical community . . . " (Hintikka [19&9) p.2*0 TABLE OF CONTENTS Page INTRODUCTION 1 PART I. HEMPEL * S THEORY OF CONFIRMATION, DIFFERENT VIEWS OF THE PARADOXES OF CONFIRMATION, AND THE CLASSICAL LOGIC 4 1. Why Have a Theory of Confirmation 5 2. Nicod's C r i t e r i o n 6 3. Hempel's Cr i t i c i s m s of Nicod's C r i t e r i o n 8 4. The Equivalence Condition 10 5. Fundamentals of Hempel*s Theory of Qualitative Confirma-t i o n 12 6. The Paradoxes of Confirmation 18 7. Max Black's Diagnosis of the Paradoxes of Confirmation 20 8. Von Wright's Treatment of the Paradoxes of Confirmation 23 9* The Bayesian Treatment of the Paradoxes of Confirmation 27 10. Quine's Solution of the Paradoxes of Confirmation 30 11. Goodman's and Scheffler's Concept of Selective Confirma-t i o n and the Paradoxes of Confirmation, and Grandy's Re-lated Discussion 33 12. Armstrong's View of the Paradoxes of Confirmation 40 13. Some Other Proposed Solutions of the Paradoxes of Confir-mation ^2 14. Hempel*s View of the Paradoxes of Confirmation ^ 15. A New View of the Paradoxes of Confirmation 48 16. Challenges to the C l a s s i c a l Logic 5 2 17. The Plausible Inadequacy of the C l a s s i c a l Logic 56 18. An Axiomatic Review of the C l a s s i c a l Sentential Logic 59 v i i PART I I . A SYNTACTIC AND FORMALLY SEMANTIC APPROACH TO THREE-VALUED CONFIRMATION THEORIES, AND A THREE-VALUED SEMANTICS 63 1. A B r i e f Introduction 64 2. A D e f i n i t i o n of "Confirmation Logic" 65 3. An Example of Minimal Confirmation Logic 67 4. The Axiomatization of MC3 71 5. In Search of more Minimal Confirmation Logics 72 6. Neo-classical Minimal Confirmation Logics 77 7. Quasi Confirmation Logics and Quasi Confirmation Theories 79 8. Some Properties of the Quasi Confirmation Logics 81 9. Some more Properties of the Quasi Confirmation Logics 84 10. The Axiomatization of a Truth-functionally Complete Quasi (Sentential) Confirmation Logic 86 11. Confirmational Valuation and a Complete Semantics f o r any Three-valued Quantificational Logic with Identity 88 12. The Failur e of the 20 Quasi-confirmation Logics to Sati s -f y a Monotonicity Condition 94 PART I I I . A CONFIRMATION LOGIC AND A PLAUSIBLE SOLUTION OF THE PARADOXES OF CONFIRMATION 98 1. A B r i e f 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 Axiomatization of L^ 102 4. Some Properties of the Three-valued Sentential Logic L^ 104 5. A Quasi-Hempelean External Confirmation Theory 107 6. Paradoxes of Confirmation Regained 108 7. An Analysis of the Fa i l u r e 110 8. Conditional Law vs. Implicational Law 112 v i i i 9. In Search of an Implication I 114 10. In Search of an Implication II 118 11. J u s t i f i c a t i o n of the Truth Rule of the Implication I 121 12. J u s t i f i c a t i o n of the Truth Rule of the Implication II 123 13. A Long Way to Reach the Truth Rule of the Implication 126 14. A B r i e f View of C3 and QC3 129 15. Elementary Internal Confirmation Theory 132 16. The General Internal Confirmation Theory 135 17. A Quasi-solution of the Paradoxes of Confirmation 136 18. Some Cr i t i c i s m s of the Quasi-solution of the Paradoxes of Confirmation 138 19. A Plausible Solution of the Paradoxes of Confirmation 140 PART IV. ADEQUACY CONDITIONS FOR CONFIRMATION AND THE GOODMAN PARADOX 145 1. Adequacy Conditions f o r Confirmation Laid down by Hem-pel 146 2. The Converse Consequence Condition and i t s Rejection 148 3. Carnap and the Consequence Condition 150 4. E x p l i c a t i o n of the Concept of Confirmation 152 5. Carnap and the Conjunction Condition 155 6. Goodman^s Paradox and the Conjunction Condition 157 7. Carnap 0s Solution of the Goodman Paradox 160 8. Salmon's Solution of the Goodman Paradox l 6 l 9. An Attempted Solution of the Goodman Paradox 163 10. The Goodman Paradox Regained 166 11. A Tentative Conclusion^of the V a l i d i t y of the Adequacy Conditions f o r Confirmation i n Light of Marsha Hanen's Study I69 ix PART V. TENTATIVE CONCLUSIONS 183 1. Introduct ion 184 2. A Summary of Hempel's Theory of Confirmation 185 3. A Summary of Goodman's and S c h e f f l e r ' s Theory of Se lec-t i v e Confirmation 188 4. A Tentat ive Conclusion for the Internal Confirmation Theory 192. BIBLIOGRAPHY 2 0 0 Acknowledgement Thanks are due to Prof . John P. Stewart who as my research superv isor has very thoroughly c r i t i c i z e d the whole content i n many long and short d iscuss ions i n the past few years and to Pro f . Howard Jackson and Prof . Thomas Patton who have very c a r e -f u l l y read some vers ions of my d ra f ts and given many valuable c r i t i c i s m s and suggestions, e s p e c i a l l y about i t s l o g i c a l and semantical aspects . Thanks are a lso due to Prof . Richard E . Robinson who discovered a ser ious mistake made i n the s e m i - f i n a l d ra f t of t h i s d i s s e r t a t i o n . This d i s s e r t a t i o n i s dedicated to my parents and my brothers and s i s t e r s . Without the i r great support I don' t know I would have enough patience to f i n i s h i t . 1 INTRODUCTION The m o t i v a t i o n o f t h i s d i s s e r t a t i o n was to c o n s t r u c t an ade -quate c o n f i r m a t i o n theo ry t h a t c o u l d so l ve " the paradoxes o f c o n -f i r m a t i o n " d i s c o v e r e d by C a r l G. Hempel. In P a r t One I t r y ma in ly t o do the t h r e e t h i n g s * ( i ) i n t r o d u c e the fundamenta ls o f Hempel ' s t h e o r y 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 as the common background f o r subsequent d i s c u s s i o n s , ( i i ) r ev iew the major v iews o f the paradoxes o f c o n f i r m a t i o n , ( i i i ) p r e s e n t a new v iew, which i s more r a d i c a l than o t h e r known v iews , and argue t h a t a s o l u t i o n to the paradoxes o f c o n f i r m a t i o n may r e q u i r e a change o f l o g i c . In P a r t Two I c o n s t r u c t a number of p r o m i s i n g t h r e e - v a l u e d l o g i c s . I employ these " q u a s i c o n f i r m a t i o n l o g i c s " as the u n -d e r l y i n g l o g i c s of some new c o n f i r m a t i o n t h e o r i e s wh ich , I had hoped, would so l ve the paradoxes o f c o n f i r m a t i o n . I t u r n to t h r e e - v a l u e d l o g i c s i n s t e a d of any o t h e r many-valued l o g i c s as the u n d e r l y i n g l o g i c f o r any p r o m i s i n g c o n f i r m a t i o n t h e o r y t h a t can s o l ve the paradoxes o f c o n f i r m a t i o n , ma in l y becuase I b e l i e v e t h a t there i s some i n t i m a t e r e l a t i o n s h i p o r , even , a one - to -one correspondence between the ( c o n t r o v e r s i a l ) t h r e e t r u t h - v a l u e s of " t r u t h " , " f a l s i t y " and " n e i t h e r t r u t h nor f a l s i t y " and, r e s p e c -t i v e l y , the ( n o n - c o n t r o v e r s i a l ) t h r e e c o n f i r m a t i o n - s t a t u s e s o f " c o n f i r m a t i o n " , " d i s c o n f i r m a t i o n " and " n e u t r a l i t y " . U n f o r t u n a t e l y , these t h e o r i e s were found to be s e m a n t i c a l l y i n adequa te . T h i s became c l e a r a f t e r a complete semant ics f o r them had been d e v e l o p e d . 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 s y n t a c t i c a l 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 . 2 However, from t h i s nega t i ve r e s u l t we l e a r n a p o s i t i v e l e s s o n : a s e m a n t i c a l approach i s more fundamenta l and d e c i s i v e than a s y n -t a c t i c a l approach, a t l e a s t t h i s i s so f o r c o n s t r u c t i n g an ade-quate t h e o r y of c o n f i r m a t i o n . I t i s reward ing to note t h a t the t h r e e - v a l u e d semant ics worked out i n t h i s Pa r t i s s imp le , complete and the f i r s t of i t s k i n d . In Pa r t Three I s h i f t the sea rch f o r a c o n f i r m a t i o n l o g i c and a c o n f i r m a t i o n theo ry tha t can s o l ve the paradoxes 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 to a s e m a n t i c a l approach because of the l e s s o n l e a r n e d i n Pa r t Two. A f t e r a sy s temat i c search through s e v e r a l p rom i s i ng t h r e e - v a l u e d l o g i c I come, a t l a s t , 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 t h e o r y tha t c o u l d s o l ve the known paradoxes of c o n f i r -ma t i on . The p rom 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 t h e o r y i s c a l l e d " 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 " . In Pa r t Four I rev iew and a p p r a i s e the adequacy c o n d i t i o n s l a i d 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 any adequate c o n f i r -mat ion t h e o r y . Under the c r i t i c i s m s of Carnap, Goodman and, e spe -c i a l l y , w i t h the h e l p of Hanen 's thorough s t u d i e s , I come to a lmost an i d e n t i c a l c o n c l u s i o n to Hanen ' s : we shou ld not impose a p r i o r i i n 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 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 , a l 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 C o n d i t i o n f o r some i n t r i n s i c r e a s o n s . In the l a s t Pa r t F i v e I t r y t o a p p r a i s e the th ree most impor tant c o n f i r m a t i o n t h e o r i e s d i s c u s s e d and /o r c o n s t r u c t e d i n t h i s d i s s e r t a -t i o n . They are Hempel*s t h e o r y 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 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 and the i n t e r n a l c o n f i r m a t i o n 3 theory. A f t e r some more v i g o r o u s c r i t i i c s m s are made and some new para-doxes of c o n f i r m a t i o n are unexpectedly d e r i v e d i n both the theory of 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 c o n f i r m a t i o n theory, I come t o , perhaps r e l u c t a n t l y , t h i s more reasonable c o n c l u s i o n : under the present s i t u a t i o n when there i s no obvious way to overcome the new d i f f i c u l t i e s the best t h i n g t h a t we can do i s to d i s s o l v e ( i . e . to l i v e with) a l l new and o l d paradoxes of c o n f i r m a t i o n , f o r Hempel may be a t f e r a l l r i g h t 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 i s to have p s y c h o l o g i c a l i l l u s i o n s . 4 PART I. HEMPEL1S THEORY OF CONFIRMATION, DIFFERENT VIEWS OF THE PARADOXES OF CONFIRMATION, AND THE CLASSICAL LOGIC 5 1. WHY HAVE A THEORY OF CONFIRMATION We need a t h e o r y o f c o n f i r m a t i o n f o r the f o l l o w i n g r e a s o n s i 1 (A) The problem of q u a l i t a t i v e c o n f i r m a t i o n . In e m p i r i c a l s c i e n c e we o f t e n r e f e r to the e p i s t e m i c s t a t u s o f a h y p o t h e s i s (or o f a t h e o r y ) . We say t h a t we can c o n f i r m or d i s c o n f i r m a h y p o t h e s i s (or a t h e o r y ) . So, an adequate t h e o r y of c o n f i r m a -t i o n must, f i r s t o f a l l , g i v e t h r e e p r e c i s e q u a l i t a t i v e d e f i n i -t i o n s « of c o n f i r m a t i o n , d i s c o n f i r m a t i o n and n e u t r a l i t y . (B) The problem of comparat i ve c o n f i r m a t i o n . Sometimes i n e m p i r i c a l s c i e n c e we compare hypotheses (or t h e o r i e s ) w i t h r e s -pec t to t h e i r r e l a t i v e s t r e n g t h o f c o n f i r m a t i o n . How to o b t a i n t h i s o r d e r i n g i s the problem of comparat ive c o n f i r m a t i o n . (C) The problem 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 . In e m p i r i -c a l s c i e n c e we a s k i to what degree does ev idence c o n f i r m a hypo-t h e s i s ? How to o b t a i n t h i s measure i s the problem 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 . So, t h e r e a re a t l e a s t t h r e e fundamenta l problems i n e m p i r i -c a l s c i e n c e , as w e l l as i n p h i l o s o p h y o f s c i e n c e , whose s o l u t i o n would f o l l o w from an (adequate) t h e o r y of c o n f i r m a t i o n . However, the main conce rn o f t h i s d i s s e r t a t i o n w i l l be an a d e -quate t h e o r y 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 , f o r the rea son t h a t we must have a c l e a r i d e 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 b e f o r e we can t a c k l e the problems o f q u a n t i t a t i v e c o n f i r m a t i o n and o f compara-t i v e c o n f i r m a t i o n ; j u s t as we must have a c l e a r i d e a of " t empera -t u r e " (or "warmth") b e f o r e we can t a l k about "warmer" and "degree o f t e m p e r a t u r e " . 1. C f . C a r l G.Hempelt " S t u d i e s i n the l o g i c o f c o n f i r m a t i o n " , s e c -t i o n 1, i n h i s A spec 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 . 3 f f . 6 2. NICOD'S CRITERION T h i s d i s s e r t a t i o n w i l l be ma in l y c o n c e r n e d w i t h the i d e a o f c o n f i r m a t i o n as deve l oped by Hempel . S i n c e J e a n N i c o d * s i d e a o f c o n f i r m a t i o n i n the c a s e o f u n i v e r s a l c o n d i t i o n a l s e n t e n c e s p l a y s an i m p o r t a n t r o l e i n the deve lopment o f Hempel*s t h e o r y o f c o n f i r m a t i o n , we w i l l s t a r t a r e v i e w o f N i c o d ' s i d e a o f c o n -f i r m a t i o n on t h i s p o i n t b e f o r e we t u r n t o Hempe l * s . N i c o d ' s i d e a o f c o n f i r m a t i o n about u n i v e r s a l c o n d i t i o n a l sen tences i s e x p r e s s e d i n the f o l l o w i n g wayt^" C o n s i d e r the f o r m u l a o r the law* 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 i t s p r o b a b i l i t y ? I f t h i s f a c t c o n s i s t s o f the p re sence o f B i n ca se o f A, i t i s f a v o r a b l e t o the law 'A e n t a i l s B_*; on the c o n t r a r y , i f i t c o n s i s t s o f the absence o f B i n a ca se 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 . I t i s c o n -c e i v a b l e t h a t we have he re 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, the 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 the 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 laws would o p e r a t e by 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 wh ich we s h a l l c a l l c o n f i r m a t i o n and i n v a l i d a t i o n . L e t us w r i t e a u n i v e r s a l c o n d i t i o n a l s en tence i n i t s s i m p l e s t form as f o l l o w s » (x) (Px =>Qx). Then, N i c o d ' s i d e a o f c o n f i r m a t i o n about u n i v e r s a l c o n d i t i o n a l 2 sen tences e x p r e s s e d above i s summed up by Hempel as belowt 1. "An 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 the a n t e c e d e n t ( h e r e « * P x * ) and the consequent ( 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 the 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 the a n t e c e d e n t , bu t no t the c o n s e q u e n t o f 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 Geometry and I n d u c t i o n ( t r a n s , by P.P. W i e n e r ) , 1930, p . 2 1 9 ; a l s o quo ted by Hempel i n h i s A spec t s o f S c i e n t i f i c E x p l a n a t i o n , p . 1 0 . 2. H e m p e l , o p . c i t . , p . 1 1 . 7 3 . "and (we add t h i s to N i c o d ' s s ta tement) 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 i d e a o f c o n f i r m a t i o n , d i s c o n f i r m a t i o n and n e u t r a l i t y about u n i v e r s a l c o n d i t i o n a l sentences expres sed by N i c o d i s 3 c a l l e d " N i c o d ' s c r i t e r i o n " by Hempel . A c c o r d i n g to Hempel , i t s t a t e s e x p l i c i t l y what i s perhaps the most common i n t e r p r e t a t i o n o f the concept o f c o n f i r m a t i o n . However, Hempel has some i m p o r t a n t c r i t i c i s m s o f N i c o d ' s c r i t e r i o n to be g i v e n i n the next s e c t i o n . 3 . Hempel , op . c i t . , p . 1 1 . 3. HEMPEL 1 S CRITICISMS OF NICOD'S CRITERION Hempel t h i n k s t h a t N i c o d ' s c r i t e r i o n , w h i l e seemingly p l a u -s i b l e , s u f f e r s from the f o l l o w i n g two s e r i o u s shortcomings:"^ (I) F i r s t , the a p p l i c a b i l i t y o f the c r i t e r i o n i s too r e s -t r i c t e d . I t a p p l i e s o n l y to hypotheses o f u n i v e r s a l c o n d i -t i o n a l form and i t p r o v i d e s no s t a n d a r d s o f c o n f i r m a t i o n , say , f o r e x i s t e n t i a l h y p o t h e s e s . Nor does i t p r o v i d e any s t a n d a r d o f c o n f i r m a t i o n f o r any sentence c o n t a i n i n g mixed q u a n t i f i e r s such as would be needed to f o r m u l a t e the f o l l o w i n g s en tence : H : You can f o o l a l l o f the people some of the t ime and some o f the people a l l o f the t i m e , but you c a n -not f o o l a l l of the people a l l o f the t i m e . 2 O r , i n symbols , a c c o r d i n g to Hempeli H i ( x ) ( 3 t ) F x t . ( 3 x ) ( t ) F x t . - ( x ) ( t ) F x t where, F x t i you c a n f o o l p e r s o n x a t t ime t . In s h o r t , Hempel says t h a t t h e r e a r e many hypotheses o t h e r t h a n u n i v e r s a l c o n d i t i o n a l ones whose s t a n d a r d s o f c o n f i r m a t i o n are not c o v e r e d by N i c o d ' s c r i t e r i o n . I t i s obv ious t h a t an adequate account o f c o n f i r m a t i o n would a p p l y to any form o f h y p o t h e s e s . Thus N i c o d ' s c r i t e r i o n i s too r e s t r i c t e d . ( I I ) The second shor tcoming of N i c o d ' s c r i t e r i o n i s t h a t i t makes c o n f i r m a t i o n depend not o n l y on the c o n t e n t o f the h y p o t h e s i s but a l s o on i t s f o r m u l a t i o n . To see t h i s , c o n s i d e r the f o l l o w i n g t h r e e hypotheses : 1. Hempel: A s p e c t s o f S c i e n t i f i c E x p l a n a t i o n , p . l l f f . 2 . I b i d . . p . 1 2 . 9 HI i ( x ) ( R x O B x ) , H2t ( x ) ( -Bxr> -Rx) , H 3 « (x ) (Rx. -Bx:>Rx. -Rx) , where Rt @ i s a r a ven , B i ® i s b l a c k . Then, l e t us make the f o l l o w i n g a b b r e v i a t i o n s . a i a b l a c k r a v e n , bi a non -b l ack r a v e n , c i a b l a c k non - r aven , d» a n o n - b l a c k n o n - r a v e n . Now a c c o r d i n g to N i c o d 1 s c r i t e r i o n we would have the f o l l o w i n g r e s u l t s * A l though we h a v e ; ( i ) o b j e c t b d i s c o n f i r m s HI, H2 and H3i and ( i i ) o b j e c t c i s n e u t r a l w i t h r e s p e c t to HI , H2 and H3 , we a l s o have i ( i i i ) o b j e c t a c o n f i r m s HI but i s n e u t r a l w i t h r e s p e c t t o H2 and to H3» and ( i v ) o b j e c t d c o n f i r m s H2 but i s n e u t r a l to HI and to H3, and (v) no o b j e c t whatever can c o n f i r m H3 . And y e t HI, H2 and H3 a re l o g i c a l l y e q u i v a l e n t s e n t e n c e s ! Thus we have the second shor tcoming o f N i c o d ' s c r i t e r i o n ; t h a t c o n -f i r m a t i o n depends not o n l y on the con ten t o f a h y p o t h e s i s but a l s o on i t s f o r m u l a t i o n "which appears ab su rd "^a s Hempel s ay s . From the above r e s u l t s Hempel conc lude s t h a t N i c o d ' s c r i t e r i -on i s inadequate as a s tandard o f c o n f i r m a t i o n . 3. Hempelt A spec 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 .13 . 4. THE EQUIVALENCE CONDITION In c r i t i c i z i n g N i c o d ' s c r i t e r i o n t h a t makes the c o n f i r m a -t i o n o f a h y p o t h e s i s depend not o n l y on the c o n t e n t o f the hypo-t h e s i s but a l s o on i t s f o r m u l a t i o n t h e r e i s an i m p o r t a n t p r i n -c i p l e t a c i t l y assumed by Hempel . The p r i n c i p l e i s the f o l l o w i n g condi t ion:^" E q u i v a l e n c e c o n d i t i o n . "Whatever c o n f i r m s ( d i s c o n f i r m s ) one o f two C l o g i c a l l y ] e q u i v a l e n t s en tences , a l s o c o n f i r m s ( d i s c o n f i r m s ) the o t h e r . " Only w i t h the h e l p o f t h i s e q u i v a l e n c e c o n d i t i o n i s the 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 ) of a h y p o t h e s i s independent of i t s f o r m u l a t i o n . So, i t i s n i c e to l a y i t down e x p l i c i t l y . Hempel c a l l s a t t e n t i o n to the above e q u i v a l e n c e c o n d i t i o n and t h i n k s t h a t the e q u i v a l e n c e c o n d i t i o n i s "a n e c e s s a r y 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 . " 2 (my i t a l i c s ) There are two reasons g i v e n by Hempel to suppor t the neces -s i t y o f 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 s t r u c t i n g an adequate t h e o r y o f c o n f i r m a t i o n . The two reasons aret (a) F i r s t , f u l f i l l m e n t o f the e q u i v a l e n c e c o n d i t i o n makes the c o n f i r m a t i o n o f a h y p o t h e s i s independent o f i t s f o r m u l a t i o n . O t h e r w i s e , a c c o r d i n g to Hempe l ,^ the q u e s t i o n as to whether 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 would have to be answered by s a y i n g i "that depends on which of the d i f f e r e n t e q u i v a l e n t f o r m u l a t i o n s o f the h y p o t h e s i s i s c o n s i d e r e d " 1. Hempel . A s p e c t s o f S c i e n t i f i c E x p l a n a t i o n , p . 1 3 . 2. I b i d . , p . 1 3 . 3. I b i d . , p . 1 3 . 11 which appears a b s u r d . (b) The second rea son i s t h a t i t does j u s t i c e to s c i e n t i f i c p r a c t i c e , e s p e c i a l l y when hypotheses are used f o r purposes o f e x p l a n a t i o n o r p r e d i c t i o n . In t h i s f u n c t i o n they serve as p r e -mises i n a d e d u c t i v e argument whose c o n c l u s i o n i s a d e s c r i p t i o n k of the event to be e x p l a i n e d or p r e d i c t e d . And we know t h a t 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 to the l a t t e r , a d e d u c t i o n which i s v a l i d w i l l remain so i f some or a l l of the premises are 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 s t a tement s ; 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 r e a s o n i n g 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 o f t h e i r e q u i v a l e n t f o r m u l a -t i o n s a re most conven ien t f o r the development of h i s c o n c l u s i o n s . (my i t a l i c s ) S ince 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 c r u c i a l as w e l l as a c o n t r o v e r s i a l r o l e i n the development o f a l l major t h e o r i e s of c o n f i r m a t i o n , we w i l l l a t e r t r y t o a s se s s whether i t i s a good adequacy c o n d i t i o n , a f t e r we have c o n s i d e r e d the major t h e o r i e s o f c o n f i r m a t i o n . 4. I b i d . . p .13 . 12 5. FUNDAMENTALS OF HEMPEL'S THEORY OF QUALITATIVE CONFIRMA-TION I t i s t ime to i n t r o d u c e Hempel*s t h e o r y 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 and examine i t s impor tance, i t s comprehen-s i o n and i t s p romi se . However, I w i l l touch o n l y the more fundamenta l p a r t s o f i t , which c o n s i s t of the f o l l o w i n g th ree p a r t s i ( I) The c l a s s i c a l two-va lued q u a n t i f i c a t i o n a l l o g i c , ( de -s i gna ted as QC2} as i t s u n d e r l y i n g l o g i c . ( I I ) Seven b a s i c concept s and one adequacy c o n d i t i o n , f o r m i n g the t h e o r y p r o p e r . ( I I I ) The development o f the t h e o r y , i n c l u d i n g some r e l a t e d concep t s and some a d d i t i o n a l adequacy c o n d i t i o n s . In t h i s s e c t i o n we are go ing to d e a l o n l y w i th the seven b a s i c concept s and the one adequacy c o n d i t i o n , t a k i n g QC2 f o r g r a n t e d . The seven b a s i c concept s and the one adequacy c o n d i t i o n a r e : ( i ) H y p o t h e s i s , ( i i ) development of a h y p o t h e s i s , ( i i i ) o b s e r v a t i o n r e p o r t , ( i v ) d i r e c t c o n f i r m a t i o n , (v) c o n f i r m a t i o n , ( v i ) d i s c o n f i r m a t i o n , ( v i i ) n e u t r a l i t y , and ( v i i i ) the e q u i v a -l e n c e c o n d i t i o n . They are d e f i n e d or e x p l a i n e d by Hempel i n the f o l l o w i n g wayi^" H y p o t h e s i s . "We s h a l l . . . unders tand by a h y p o t h e s i s any sentence which can be expressed i n the assumed language of s c i e n c e , no mat te r whether i t i s a g e n e r a l i z e d sen tence , 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 to a f i n i t e number of p a r t i c u l a r o b j e c t s . " 1. C f . Hempel: A spec t s of S c i e n t i f i c E x p l a n a t i o n , pp.2 2 - 3 7 . 1 3 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, can he d e f i n e d p r e c i s e l y by r e c u r s i o n ; hence i t w i l l s u f f i c e to say tha t the development of H f o r C s t a t e s what H would a s s e r t i f t he re 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 e lements o f C . " Thus, e . g . , the development of the h y p o t h e s i s , H i t ( x ) ( F x v G x ) , f o r the c l a s s / a , b} i s : ( F a v G a ) . ( F b v Gb); and the development of the h y p o t h e s i s , H2: (3y )Gy, f o r the c l a s s fb,c} i s : Gbv Gc. O b s e r v a t i o n Repor t . "An 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 as a f i n i t e c l a s s (or a c o n j u n c t i o n o f a f i n i t e number) of o b s e r v a t i o n sen tences ; and an o b s e r v a t i o n sentence as a s e n -tence which e i t h e r a s s e r t s or den ie 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 ob se rvab le 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 s tand i n a c e r t a i n obse rvab le r e l a t i o n . " Or, e q u i v a l e n t l y , " o b s e r v a t i o n sentences are d e f i n e d as s e n -tences c o n t a i n i n g no q u a n t i f i e r s . " 2 D i r e c t C o n f i r m a t i o n . "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 the development of 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 which are mentioned e s s e n t i a l l y i n B" means t h a t the o b j e c t s are mentioned i n a n o n - a n a l y t i c component of B."3 Thus l e t ' s take one example from Hempel ' s to i l l u s t r a t e the above p o i n t o f " 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 the f o l -l ow ing h y p o t h e s i s : H3« (x)Px, and the two o b s e r v a t i o n r e p o r t s . B I : Pa, 2. I b i d . Note 4 6 , p p . 3 7 - 3 8 . 3 . I b i d , Note 4 6 , p . 3 8 . 1** B 2 : Pa . (Pb v - P b ) . Now, by d e f i n i t i o n , B l d i r e c t l y con f i rms H3, f o r we have* (1) Pa II-Pa, where " )(•" i s " t he semantic e n t a i l m e n t r e l a t i o n " , i . e . we have: B l e n t a i l s Pa, where Pa i s the development of H 3 f o r the c l a s s of those o b j e c t s mentioned (as w e l l as e s s e n t i a l l y mentioned) i n B l . But B 2 would not d i r e c t l y c o n f i r m H 3 i f we n e g l e c t e d the p a r t of " e s s e n t i a l l y ment ioned" p l a yed i n the d e f i n i t i o n of d i r e c t c o n f i r m a t i o n , f o r B 2 does not e n t a i l Pa.Pb, i . e . we do not have: ( 2 ) Pa . (Pb v -Pb) IP Pa . Pb, where Pa.Pb i s the development of H3 f o r the c l a s s of those o b j e c t s mentioned however, not e s s e n t i a l l y mentioned i n B 2 . But B l and B 2 a re l o g i c a l l y e q u i v a l e n t . T h i s 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 d i s c u s s e d e a r l i e r . T h i s d e f i c i e n c y i s remedied by the q u a l i f i c a t i o n of "es-s e n t i a l ment ion " i n the d e f i n i t i o n of d i r e c t c o n f i r m a t i o n , f o r " P b v - P b " o f B 2 i s an a n a l y t i c component of B 2 . So, the " b " does not appear , by d e f i n i t i o n , " e s s e n t i a l l y " t h e r e . Hence, B 2 i s n o t , so to speak, a " p r o p e r " o b s e r v a t i o n r e p o r t . Thus no v i o l a t i o n of the e q u i v a l e n c e c o n d i t i o n occur s h e r e , once we have the q u a l i f i c a t i o n o f " e s s e n t i a l ment ion " i n the d e f i n i t i o n o f d i r e c t c o n f i r m a t i o n about the o b j e c t s appea r i n g i n the d e -velopment o f a h y p o t h e s i s . C o n f i r m a t i o n . "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-thes i s H i f H i s enta i led 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." D isconf i rmat ion . "An observation report B disconf i rms a hypothesis H i f i t confirms the denia l of H." k Neut ra l i t y . "An observation report B i s neut ra l with respect to a hypothesis H i f B nei ther confirms nor disconf i rms H." Equivalence Condi t ion . "Whatever confirms (disconfirms) one of two ( logical ly" ) equivalent sentences, a lso confirms (disconfirms) the other . " Below are some examples^ taken from Hempel to i l l u s t r a t e some of the above newly defined concepts. Example 1. The development of the hypothesis H4J (x) (PxvQx) fo r the c l a s s -fa.bj- i s : (Pa v Qa). (Pb v Qb). The observation r e -port B3". Pa.Pb d i r e c t l y confirms the development of H4 f o r the c l a s s {a,b},i.e. we havei (3) (Pa.Pb) |K (Pav Qa) . (Pbv Qb)). Because of ( 3 ) , B3 a lso confirms H*K Example 2. The development of the hypothesis H5» Pc v Qc i s , by d e f i n i t i o n , that hypothesis i t s e l f . Now the observation report B3 of the above example does not d i r e c t l y confirms H 5 t because i t s development, v i z . H5 i t s e l f , i s not enta i led by B3, 4. Hempel c a l l s " n e u t r a l i t y " sometimes a lso as " i r re levancy" , c f . i b i d , p .5. p.6 and p.11. 5. Ib id , p p .36-37. However, some mispr ints contained there have been corrected here. 16 i . e . we do not have. (4) (Pa.Pb) \\- (Pc v Qc ) . However, H4 e n t a i l s H5» i . e . , we have. (5) ( x ) ( P x v Qx) ||-(Pc v Qc ) . Hence, by d e f i n i t i o n of c o n f i r m a t i o n , B3 c o n f i r m s H5» s i n c e B3 d i r e c t l y con f i rms H4. F i n a l l y i n t h i s s e c t i o n we have an impor tant o b s e r v a t i o n to make. Note t h a t i n a l l the examples g i v e n above we have i n t e r -p r e t e d Hempel ' s concept of " e n t a i l m e n t " r e l a t i o n by "|f-". which i s the " semant ic e n t a i l m e n t " r e l a t i o n of the c l a s s i c a l l o g i c . But i s n ' t i t t r u e t h a t Hempel ' s t heo ry of c o n f i r m a t i o n i s framed i n terms o f syntax? I f so, how can we use a semantic concept to c a p t u r e h i s i d e a ? I t h i n k the j u s t i f i c a t i o n i s t h i s . F i r s t , i n the c l a s s i c a l two-va lued q u a n t i f i c a t i o n a l l o g i c QC2 we have the f o l l o w i n g two theorems t Completeness Theorem. I f A 1|-B, then A |-B, (where " (-" i s the " s y n t a c t i c consequence" r e l a t i o n of QC2.) Soundness Theorem. I f A |-B, then A ||"B. So, t o g e t h e r we have the f o l l o w i n g theorem. Correspondence Theorem. A |-B i f f A|(-B. So, by the above theorem we know t h a t the semantic e n t a i l -ment r e l a t i o n and the s y n t a c t i c consequence r e l a t i o n o f the c l a s s i c a l 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 . Hence, a l though my i n t e r p r e t a t i o n o f Hempel ' s i d e a of " e n t a i l m e n t " i s i n terms of the semantic e n t a i l m e n t , i t c an be swi tched to i t s c o r r e s -17 ponding syntactic consequence r e l a t i o n any time we l i k e . Furthermore, i n t h i s d i s s e r t a t i o n we give examples again and again, and i t i s easier to do t h i s i n semantical terms and, hence, i n terms of the semantic entailment r e l a t i o n . 6 . THE PARADOXES OF CONFIRMATION 18 One n i c e , neat r e s u l t of Hempel*s 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 i s t h a t i t d i v i d e s a l l p o s s i b l e hypotheses w i t h r e s p e c t to any 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 n t o one of the t h r e e m u t u a l l y e x c l u s i v e c l a s s e s : those which a re c o n f i r m e d , those which are d i s c o n f i r m e d and those which a re neutral."*" However, t h e r e are a l s o some c o u n t e r - i n t u i t i v e r e s u l t s t h a t can be d e r i v e d from Hempel*s t h e o r y . Fo r example, from the Raven Hypo the s i s , RH» A l l ravens are b l a c k , o r , i n symbols, RH: (x)(Rx=>Bx), we can d e r i v e the f o l l o w i n g r e s u l t s : (A) Any b l a c k r aven con f i rms the Raven H y p o t h e s i s ; (B) Any non -b l a ck raven d i s c o n f i r m s the Raven H y p o t h e s i s ; (C) Any non -b l a ck non- raven c o n f i r m s the Raven H y p o t h e s i s ; (D) Any th ing t h a t i s b l a c k or a non - raven a l s o c o n f i r m s the Raven H y p o t h e s i s ; and, i n p a r t i c u l a r , (D*) Any b l a c k non - raven c o n f i r m s the Raven H y p o t h e s i s . P r o o f . F i r s t , l e t ' s i n t r o d u c e the f o l l o w i n g names: a : a b l a c k r aven , b: a non -b l ack r a v e n , c i a non -b l ack non - r aven , d i an o b j e c t t h a t i s b l a c k or a n o n - r a v e n , d * : a b l a c k n o n - r a v e n . Second, note t h a t we have the f o l l o w i n g r e l a t i o n s : 1. C f . 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.3?T where, 19 ( i ) (Ra.Ba) ||- ( R a o B a ) ; ( i i ) (Rb.-Bb) ||--(Rb^Bb); ( i i i ) ( -Rc . -Bc) ||-(Rc r>Bc); ( i v ) ( - R d v B d ) ({-(Rd^Bd); and, i n p a r t i c u l a r , (v) ( -Rd*.Bd*) II-(Rd* P B d * ) . From the above f i v e r e s u l t s w i t h the h e l p of Hempel ' s d e f i n i t i o n s of 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 , we o b t a i n r e s p e c t i v e l y our d e s i r e d r e s u l t s o f (A) , (B), (C), (D) and (D*) . 'QED. The r e s u l t s (A) and (B) a re expec ted ; but the r e s u l t s (C) , (D) and (D*) are n o t - — a t l e a s t not expected by many. How c o u l d , one might ask, a wh i te swan, a b l u e eye , a p ink c a r , a b l a c k h o l e , or a red h e r r i n g , e t c . , which appear to have n o t h i n g to do w i t h ravens , c o n f i r m the Raven Hypo the s i s ? These seeming ly c o u n t e r - i n t u i t i v e r e s u l t s (and some r e l a t e d ones) are 2 c a l l e d " the paradoxes of c o n f i r m a t i o n " by Hempel. Hempel ' s c a l l i n g a t t e n t i o n to the paradoxes of c o n f i r m a t i o n and r e l a t e d problems has provoked an e x t e n s i v e s tudy and l i t e r a t u r e about them. Some t r y to s o l ve the paradoxes of c o n f i r m a t i o n ; a few t r y to d i s s o l v e them. In the next few s e c t i o n s we w i l l t r y to rev iew, d i agnose , and perhaps a l s o c r i t i c i z e , some of the most i n t e r e s t i n g and impor tant s o l u t i o n s and 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 . 2. I b i d . . p p . l ^ f f . 20 7. MAX BLACK'S DIAGNOSIS OF THE PARADOXES OF CONFIRMATION Perhaps i t i s wiser and more desirable to f i r s t diagnose the paradoxes of confirmation, before coming to any solution or d i s s o l u t i o n . With regard to a diagnosis of the paradoxes of confirmation, i t seems to me that Max Black has the most p e r t i -nent things to say. Thus i n t h i s section we devote ourselves to some of his more important ideas. Consider, again, the raven hypothesis that a l l ravens are black. According to Black common sense would hold thati ( i ) The existence of a black raven supports (or confirms) the raven hypothesis; ( i i ) The raven hypothesis i s shown to be f a l s e (or discon-firmed) by the existence of a single non-black raven; ( i i i ) Not a l l objects bear upon the raven hypothesis, p o s i -t i v e l y or negatively, i n t h i s way* f o r instance, the existence of Halley's comet neither supports (confirms) nor f a l s i f i e s ( d is-confirms) the raven hypothesis. In short, according to Black "the common sense p o s i t i o n i s that the existence of some, but not a l l , things i s relevant to the raven hypothesis." 1 This i s c a l l e d "the p r i n c i p l e of l i m -i t e d relevance" by Black. Thus i t follows from Black's terminology and discussion that 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 holds or advocates the p r i n c i p l e of l i m i t e d relevance. Moreover, there are three d i f f e r e n t types of the p r i n c i p l e of l i m i t e d relevance that can be d i f f e r e n t i a t e d , I think, when 1. Max Blackt "Notes on the "paradoxes of confirmation"" i n As- pects of Inductive Logic, (ed.) J . Hintikka and P. Suppes, 1966, p.175. 2. Ibid., p.175. i, 21 i t a p p l i e s to the r a v e n h y p o t h e s i s ! ( i i i . 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 n o n - r a v e n are n e u t r a l t o the r a v e n h y p o t h e s i s . ( i i i . b ) Any n o n - b l a c k n o n - r a v e n i s n e u t r a l to the r a v e n h y -p o t h e s i s ; w h i l e a b l a c k n o n - r a v e n c o n f i r m s i t . ( i i i . c ) Any b l a c k n o n - r a v e n i s n e u t r a l to the r a v e n h y p o t h e -s i s ; w h i l e a n o n - b l a c k n o n - r a v e n c o n f i r m s i t . Among the above three types the f i r s t one i s the most i n t e r -e s t i n g and i m p o r t a n t one . I t w i l l be c a l l e d "the 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 o f N i c o d ' s type"; w h i l e the o t h e r two w i l l be s i m p l y c a l l e d "the 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 o f n o n - N i c o d ' s t y p e s " , f o r they are o f minor i m p o r t a n c e . On the o t h e r hand, from Hempel ' s t h e o r y o f c o n f i r m a t i o n we c a n d e r i v e , b e s i d e s ( i ) and ( i i ) , when a p p l i e d to the r a v e n h y -t h e s i s , a l s o the f o l l o w i n g r e s u l t s . ( i v . a ) Any n o n - b l a c k n o n - r a v e n c o n f i r m s the r a v e n h y p o t h e -s i s . ( i v . 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 r a v e n hypothe -s i s . In s h o r t , i n Hempel 's t h e o r y o f c o n f i r m a t i o n any o b j e c t i n the u n i v e r s e e i t h e r c o n f i r m s o r e l s e d i s c o n f i r m s the r a v e n h y -3 p o t h e s i s ; no o b j e c t i s n e u t r a l t o the r a v e n h y p o t h e s i s . S i n c e n o t h i n g i s n e u t r a l , o r i r r e l e v a n t , t o the r a v e n h y p o t h e s i s i n Hempel ' s t h e o r y o f c o n f i r m a t i o n , we would say t h a t Hempel ' s t h e -o r y o f c o n f i r m a t i o n h o l d s o r a d v o c a t e s a " 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 " , a l a Max B l a c k . 3. Here a complete i n f o r m a t i o n about the o b j e c t r e l a t i v e to the h y p o t h e s i s i s assumed. 4 . I b i d . , p . 1 7 6 . In the following we w i l l t r y to discuss some more important and representative treatments, solutions or dissolutions of the paradoxes of confirmation: and we w i l l t r y to discuss them, i f possible, i n the l i g h t of Max Black's two p r i n c i p l e s of l i m i t e d relevance and universal relevance. 23 8 . VON WRIGHT'S TREATMENT OF THE PARADOXES OF CONFIRMATION Von Wright has two in t e r e s t i n g things to say concerning the paradoxes of confirmation i (I) The problem of "relevant range". According to von Wright before we t r y to confirm or disconfirm any universal hypothesis such as the raven hypothesis, we have i n the f i r s t place to ask the following question i What i s the relevant range of app l i c a t i o n of the hypothesis? Thus, i f the relevant range of app l i c a t i o n of the raven hypothesis i s a l l objects i n the world, then Hempel's di s s o l u t i o n of the paradoxes of confirmation i s the correct ones and, i f the relevant range of app l i c a t i o n of the raven hypothesis i s only about ravens i n the world, then Nicod's answer to the r a -ven hypothesis i s the correct one. And i n between, there are o-ther possible solutions i f o r instance, the relevant range of ap-p l i c a t i o n of the raven hypothesis could be about birds, or about other animals, etc. Which one i s accepted as the correct solu-t i o n to the raven hypothesis w i l l depend upon which relevant range of the hypothesis i s selected. Thus the f i r s t important thing f o r von Wright to say about the confirmation or disconfirmation of a universal generaliza-t i o n as a hypothesis i s the following thesis i^" Von Wright's Thesis of Range of Relevance. A l l things i n the range of relevance of a generalization may c o n s t i -tute genuine confirmations or disconfirmations of the ge-n e r a l i z a t i o n . The things outside the range are ir r e l e v a n t to the generalization. They cannot confirm i t genuinely. 1. See von Wright i "The paradoxes of confirmation", i n Aspects of Inductive Logic, (ed.) J . Hintikka and P. Suppes, p.211. 2k The above t h e s i s of von W r i g h t ' s g i v e s him a c e r t a i n f l e x i -b i l i t y i n h i s t rea tment of the paradoxes o f c o n f i r m a t i o n . He a l s o has the f o l l o w i n g second i n t e r e s t i n g t h i n g to s a y i ( I I ) The problem of " n a t u r a l range" . A c c o r d i n g to von W r i g h t , 2 "When 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 of the type t h a t a l l A are B i s no t s p e c i f i e d , t h e n the range i s . . . u s u a l l y unders tood to be the c l a s s o f t h i n g s which f a l l under the antecedent term A . The g e n e r a l i z a t i o n t h a t a l l ravens 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 to be a g e n e r a l i z a t i o n about ravens and not about b i r d s or about a n i m a l s or about e v e r y t h i n g t h e r e i s . " Thus , von Wright c a l l s the c l a s s o f t h i n g s t h a t are A "the n a t u r a l range o f r e l e v a n c e " o f the g e n e r a l i z a t i o n t h a t a l l A are B . With the above d e f i n i t i o n o f n a t u r a l range o f r e l e v a n c e von Wright comes to the f o l l o w i n g conc lus ion*-^ " W i t h i n 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 de termined by N i c o d ' s C r i t e r i o n . " T h i s second c o n c l u s i o n o f von W r i g h t ' s i s , t h u s , i d e n t i c a l 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 o f N i c o d ' s t y p e . T h i s i d e n t i f i c a t i o n i n c r e a s e s the p l a u s i b i l i t y o f s o l v i n g the paradoxes o f c o n f i r m a t i o n i n the l i g h t o f 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 o f N i c o d ' s t y p e , by way o f von Wright* s p r o -posed n o t i o n o f "the n a t u r a l range o f r e l e v a n c e " o f t h » r a v e n h y p o t h e s i s . 2 . See i b i d . p . 2 l 6 . 3. See i b i d , p . 2 1 ? . However, according to Hempel there are a few d i f f i c u l t i e s f acing von Wright's treatment of the paradoxes of confirmation by consideration of the natural range (or any r e s t r i c t e d range) of relevance. The d i f f i c u l t i e s are. (a) According to Hempel science involves no " f i e l d of appli cation" (such as the "natural range of relevance"), which would have to be a r b i t r a r y at best. For Hempel thinks. "the way i n which general hypotheses are used i n science never involves the statement of a f i e l d of application} and the choice of the l a t t e r i n a symbolic formulation of a given hypothesis thus introduces ... a considerable 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 scien-t i f i c hypothesis to the e f f e c t that a l l P's are Q's, the f i e l d of application cannot simply be said 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 relevance i s , according to Herapel , a confusion of l o g i c a l and p r a c t i c a l considerations. Thus Hempel has the following to say:-* "the view ... that a hypothesis of the simple form 'Every P i s a Q' asserts something about a l i m i t e d class of objects only, namely, the c l a s s of a l l P's ... involves a confusion of l o g i c a l and p r a c t i c a l considerations: Our in 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 that p a r t i c u l a r class of objects, but the hypo thesis nevertheless asserts something about, and indeed imposes r e s t r i c t i o n s upon, a l l objects." (c) The idea of "a f i e l d of a p p l i c a t i o n " of a hypothesis cannot be defined i n terms of syntax; besides, the idea also v i o l a t e s the equivalence condition. Hence, i t i s only an ad hoc device. 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 following paragraph:^ 4. See Hempel: Aspects of S c i e n t i f i c Explanation, pp.17-18. 5. See i b i d , p.18. 6. Ibid, p.18. 26 "the consistent use of a f i e l d of a p p l i c a t i o n i n the formulation of general hypotheses would involve con-siderable l o g i c a l complications, and yet would have no counterpart i n the t h e o r e t i c a l procedure of s c i -ence where hypotheses are subjected to various kinds of l o g i c a l transformation and inference without any consideration that 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 . This method of meeting the paradoxes would therefore amount to do-dging the problem by means of an ad hoc device which cannot be j u s t i f i e d by reference to actual s c i e n t i -f i c procedure." (my i t a l i c s ) 27 9. THE BAYESIAN TREATMENT OF THE PARADOXES OF CONFIRMATION The Bayesian treatment of the paradoxes of confirmation i s "astonishingly popular",or at l e a s t i t was f o r a period of time, as pointed out by Max Black. 1 I t i s represented by Hosiasson-Lin-2 3 4 ^ denbaum, J.K. Mackie, Patrick Suppes, Paul Horwich, and some others. Let us take Mackie's approach as an example. Mackie proposes, to begin with, the following p r i n c i p l e i The Inverse P r i n c i p l e . A hypothesis H i s confirmed by an observation report B i n r e l a t i o n to background knowledge K i f and only i f the observation report i s made more probable by adding the hypothesis to the background knowledge. Then, he a v a i l s himself of the background knowledge that there are l e s s ravens (and, respectively, black things) than non-ravens (and, respectively, non-black things), and thus assigns i n s i g n i -f i c a n t p r i o r p r o b a b i l i t y to those things under the categories of non-black non-ravens as well as black non-ravens such that i t turns out that the things under these two categories become i r -relevant to the hypothesis. In t h i s approach the common strategy i s not to r e j e c t the paradoxes of confirmation, but to render them to be harmless, to dissolve them rather than to solve them. 1. See Max Black t "Notes on the •Paradoxes of Confirmation'", i n Aspects of Inductive Logic, (ed.) J . Hintikka and P. Sup-pes, p.195.. 2. Hosiasson-Lindenbaum, J« "On confirmation", the Journal of Symbolic Logic, 19^0, pp.133-168. 3. J.K. Mackie» "The paradox of confirmation", B r i t i s h Journal f o r the Philosophy of Science, 1963, pp.263-277. 4. P. Suppest "A Bayesian approach to the paradoxes of confirma-t i o n " , i n Aspects of Inductive Logic, (ed.) J . Hintikka and P. Suppes, pp.198-207. 5. P. Horwichi P r o b a b i l i t y and Evidence, 1982. 2 8 Hempel has a c r i t i c i s m of t h i s approach. He raises doubts about the numerical assumption f o r the p r i o r p r o b a b i l i t i e s as-signed by the Bayesians, f o r he says*^ The answer depends i n part upon the l o g i c a l structure of the .language of science. I f a "coordinate language" i s used, i n which, say, f i n i t e space-time regions figure as indivi d u a l s , then the raven hypothesis assumes some such form as 'Every space-time region which contains a raven contains something black ' s and even i f the t o t a l number of ravens ever to exis t i s f i n i t e , the cla s s of space-time regions containing a raven has the power of the con-tinuCu)m, and so does the class of space-time regions containing something nonblackj thus, f o r a coordinate language of the type under consideration.the above numeri-c a l assumption i s not warranted. Now the use of a coor-dinate language may appear quite a r t i f i c i a l i n t h i s par-t i c u l a r i l l u s t r a t i o n j but i t may seem very appropriate i n many other contexts, such as, e.g.,that of physical f i e l d theories. On the other hand, on the basis of a "thing language", Hempel thinks that the numerical assumption may be j u s t i f i e d . But even here, i t remains an empirical question. Besides the above c r i t i c i s m made by Hempel, I think that the Bayesian approach also contains the following two d i f f i c u l t i e s ! 1) How to incorporate the "background knowledge" into Hempel's theory? For Hempel requires that any confirmation theory must be expressible, i n the f i r s t place, i n terms of syntax, where there i s no room f o r any background knowledge to be b u i l t i n . 2) I t seems that the Bayesians never say whether the paradoxes of confirmation are a genuine d i f f i c u l t y or no^ fc. They may have worked out a theory of prob a b i l i t y ; but, i n fact, they do not r e a l l y have a theory of confirmation. Perhaps t h i s i s what makes Max Black say that, 6 . See Hempel: Aspects of S c i e n t i f i c Expalnation, pp.20-21, foot-note 25. 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 i t a l i c s ) 7. Max Black: "Notes on the 'Paradoxes of Confirmation'", i n As^ pects of Inductive Logic, (ed.) J . Hintikka and P. Suppes, p.197. 3 0 10. QUINE'S SOLUTION OF THE PARADOXES OF CONFIRMATION Q u i n e ' 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 can be d e s c r i b e d as f o l l o w s : (I ) F i r s t , we need a new c o n d i t i o n s The P r o . j e c t i b i l i t y C o n d i t i o n . E v e r y p r e d i c a t e o f a hypo-t 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 the h y p o t h e s i s to 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 o t h e r 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 ) o f a h y p o t h e s i s presupposes t h a t every p r e d i c a t e o f 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 are p r e d i c a t e s P and Q whose shared i n s t a n c e s a l l do c o u n t , f o r whatever r e a s o n , toward c o n f i r m a t i o n o f " a l l P are Q". ( I I ) S e c o n d , t h e p r e d i c a t e s " . . . i s a raven" and " . . . i s b l a c k " are p r o j e c t i b l e , w h i l e the p r e d i c a t e s " . . . i s a n o n - r a v e n " and " . . . i s n o n - b l a c k " are not p r o j e c t i b l e , a t l e a s t a c c o r d i n g to Q u i n e . ( I I I ) I t f o l l o w s t h a t we have: 1. Any b l a c k r a v e n c o n f i r m s the r a v e n h y p o t h e s i s . 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 the r a v e n h y p o t h e s i s . 3 . Any b l a c k n o n - r a v e n i s n e u t r a l to the r a v e n h y p o t h e s i s . 4 . Any n o n - b l a c k n o n - r a v e n i s a l s o n e u t r a l to the r a v e n hypo-t h e s i s . Quine has thus "so lved" the paradoxes o f c o n f i r m a t i o n . Nov/ we would l i k e to r a i s e some q u e s t i o n s c o n c e r n i n g h i s s o -l u t i o n : Why does Quine t h i n k t h a t the p r e d i c a t e s " . . . i s a raven" and " . . . i s b l a c k " are p r o j e c t i b l e w h i l e the p r e d i c a t e s " . . . i s a n o n - r a v e n " and " . . . i s n o n - b l a c k " are not p r o j e c t i b l e ? And, i n g e n e r a l , what i s the c r i t e r i o n o f p r o j e c t i b i l i t y o f a p r e d i c a t e ? Q u i n e ' s answers a r e : (a) A l l ravens are o f the same k i n d , so are a l l b l a c k t h i n g s ; but i t i s not t r u e t h a t a l l non-ravens are o f the same k i n d , nor i s 31 i t the case that non-black things are of the same kind, and only things (or properties) of the same kind 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 predicate i s one that i s true of a l l and 2 only the things of a kind." (c) Moreover, p r o j e c t i b i l i t y , natural kind, s i m i l a r i t y and s i m p l i c i t y are, at lea s t to Quine, a l l of a p i e c e . J (d) And these concepts are i n t u i t i v e l y c l e a r , however sub-. 4 j e c t i v e . (e) We need these concepts because we want to do induction, e s p e c i a l l y simple and successful induction, and " i n induction nothing succeeds l i k e success."-* Here are some comments on Quine*s solution of the paradoxes of confirmation! 1) Quine's solution of the paradoxes of confirmation 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 involved i n h i s solution such as " p r o j e c t i b i l i t y " , "natural kind", " s i m i l a r i t y " , " s i m p l i c i t y " are s t i l l too vague or "subjective" as Quine himself admits. 2) Quine's P r o j e c t i b i l i t y Condition v i o l a t e s Hempel*s equi-valence condition. In consequence, i t also 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. So, I would aski What i s h is c r i t e r i o n of confirmation? And the answer i s i Quine does not r e a l l y have a theory of confirmation Juet as he himself ex-1. Cf. Quine: "Natural kinds", i n his Ontological R e l a t i v i t y and Other Essays. 19&9* 2. Ibid, p.116. 3. Cf. Quine: "Reply to Chihara", i n Midwest Studies of Philosophy, 1981, pp.453-454. 4 . See Quine: "Natural kinds", i n hi s Ontological R e l a t i v i t y and Other Essays, p . l l 6 . 5. Ibid, p.129. 32 p l a i n s i I f o l l o w the 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 seeks a se t o f hypotheses t h a t w i l l j o i n t l y i m p l y the d a t a , and one chooses among such s e t s w i t h an eye to s i m p l i c i t y and c o n s e r v a t i s m and perhaps o t h e r v i r t u e s . S u c c e s s f u l p r e -d 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 -erage o f new d a t a . So Q u i n e ' s " s o l u t i o n " to the paradoxes o f c o n f i r m a t i o n i s to i g n o r e them, i . e . to o f f e r no s o l u t i o n a t a l l . An odd response by a p h i l o s o p h e r . 6 . Q u i n e . "Reply to C h i h a r a " , Midwest S t u d i e s o f P h i l o s o p h y , 1971. p p . 453-^5^ . 11. GOODMAN'S AND SCHEFFLER'S CONCEPT OF SELECTIVE CONFIRMATION AND THE PARADOXES OF CONFIRMATION, AND GRANDY'S RELATED DISCUSSION Concern ing the paradoxes of c o n f i r m a t i o n and t h a t a g i v e n ob -j e c t which i s n e i t h e r b l a c k nor a r aven can c o n f i r m the r a v e n hypo-t h e s i s , Goodman has t h i s i n t e r e s t i n g and remarkable t h i n g to say:"*" The p r o s p e c t o f b e i n g ab le to i n v e s t i g a t e o r n i t h o l o g i c a l t h e o r i e s w i thout go ing out i n the r a i n i s so a t t r a c t i v e t h a t we know the re must be a c a t c h i n i t . So what i s the c a t c h meant by Goodman? To see the c a t c h , l e t us c o n s i d e r a ga i n the r aven h y p o t h e s i s * (1) ( X ) ( R X P B X ) . Now suppose t h a t we have the o b s e r v a t i o n r e p o r t i (2) - R c . B c , where £ i s any b l a c k n o n - r a v e n . Then, by Hempel ' s 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 ( 1 ) . But (2) c o n f i r m s a l s o the f o l l o w i n g h y p o t h e s i s * (3) (x ) (Rx3 -Bx), which i s , a c c o r d i n g to Goodman, a " c o n t r a r y " o f ( 1 ) . So, we have t h a t (2) c o n f i r m s bo th the raven h y p o t h e s i s and i t s c o n t r a r y . T h i s i s the c a t c h meant by Goodman. Goodman t h i n k s t h a t an adequate concept of c o n f i r m a t i o n must enab le the i n s t a n c e s of a h y p o t h e s i s to p r e f e r the h y p o t h e s i s to i t s c o n t r a r y . T h i s i d e a o f Goodman's can be c a l l e d " (Goodman's) P r e f e r e n c e C o n d i t i o n " . I s r a e l S c h e f f I e r n o t i c e s i t s importance and 1. Goodman, Ne l son : F a c t , F i c t i o n , and F o r e c a s t , 3rd e d . , 1955, p .70. 2. Thus, a c c o r d i n g to Goodman, two sentences are " c o n t r a r i e s " i f f i n any non-empty domain they may be bo th t r u e but cannot be both f a l s e . Note t h i s i s not the t r a d i t i o n a l d e f i n i t i o n , which would not 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 of s e l e c t i v e c o n -f i r m a t i o n t o s o l v e the paradoxes of c o n f i r m a t i o n . We s h a l l em-p l o y Goodman's sense of " c o n t r a r y " h e r e a f t e r . 34 develops into a new concept of " s e l e c t i v e conf i rmat ion" , which i s def ined as f o l l o w s : ^ The d e f i n i t i o n of se lec t i ve conf i rmat ion . An observation report B s e l e c t i v e l y confirms a hypothesis H, i f 3 confirms H and disconf i rms the contrary of H, (where both conf i rmat ion and d isconf i rmat ion are to be understood i n the Hempelian senses. ) Once we have t h i s new concept of se lec t i ve conf i rmat ion, the paradoxes of conf i rmation and "the prospects of indoor or -n i thology vanish" . To see t h i s , we have to consider the fo l lowing four casest ( i ) Any black raven s e l e c t i v e l y confirms the raven hypothe-s i s . This i s because we have: Ra • Ba |r (Ra o3a), and Ra • 3a ||- - (Ra = » - B a ) , where a i s any black raven. Thus we have any black raven con-f i rms the raven hypothesis and disconf i rms i t s contrary . So, by d e f i n i t i o n any black raven s e l e c t i v e l y confirms the raven hypothesis . ( i i ) Any non-black raven disconf i rms the raven hypothesis . This i s because we have: Rb« -Bb f|- -(Rb PBb) , where b i s any non-black raven. ( i i i ) Any black non-raven does not s e l e c t i v e l y confirm the raven hypothesis . This i s because we have: -Re* 3c \Y (Rc =>3c), but we do not have: -Rc - Be / L - ( R c =>-Be), 3. Cf . I s rae l S c h e f f l e r : Anatomy of Inquiry, I963. p p . 2 8 6 -291. 4 . See Goodman: Fact , F i c t i o n , and Forecast , 3 r d , p .71. 35 where c_ i s any black non-raven. (iv) Any non-black non-raven does not s e l e c t i v e l y confirm the raven hypothesis. This i s becausei although we have: -Rd.-Bd IKRdoBd), we do not havei -Rd.-Bd \[ -(RdD -Bd), where d i s any non-black non-raven. Although sele c t i v e confirmation can avoid the paradoxes of confirmation, Grandy offers three important objections:^ F i r s t , s e l e c t i v e confirmation rejects the equivalence condi-t i o n . Perhaps selective confirmation can supply a reason for re j e c t i n g the equivalence condition} but i s n ' t i t a price too high to pay f o r i t ? For, then, we would have no more r i g h t to make transformation of equivalent statements i n theory and i n practice without paying attention to t h e i r forms. Second, se l e c t i v e confirmation cannot apply to any hypothe-s i s which does not have the form of a generalized c o n d i t i o n a l . Thus i t i s not applicable to a hypothesis of the form: (x)Gx, which has no contrary at a l l . Third, some p e r f e c t l y ordinary hypotheses cannot be selec-t i v e l y confirmed. This i s odd. For instance, the following hypothesis, (4) (x)(Rx.-Bx=>Rx.-Rx), which i s l o g i c a l l y equivalent to the raven hypothesis, and which has 5, Richard E. Grandy: "Some comments on confirmation and selec-t i v e confirmation", i n Readings i n the Philosophy of Science, (ed.) Baruch A. Brody, 1970, pp.428-432. 36 the f o l l o w i n g c o n t r a r y , (5) (x) ((Rx.-Bx)=> - ( R x . - R x ) ) , can never be s e l e c t i v e l y con f i rmed , because (5) i s a v a l i d s e n -tence and can never be d i s c o n f i r m e d . G randy ' s t h r e e c r i t i c i s m s of the t h e o r y of s e l e c t i v e c o n f i r -mat ion can be f u r t h e r e l u c i d a t e d as f o l l o w s » (a) G randy ' s f i r s t c r i t i c i s m p o i n t s out t h a t 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 v i o l a t e s Hempel ' s e q u i v a l e n c e c o n d i t i o n . I t has at l e a s t the f o l l o w i n g two s i g n i f i c a n c e s ! ( a . l ) 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 may o f f e r a rea son to r e j e c t Hempel ' s e q u i v a l e n c e c o n d i t i o n . But i s i t a good reason? T h i s would s t i l l remain a c o n t r o v e r s i a l i s s u e , f o r the s e l e c t i v e c o n f i r m a t i o n o f 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 such as the f o l l o w i n g one, H: (x ) (Fx 3Gx), means t h a t ( i ) H i s con f i rmed and ( i i ) the c o n t r a r y o f H, CH: ( X ) ( P X D - G X ) , i s d i s c o n f i r m e d , and t h i s , i n t u r n , means t h a t : H* : - ( x ) ( F x 3 - G x ) , i s conf i rmed? and H* i s l o g i c a l l y e q u i v a l e n t t o : H* * : ( 3 x ) ( F x . G x ) . Hence, to s e l e c t i v e l y c o n f i r m H i s t o c o n f i r m H & H * * ; and to s e l e c t i v e l y c o n f i r m the f o l l o w i n g h y p o t h e s i s , K i ( x ) ( - G x O - F x ) , i s to c o n f i r m bo th K and the f o l l o w i n g h y p o t h e s i s ! K** i (3x) ( -Gx.-Fx). 37 T h u s w h e n we s a y t h a t t h e 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 d o e s n o t p r e s e r v e H e m p e l ' s e q u i v a l e n c e c o n d i t i o n we mean t h a t H A H * * i s n o t l o g i c a l l y e q u i v a l e n t t o K & K * * , t h o u g h H i s s t i l l l o g i c a l l y e q u i v a l e n t t o K . I n s h o r t , w h a t i s c o n t r o v e r s i a l i s w h e t h e r " a l l F s a r e G s " c a n b e e q u a l l y w e l l - r e p r e s e n t e d b y H o r K? H e m p e l ( a n d m o s t l o -g o c i a n s ) t h i n k t h a t i t c a n t b u t Goodman a n d S c h e f f l e r t h i n k t h a t i t c a n n o t . ( a . 2 ) The f o r m o f a h y p o t h e s i s b e c o m e s c r u c i a l i n t h e c o n -f i r m a t i o n o f a h y p o t h e s i s . W h e t h e r t h e f o r m o f a h y p o t h e s i s p l a y s s u c h a c r u c i a l r o l e i n s c i e n t i f i c p r a c t i c e i s d u b i o u s o r , a t l e a s t , a g a i n , c o n t r o v e r -s i a l . A n y w a y i t w o u l d c r e a t e g r e a t i n c o n v e n i e n c e i n s c i e n t i f i c p r a c t i c e i f i t i s s t i l l w o r k a b l e - ^ - e s p e c i a l l y i n t h e t r a n s -f o r m a t i o n s o f e q u i v a l e n t s e n t e n c e s a n d h y p o t h e s e s a n d a l s o i n p r e d i c t i o n . ( b ) G r a n d y ' s s e c o n d c r i t i c i s m c l e a r l y means t h a t t h e 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 i s i n a d e q u a t e , f o r t h e r e a r e h y p o t h e s e s w h i c h c o u l d n o t b e s e l e c t i v e l y c o n f i r m e d . N o t e t h a t a n y h y p o t h e s i s K , i n c l u d i n g ( x ) G x , c a n h a v e t h e f o l l o w i n g " c o n t r a r y " : C K * : ( x ) ( ( A x v - A x ) r > - K ) , w h i c h i s t h e " c o n t r a r y o f K * " , w h e r e K * i s : K * i ( x ) ( ( A x v - A x ) = > K ) . S i n c e K * i s a l w a y s s e l e c t i v e l y c o n f i r m a b l e a n d K i s l o g i c a l l y e q u i v a l e n t t o K * , K i s i n a s e n s e a l s o s e l e c t i v e l y c o n f i r m a b l e 38 i f we have the equivalence condition available i n the theory of select i v e confirmation; but the trouble i s , of course, that the equivalence condition does not hold with respect to the selec t i v e confirmation of a hypothesis i n the theory of se l e c t i v e confirma-t i o n . (c) Grandy's t h i r d c r i t i c i s m just re-emphasizes h i s f i r s t two c r i t i c i s m s . His example (4) shows that there are forms of a hypothesis that cannot be s e l e c t i v e l y confirmed, though i n the case of (4) one of i t s equivalent hypotheses, which i s the Raven Hypothesis, does turn out to be s e l e c t i v e l y confirmed. Hence, we learn the moral again that the form of a hypothesis plays a cru-c i a l r o l e i n i t s sel e c t i v e confirmation. •»*» *•>*•* »»* Grandy, a f t e r the above three c r i t i c i s m s of Goodman's and Scheffler's s e l e c t i v e confirmation, proposes the following r e-l a t i v e concept of selective confirmation to avoid the d i f f i c u l -t i e s facing Goodman's and Scheffler's theory of selective con-f irmatiom "M s e l e c t i v e l y confirms S r e l a t i v e to K i f S i s a member of K and M confirms S and M disconfirms every other mem-ber of K", where S i s a hypothesis, M i s an observation sentence and K i s a sp e c i f i e d set of hypotheses, no two of which are equivalent. Now according to Grandy his new d e f i n i t i o n of r e l a t i v e con-cept of selec t i v e confirmation retains the equivalence condition, at l e a s t i n the following sense * 6 . Grandy, Richard E.: "Some comments on confirmation and select t i v e confirmation", i n Readings i n the Philosophy of Science, (ed.) Baruch A. Brody, 1970, p.431. 39 G r a n d y ' s sense o f e q u i v a l e n c e c o n d i t i o n . ^ I f S i s s e l e c -t i v e l y c o n f i r m e d from K by M, then M w i l l s e l e c t i v e l y c o n f i r m S' from K * . where S* i s a s tatement e q u i v a l e n t to S and where K* i s o b t a i n e d from K by r e p l a c i n g s tatements by t h e i r e q u i v a -l e n t s . Indeed , Grandy*s r e l a t i v i z e d sense of s e l e c t i v e c o n f i r m a t i o n can improve 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 . Thus , f o r i n s t a n c e , the o b s e r v a t i o n r e p o r t , BI* G a , s e l e c t i v e l y c o n f i r m s the h y p o t h e s i s * (6) (x)Gx, i n s t e a d of the h y p o t h e s i s , (7) ( x ) - G x , from the s e t ?(x)Gx, ( x ) - G x | . My o n l y c r i t i c i s m of G r a n d y ' s newly proposed r e l a t i v e c o n -cept o f s e l e c t i v e c o n f i r m a t i o n i s t h i s * H i s sense of e q u i v a l e n c e c o n d i t i o n i s not r e a l l y 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 we are t a l k i n g a b o u t . I n o t h e r words , h i s sense o f e q u i v a l e n c e c o n d i -t i o n i s not Hempel 's e q u i v a l e n c e c o n d i t i o n , f o r G r a n d y ' s e q u i -v a l e n c e c o n d i t i o n can o n l y a p p l y to the s tatements mentioned i n a g i v e n se t o f hypotheses K where no two of which are e q u i v a l e n t but t h e r e i s no such r e s t r i c t i o n i n Hempel*s sense o f e q u i v a l e n c e c o n d i t i o n . 7. I b i d , p .431. 40 12. ARMSTRONG'S VIEW OF THE PARADOXES OF CONFIRMATION D.M. Armstrong's view of the paradoxes of confirmation can be b r i e f l y summarized as follows i ^ " F i r s t , i f we understand the Raven Hypothesis, RH» A l l ravens are black, as HI: (x)(RxZ>Bx), then Armstrong agrees with Hempel and thinks that the paradoxes of confirmation are genuine. Second, Armstrong thinks instead of HI the Raven Hypothesis should be understood as the following hypothesis: H2: It i s a law that Rs are Bs, which i s supposed by Armstrong to be stronger than the r e g u l a r i -ty expressed i n HI. Third, i f H2 i s the correct understanding of the Raven Hypo-thesis, then Armstrong argues that at le a s t one case of the para-doxes of conf irmation i s genuine. Armstrong argues as follows: ( i ) Suppose that we have d i r e c t evidence bearing upon an object which i s both an R and a B then t h i s observation would be explained by the law-hypothesis H2 (and here Armstrong agrees with Dretske that confirmation i s roughly the converse of expla-nation.) So, the observation confirms H2. ( i i ) Again, i f we come upon an object which i s neither an R nor a B, then i t can s t i l l be said that t h i s observation i s ex-1. D.M. Armstrong: What i s a Law of Nature? Cambridge Universi-ty Press, 1983, p . 4 l f f . 2. F.I. Dretske: "Law of nature", Philosophy of Science, 44, 1977. 41 p l a i n e d by H2. F o r , g i v e n the l a w - h y p o t h e s i s H2, we know t h a t s i n c e the o b j e c t i s not a B i t cannot be an R. ( i i i ) F i n a l l y , suppose t h a t we come upon an o b j e c t which i s not an R but i s a B . A c c o r d i n g to Armstrong there i s no case a t a l l f o r s a y i n g t h a t t h i s o b s e r v a t i o n i s e x p l a i n e d by the h y -p o t h e s i s t h a t i t i s a law t h a t Rs are B s . So, i t f o l l o w s t h a t i t does not i n any way c o n f i r m the l a w - h y p o t h e s i s H2. Here we have one case o f the paradoxes o f c o n f i r m a t i o n . Thus i f Armstrong i s r i g h t , then t h r e e r e s u l t s can be drawn* (1) the Raven H y p o t h e s i s should be unders tood as H2; (2) one case o f the paradoxes o f c o n f i r m a t i o n i s genu ine : and (3) HI and H2 are two t o t a l l y d i f f e r e n t h y p o t h e s e s . However, i n a r g u i n g f o r case ( i i i ) a b o v e , A r m s t r o n g seems to o v e r l o o k s o m e t h i n g . F o r e x a c t l y i n case ( i i i ) , when we have an o b j e c t which i s not an R but i s a B , we have the o b j e c t t h a t i s e x p l a i n e d by the f o l l o w i n g g e n e r a l i z a t i o n : H3: ( x ) ( - R x v B x ) , which i s l o g i c a l l y e q u i v a l e n t to H I . Now i t i s obv ious t h a t the o b j e c t , which i s not an R but i s a B , i s e x p l a i n e d by H 3 . Hence, i t i s a l s o e x p l a i n e d by HI and RH a f a c t c o n t r a d i c t i n g the o r i g i n a l i n t e n t i o n o f A r m s t r o n g , who t h i n k s t h a t HI and H3 are not laws nor the p r o p e r t r a n s l a t i o n o f the Raven H y p o t h e s i s , f o r they do not e x p l a i n the o b s e r v a t i o n r e p o r t which says t h a t the o b j e c t i s not an R but i s a B . Thus A r m s t r o n g ' s view o f the paradoxes o f c o n f i r m a t i o n , a l -though i n t e r e s t i n g , i s not r e a l l y t e n a b l e . 42 13. SOME OTHER PROPOSED SOLUTION OF THE PARADOXES OF CONFIR-MATION There i s one i n t e r e s t i n g contemplated solution of the para-doxes of confirmation considered by Hempel.^ The contemplated solution 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 e x i s t e n t i a l import 2 i s conferred upon any universal conditional sentence. If we adopt t h i s A r i s t o t a l i a n p o l i c y , the paradoxes of confirmation can be avoided; for, instead of expressing the Raven Hypothesis as t (1) (x)(Rx3>Bx) we add e x p l i c i t l y a sentence expressing t h i s e x i s t e n t i a l import as follows 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 derive the paradoxes of con-firmation, f o r the equivalence condition i s obviously v i o l a t e d , i . e . (2) and (4) are not l o g i c a l l y equivalent. Hempel gives 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 so-l u t i o n " i 3 (a) F i r s t of 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 Explanation, pp.16-17. 2. Cf. R u s s e l l . History of Western Philosophy. p . 2 0 6 f f . 3. Hempel, op. -ci-t.. pp.16%-17. ^3 validate many l o g i c a l inferences that appear to be v a l i d . (b) Second, i n empirical science the customary formulation of general hypotheses does not r e a l l y contain an e x i s t e n t i a l clause : nor does i t determine such a clause unambiguously. For example, consider t h i s example given by Hempeli If a person a f t e r receiving an i n j e c t i o n of a c e r t a i n test substance has a p o s i t i v e skin reaction, he has diphtheria. Where should we im-port the e x i s t e n t i a l clause? Should we import i t as r e f e r r i n g to persons? to persons a f t e r receiving the i n j e c t i o n ? or to per sons who a f t e r receiving the i n j e c t i o n show a pos i t i v e skin re-action? According to Hempel any decision made here i s quite ar b i t r a r y , and each decision gives 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 correct. (c) F i n a l l y , many universal hypotheses cannot be said to imply an e x i s t e n t i a l clause. Hempel states his view as follows " i t may happen that from a c e r t a i n astrophysical theory a universal hypothesis i s deduced concerning the character of the phenomena which would take place under c e r t a i n s p e c i f i e d extreme conditions. A hypo-thesis of t h i s kind need not (and, as a rule, does not) imply that such extreme conditions 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 point given by Hempel seems l e s s convincing, f o r a hypothesis that 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 confirmation i s d i f f e r e n t from a l l of the views so fa r we have examined. His view i s i n pa r t i c u l a r d i f f e r e n t from the views of von Wright's, Quine's, Scheffler's and Goodman's. Hempel sees nothing wrong with the paradoxes of confirmation. In other words, he does not think that the paradoxes of confirma-t i o n are genuine. He i s an upholder of the p r i n c i p l e of universal relevance (so-called by M. Black), which i s quite d i f f e r e n t from the p r i n c i p l e of li m i t e d relevance of Nicod's type as upheld by Quine, Scheffler, Goodman, etc. There are several reasons that make Hempel come to the conclu-sion that the paradoxes of confirmation are not genuine* F i r s t , he thinks that we need the Equivalence Condition, f o r 1 " F u l f i l l m e n t of t h i s condition makes the confirmation 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 that t h i s i s a necessary condition f o r the adequacy of any proposed c r i t e r i o n of confirmation." Second. Hempel re j e c t s the A r i s t o t a l i a n type of solution of the paradoxes of confirmation by conferring on any universal condi-t i o n a l sentence an e x i s t e n t i a l import. He has given three reasons fo r t h i s r e j e c t i o n , which we have just discussed i n the previous section. Third, Hempel also rejects v o n Wright's idea of "the natural range of relevance" of a universal conditionalhypothesis by saying that the way i n which general hypotheses are used i n science never involves the statement of a " f i e l d of ap p l i c a t i o n " or any "natural range of relevance", 1. Hempel: Aspects of S c i e n t i f i c Explanation, p. 13. 2. Ibid, p.17. 4 5 A f t e r examining the above two a l t e r n a t i v e s to the customary-method o f r e p r e s e n t i n g g e n e r a l hypotheses by means o f u n i v e r s a l 3 c o n d i t i o n a l s , Hempel c o n c l u d e s ! "ne i ther o f them proved an adequate means o f p r e c l u d i n g the paradoxes o f c o n f i r m a t i o n . We s h a l l now t r y to show t h a t what i s wrong does not l i e i n the customary way of 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 hypotheses , but r a t h e r i n our 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 the 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 not 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 ) A c c o r d i n g to Hempel there are two sources o f m i s u n d e r s t a n d i n g ! (a) One source o f m i s u n d e r s t a n d i n g i s the view t h a t a hypo-t h e s i s o f the s imple form, H : E v e r y P i s a Q, such as " a l l sodium s a l t s b u r n ye l low", a s s e r t s something about a c e r t a i n l i m i t e d c l a s s o f o b j e c t s o n l y , v i z . , the c l a s s o f a l l P ' s . T h i s i d e a , a c c o r d i n g t o Hempel , i n v o l v e s (as a l r e a d y ment ioned 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 the 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 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 a b o u t , and indeed 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 source o f the appearance o f p a r a d o x i c a l i t y i n c e r t a i n cases o f c o n f i r m a t i o n i s e x h i b i t e d by Hempel by the f o l -4 l o w i n g c o n s i d r a t i o m Suppose t h a t i n suppor t o f the a s s e r t i o n ' A l l sodium s a l t s b u r n y e l l o w ' somebody were to adduce an exper iment i n which a p i e c e o f pure i c e was h e l d i n t o a c o l o r l e s s f lame and d i d not t u r n the f lame 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 a s s e r t i o n , 'Whatever does not 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 , by v i r t u e o f the 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 does t h i s 3 . I b i d , p . 1 8 . 4 . I b i d , p . 1 9 . 4 6 impress us as paradoxical? The reason becomes cle a r when we compare the previous s i t u a t i o n with the case where an object whose chemical c o n s t i t u t i o n i s as yet unknown to us i s held into a flame and f a i l s to turn i t yellow, and where subsequent analysis reveals i t to contain no sodium s a l t . This outcome... i s what was to be expected on the basis of the hypothesis that a l l sodium s a l t s burn yellow... Now the only difference between the two situations here considered i s that i n the f i r s t case we are t o l d beforehand the test substance i s i c e , and we happen to "know anyhow" that ice contains no sodium s a l t ; t h i s has the consequence that the outcome of the flame-color test becomes e n t i r e l y i r r e l e v a n t f o r the confirma-t i o n of the hypothesis and thus can y i e l d no new evidence fo r us. Indeed, i f the flame should not turn yellow, the hypothesis requires that the substance contain no sodium s a l t and we know beforehand that ice does not; and i f the flame should turn yellow, the hypothesis would impose no further r e s t r i c t i o n s on the substance: hence, either of the possible outcomes of the experiment would be i n accord with the hypothesis. Thus Hempel comes to a general point from the analysis of the above example:^ In the seemingly paradoxical cases of confirmation, we are often not a c t u a l l y judging the r e l a t i o n of the given evidence E alone to the hypothesis H ...; instead, we t a c i t l y introduce a comparison of H with a body of e v i -dence which consists of E i n conjunction with additional information that we happen to have at our disposal; i n our i l l u s t r a t i o n , t h i s information includes the knowledge (1) that the substance used i n the experiment i s i c e , and (2) that ice contains no sodium s a l t . I f we assume t h i s additional information as given, then, of course, the out-come of the experiment can add no strength to.the hypothe-s i s under consideration. But i f we are c a r e f u l co avoid t h i s t a c i t reference to additional knowledge... i t i s c l e a r that ... the paradoxes vanish. So far, i n (b), what Hempel has considered mainly i s the type of paradoxical case which i s i l l u s t r a t e d by the assertion that any non-black non-raven constitutes confirming evidence f o r the hypothesis , " a l l ravens are black". According to Hempel,6" Other paradoxical cases of confirmation may be dealt with analogously. Thus i t turns out that the paradoxes of con-5. Ibid, p.19. 6. Ibid, p.20. 47 firmation ... are due to a misguided i n t u i t i o n i n the matter rather 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 derived. (my i t a l i c s ) Thus Hempel thinks, i n conclusion, that the paradoxes of con firmation are not r e a l l y genuine. By the way, Prof. Stewart a t t r i b u t e s the following view to Goodman, and the view i s . Goodman thinks that the paradoxes of confirmation are indeed not genuine as Hempel t r i e s to say here, and Goodman's idea of selective confirmation i s only one way to show how the i l l u s i o n a r y mechanism works i n our mind. Unfortu-nately, I have not found these claims of Goodman's i n the l i t e r -ature, and, consequently cannot o f f e r a more c a r e f u l analysis of t h i s i n t e r e s t i n g view. 48 15. A NEW VIEW OF THE PARADOXES OF CONFIRMATION From our p r e v i o u s rev iew and d i s c u s s i o n s of the paradoxes of c o n f i r m a t i o n e s p e c i a l l y t h e i r s o l u t i o n s and d i s s o l u t i o n a l o t o f t h i n g s can be l e a r n e d . To sum up my own o p i n i o n s : (I) 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 genu-i n e . 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 they are not p s y c h o l o g i c a l i l l u s i o n s as Hempel persuades us t o b e l i e v e . ( I I ) I agree w i th Hempel*s o p i n i o n , on the o t h e r hand, t h a t a l l proposed s o l u t i o n s of the paradoxes o f c o n f i r m a t i o n c o n t a i n some d i f f i c u l t i e s and are not t o t a l l y s a t i s f a c t o r y o r , even, s im -p l y not s a t i s f a c t o r y . However, t h a t they support a p r i n c i p l e of l i m i t e d r e l e v a n c e of N i cod* s type i s , I b e l i e v e , c o r r e c t o r , a t l e a s t , more p l a u s i b l e . N e v e r t h e l e s s t h e r e i s something we can l e a r n from the proposed s o l u t i o n s , f o r i n s t a n c e , von W r i g h t ' s concept 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 I I ) I b e l i e v e t h a t 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 f a c e s a g rea t d i f f i c u l t y : i t commits the " f a l l a c y of i ndoo r o r n i t h o l o g y " as p o i n t e d out by Goodman."1' Thus c o n s i d e r the f o l l o w i n g Indoor Raven H y p o t h e s i s : IRH 1 : A l l ravens i n my o f f i c e a t UBC are b l a c k , o r i n symbols, (x) (Rx . Ox =>Bx), where R: (T) i s a r a v e n , 0:(J) i s i n my o f f i c e a t UBC, B: (J) i s b l a c k . 1. C f . Goodman: F a c t , F i c t i o n , and F o r e c a s t , 3rd e d . , 70-71. 49 Now i n my o f f i c e at UBC there are one desk, two chairs and some books. It follows that i n Hempel's theory of confirmation IRH^ i s confirmed by the observation report about the development of IRH.^ with respect to the reference class of indi v i d u a l s I, which are the desk, the two chairs and the books i n my o f f i c e at UBC. So are the following hypotheses confirmed by the same class of individuals I: IRH 2« A l l ravens i n my o f f i c e at UBC are white. IRH^: A l l ravens i n my o f f i c e at UBC are red. IRH^: A l l ravens i n my o f f i c e at UBC are both black and white. In f a c t the f a l l a c y of indoor ornithology i s not confined to indoor nor only to ornithology. So consider the following Local Panda Hypothesess LPH^: Each panda i n Alaska has one eye. LPH 2: Each panda i n Alaska has two eyes. LPH^: Each panda i n Alaska has 1,001 eyes. LPH^: Each panda i n Alaska has both three and four eyes. Now each of the above Local Panda Hypotheses i s also confirmed i n Hempel's theory of confirmation, simply because there i s not a panda i n Alaska. That Hempel's theory of confirmation has committed the f a l l a c y of "indoor ornithology" as shown i n the above examples always de-pends the same device: the development of each of the above hypo-theses always has a f a l s e antecedent. It i s important to r e a l i z e that the "device" i s also used i n the Raven Hypothesis. Thus when people say that i t i s absurd to 50 say that ( i ) a non-black non-raven, or ( i i ) a black non-raven confirms the Raven Hypothesis, the reason i s simply that the development of the Raven Hypothesis with respect to that e n t i t y also has a f a l s e antecedent. In consequence, Hempel cannot show us any object which s a t i s f i e s both the antecendent and the conse-quent of IRH l t IRH ? i IRHy IRH^, LPH^ LPH 2, LPH 3 or LPH^ or else what he can show to us i s something that s t r i k e s us as i r r e l e v a n t . In short, there i s one important point made by the c r i t i c i s m that Hempel's theory of confirmation has committed the f a l l a c y of "indoor ornithology": there are some s c i e n t i f i c f i c t i o n s or mythologies such as the confirmation of IRH^- IRH^ and LPH^-LPH^ b u i l t into Hempel's theory of confirmation. These hypo-theses are confirmed i n Hempel's theory of confirmation and, yet, he cannot show us even one such object (a raven or a panda) that s a t i s f i e s any of the hypotheses (under discussion). (IV) I believe that von Wright has an important point to make when he says that the "natural" reading of the Raven Hypothesis i s confined to ravens and nothing else. In other words his notion of "the natural range of relevance" of a universal conditional hypothesis makes senset although the standard reading of the Raven Hypothesis^,as about everything i n the universe including ravens and non-ravens, also make sense. (V) I believe f i n a l l y that both readings of the Raven Hypothe-s i s can only be accommodated i n a l o g i c other than c l a s s i c a l l o g i c . In other words, the whole point of t h i s i s to suggest that a change of l o g i c may help solve the Raven Paradox and the paradoxes of con-firmation. Thus we come to the suggestion that the source of the paradoxes of confirmation and the f a l l a c y of "indoor ornitholo-gy" may derive from c l a s s i c a l l o g i c . Since I am not the f i r s t to challenge the absolute status of the c l a s s i c a l l o g i c , i n the following I s h a l l have to turn to a review of the c r i t i c i s m s con-cerning 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 c l a s s i c a l l o g i c has enjoyed the unique p o s i t i o n i n the l o g i c a l w o r l d f o r more than two thousand y e a r s . I t i s somewhat l i k e t h a t o f E u c l i d e a n geometry i n the mathemat i ca l and s c i e n -t i f i c w o r l d , or even more s o . E u c l i d e a n geometry was f i r s t c h a l -l e n g e d i n the 18th 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 the b e g i n -n i n g o f t h i s c e n t u r y when he adopted Riemannean geometry as the p r o p e r t o o l f o r h i s t h e o r y o f r e l a t i v i t y , w h i l e c l a s s i c a l l o g i c was not c h a l l e n g e d u n t i l l a t e l a s t c e n t u r y and i t s unique p o s i -t i o n i s s t i l l b e l i e v e d by the g r e a t m a j o r i t y o f l o g i c i a n s . In t h i s s e c t i o n we w i l l d e s c r i b e the t h r e e most i m p o r t a n t c h a l l e n g e r s to c l a s s i c a l l o g i c and t h e i r b a s i c i d e a s . (I ) B r o u w e r ' s c h a l l e n g e . Brouwer c h a l l e n g e s the u n i v e r s a l v a l i d i t y o f the law of e x c l u d e d - m i d d l e and the law of d o u b l e -n e g a t i o n from a c o n s t r u c t i v i e t p o i n t o f v i ew , e s p e c i a l l y when the laws i n v o l v e i n f i n i t e s e t s o f (mathemat ica l ) e n t i t i e s . Thus i n h i s s o - c a l l e d " i n t u i t i o n i s t i c l o g i c " we have these prominent f e a t u r e s J 1 ( i ) The d o u b l e - n e g a t i o n law does not h o l d ; more s p e c i f i c a l l y , the double n e g a t i o n of a sentence i s not e q u i v a l e n t to the s e n -tence i t s e l f . ( i i ) The law o f e x c l u d e d - m i d d l e does not h o l d . ( I I ) C . I . L e w i s ' c h a l l e n g e . L e w i s ' c h a l l e n g e i s concerned m a i n l y w i t h the i n t e r p r e t a t i o n o f the c l a s s i c a l "horseshoe" o p e r -a t o r . He t h i n k s t h a t i t i s wrong to i n t e r p r e t the "horseshoe" 1. C f . A . H e y t i n g : I n t u i t i o n i s m , 1966, 0 r S . C . K l e e n e i I n t r o d u c - t i o n to Metamathematics . p p . 4 6 - 5 3 . 53 as " i m p l i c a t i o n " , and he l i s t s the f o l l o w i n g t h i r t e e n theorems 2 of the c l a s s i c a l l o g i c as "the paradoxes o f i m p l i c a t i o n " * 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 ( Q 3 P ) 7. ( - P . Q ) =>(P^Q) 8. - ( P 3 Q ) =>P 9. - ( P O Q ) a - Q 10. - ( P 3 Q ) P ( P 5 - Q ) 11. - ( P ? Q ) P ( - P 3 Q ) 12. - ( P P Q ) ^ (-P3-Q) 13. - ( P P Q ) ? ( Q 5 P ) The main r e a s o n C . I . Lewis t h i n k s t h a t i n t e r p r e t i n g "horse -shoe" as " i m p l i c a t i o n " i s wrong i s t h i s : the i n t e r p r e t a t i o n e n -a b l e s two i n t r i n s i c a l l y (or 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 o f a c e r -t a i n k i n d t o e n j o y 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 o f " i m p l i c a t i o n " . H i s c h a l l e n g e r e s u l t s i n a whole s e r i e s o f new modal systems, S1 -S5 . U n f o r t u n a t e l y , some of the "paradoxes o f s t r i c t i m p l i c a -t i o n " such as the f o l l o w i n g two j ( i ) Q - J ( P v - P ) ( i i ) ( P . - P M Q (where " A - ? B " i s s h o r t f o r "Cj(A=>B)"), appear , even i n the weak-e s t o f h i s modal systems S I . 2. See, C . I . Lewis and C . H . L a n g f o r d * Symbol ic L o g i c , p p . 8 6 - 8 8 . 3. I b i d , p . 5 0 4 . 54 ( I I I ) - L u k a s i e w i c z ' s c h a l l e n g e . L u k a s i e w i c z ' s c h a l l e n g e to c l a s s i c a l l o g i c i s , a t once , b o t h v e r y modern and v e r y a n c i e n t . H i s i n t e n t i o n i s to s o l v e the A r i s t o t a l i a n problem o f f u t u r e c o n t i n g e n t s . The A r i s t o t a l i a n problem o f f u t u r e c o n t i n g e n t s i s , to be v e r y b r i e f , t h i s . - ' What i s the t r u t h - v a l u e o f a f u t u r e c o n t i n g e n t sentence such as the f o l l o w i n g one, "there 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 can 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 presence i n Warsaw a t a c e r t a i n moment o f next y e a r , e . g . a t noon on 21 December, i s a t the p r e s e n t t ime de termined n e i -t h e r p o s i t i v e l y nor n e g a t i v e l y . Hence i t i s p o s s i b l e , but not 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 the the g i v e n t i m e . On t h i s a s sumpt ion the p r o p o s i t i o n *I s h a l l be i n Warsaw a t noon on 21 December o f next y e a r ' , can a t the p r e s e n t t ime be n e i t h e r t r u e nor 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 presence i n Warsaw would have to be n e c e s s a r y , which i s 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 . I f i t w e r e f a l s e now, on the o t h e r hand, my f u t u r e presence i n Warsaw would be i m p o s s i b l e , which 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 . T h e r e f o r e 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 the moment n e i t h e r t r u e nor f a l s e and must posses s a t h i r d v a l u e . . . " From the above l i n e o f thought L u k a s i e w i c z i n v e n t e d a new system of t h r e e - v a l u e d l o g i c i n 1920. A c c o r d i n g to one i n t e r p r e t a t i o n , w i t h which I a g r e e , 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 ( i ) we c a n i n v e n t a l o g i c such t h a t the p r i n c i p l e o f b i -v a l e n c e i s r e j e c t e d i n the l o g i c , w h i l e ( i i ) the law o f e x c l u d e d - m i d d l e i s r e t a i n e d i n the l o g i c , 4. 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 . , ( e d . ) , P o l i s h L o g i c . 1920-1939. 5. 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 . 6 . Haack, S» D e v i a n t L o g i c , p . 7 3 . 7 . 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 Review (71) , 1962. 55 In other words, the problem of future contingents can be solved i f we have» ( i * ) t f ( T p v F p ) , and ( i i * ) |(-(Pv-P), where "T" and "F" are respectively the pre-dicates "... i s true" and "... i s f a l s e " . It turns out that Lukasiewicz's three-valued l o g i c L3 i s , indeed, not bi-valent: but, unfortunately, the law of excluded-middle does not hold either. Thus Lukasiewicz does not solve the problem of future con-tingents. However, that does not mean that his challenge does not make sense. 56 17. THE PLAUSIBLE INADEQUACY OF THE CLASSICAL LOGIC Besides the plausible points made by Brouwer, Lewis and •Lukasiewicz there are at l e a s t three other points that w i l l show the plausible inadequacy of c l a s s i c a l l o g i c . (I) I t f a i l s to express some important senses of negation. In natural language such as Chinese, E n g l i s h , etc. there are d i s t i n c t senses of negation such as ( i ) Negation as contrary, ( i i ) negation as complementation, and ( i i i ) negation as contradiction. But i n c l a s s i c a l l o g i c a l l of them are taken care of by a single negation operation. How many d i f f i c u l t i e s and paradoxes i n l o g i c are derived from t h i s awkward "squeezing"? In contrast, i n cer-t a i n three-valued l o g i c there are, t h e o r e t i c a l l y , 26 possible d i f f e r e n t senses of negation that can be d i f f e r e n t i a t e d . (II) It f a i l s to express the concept of presupposition, which requires the additional truth-value "neither true nor f a l s e " , Consider the well-known example of Russell's» (1) The present king of France i s bald. According to Russell's theory of d e f i n i t e descriptions i t should be translated a s i 1 (2) (3x)(Kx.Bx.(y)(Ky3 x=y)), where Ki (I)is a present king of France, Bt Q i s bald. Since there i s no present king of France,(2) i s f a l s e . Hence, 1. Cf Russelli "On denoting", Mind, 1905, PP. k79- k93. 57 (1) i s f a l s e according to Russell's theory of d e f i n i t e descrip-tions . But according to P.F. Strawson, (1) presupposes the follow-ing statement: (3) The present king of France exists, or, i n symbols, (3*) (3x)Px, where P: (I) i s the present king of France. Then, according to Strawson's analysis a statement i s "nei-2 ther true nor f a l s e " i f i t s presupposition i s not f u l f i l l e d . Thus i n the present case the truth value of (1) i s "neither true nor f a l s e " , since (3) i s not f u l f i l l e d . So, whose analysis i s the correct fit more s a t i s f a c t o r y , one? My present understanding i s t h i s : ( i ) Russell's theory of d e f i n i t e descriptions seems tenable, ( i i ) On the other hand, the truth-value of (1) should be "nei-ther truth nor f a l s i t y " , f o r (1) indeed presupposes (3) and here Strawson appears to me to be r i g h t to say that a sentence i s "nei ther true nor f a l s e " when i t s presupposition i s not f u l f i l l e d . If my understanding i s r i g h t or of f e r s another plausible view then c l a s s i c a l l o g i c i s l i k e l y to be inadequate f o r both the con-cepts of "neither truth nor f a l s i t y " and "presupposition" cannot be expressed i n c l a s s i c a l l o g i c . ( I l l ) I t f a i l s to express the concept of "causality". 2. Cf. Strawson: "On r e f e r r i n g " , Mind. 1950. 3. My i m p l i c i t solution of t h i s problem i s contained below i n sec-t i o n 11, Part I I . L e t ' s f o l l o w D a v i d L e w i s a n d w r i t e " a i s a n e v e n t " a s " O a " . T h e n , t h e r e a r e l o g i c i a n s who h a v e p r o p o s e d t o i d e n t i t y " e v e n t a c a u s e s e v e n t b " w i t h "0a«^0b".k B u t t h i s i d e n t i f i c a t i o n i s n o t a c c e p t a b l e . To s e e t h i s , c o n s i d e r t h e f o l l o w i n g e x a m p l e * - ' (5) I f t h e m a t c h i s s t r u c k t h e n i t w i l l l i g h t . T h e r e f o r e , i f t h e m a t c h i s s t r u c k a n d d u n k e d i n w a t e r t h e n i t w i l l l i g h t . O r , i n s y m b o l s , (6) O s ^ O l / / . ( O s . O d ) ^ O l Now i t i s i n t u i t i v e l y c l e a r t h a t t h e i n f e r e n c e o f (5) i s u n -a c c e p t a b l e ; b u t i n c l a s s i c a l l o g i c t h e i n f e r e n c e o f (6) h o l d s , b e c a u s e i n c l a s s i c a l l o g i c we h a v e t h e f o l l o w i n g l a w * The Law o f W e a k e n i n g * I n c l a s s i c a l l o g i c , f r o m ( P ^ Q ) we c a n i n f e r : ( P . R ) D Q . T h u s t h e c o n c e p t o f c a u s a l i t y c a n n o t b e d e f i n e d i n t h e c l a s -s i c a l l o g i c i n t h e a b o v e p r o p o s e d w a y , a n d n o o t h e r way t h a t i s p l a u s i b l e a n d d e f i n a b l e i n c l a s s i c a l l o g i c i s k n o w n i n t h e l o n g h i s t o r y o f p h i l o s o p h y a n d l o g i c 4. C f . S o s a , E r n e s t ( e d . ) : C a u s a t i o n a n d C o n d i t i o n a l s , e s p e c i a l l y t h e I n t r o d u c t i o n . 5. 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 . l l k f f . 6. S o s a , o p . c i t . , e s p e c i a l l y t h e I n t r o d u c t i o n . w h e r e s : t h e s t r i k i n g o f t h e m a t c h , 1: t h e m a t c h b e i n g l i g h t , d* t h e m a t c h b e i n g d u n k e d i n w a t e r . 59 18. A\T AXIOMATIC REVIEW OF THE CLASSICAL SENTENTIAL LOGIC Since I think that c l a s s i c a l l o g i c i s l i k e l y the source of the paradoxes of confirmation and since there are a few l o g i c i a n s who t r y to challenge i t s adequacy, i t would be quite proper f o r us to have a b r i e f review before we t r y to announce and pronounce our f i n a l judgement. The f i r s t modern rigorous treatment of the c l a s s i c a l l o g i c was given by Frege i n 1879. I t contains both the sentential c a l -culus and the predicate calculus. Its sentential part has six axioms. In terms of the current notations they are:''" F l . P 3 ( Q OP) F2. ( P 3 ( Q 3 R ) ) 5 ( ( P a a ) 3 ( P 5 R ) ) ) F3. (Ps» (Q=>R)) => (Q =>(?=>R)) ? 4 . (?»Q) » ( - Q » - P ) F 5 . - P a P ?6. P => - - P In t h e i r monumental work P r i n c i p i a Mathematica, published in 1910, Whitehead and Russell reduced the number of the axioms of the sentential calculus to f i v e . They a r e : 2 PM1. (Pvp)=>? PM2. Q 3 ( P v 3 ) PM3. (P v Q) = (Qv P) PM4. ( Q 3 R ) 3 ( ( P V Q ) 3 ( P V R ) ) PM5. (Pv ( Q v R ) ) a (Qv (P v R)) It was found i n 1928 by H i l b e r t and Ackermann, among others, that the f i f t h axiom of PM5 i s redundant. In the following we 1. Frege, G.: E e g r i f f s c h r i f t . 2. Cf. Russell, B. and A.M. Whitehead: P r i n c i p i a f-lathematica, v o l . 1. Note that i n t h e i r system • and 'v' are the primi-t i v e operators and, hence, 'PaQ' i s short f o r UPvQ', 60 give i n out l ine the axiomatic system of H i l b e r t and Ackermann 3 below fo r l a t e r references I. Formation r u l e s . 1. P r imi t i ve operators: - , v . 2. A u x i l i a r y symbols: (, ). 3. Sentent ia l l e t t e r s : X , Y , Z , . . . . 4. Rules of wffs . 5. Axioms: HA1. ( X v X ) 3 X HA2. X o ( X v Y ) HA3- ( X v Y ) ^ ( Y v X ) HA k . (X=>Y) =» ((Z v X) => (Z v Y)) II . Transformation r u l e s . 1. D e f i n i t i o n s : Df 1. X-Y =df - ( - X v - Y ) Df 2. X =>Y =df - X v Y Df 3. X = Y =df ( X 3 Y ) - ( Y 3 X ) 2. Rule of in ference : modus ponens. 3. Rules of s u b s t i t u t i o n . In 1929 -Lukasiewicz discovered that the c l a s s i c a l s e n t e n t i a l ca lcu lus needs only three axioms i f we adopt - and z> as the p r imi t i ve operators. Lukasiewicz 's three axioms are : JL1. ( - P » P ) = > P JL2. p a ( - P D Q ) JL3. ( P 3 Q ) 3 ( ( Q 3 R ) 3 ( P 3 R ) ) In f a c t , i f we express them in P o l i s h notat ion, we do not even need the parentheses. In other words, the parentheses are superf luous, as shown by the fo l lowing compromised P o l i s h 3. See, H i l b e r t D. and W. Ackermann: Mathematical Logic , p . 2 7 f f . Chelsea, 1950. 4. -Lukasiewicz, J . : Elements of Mathematical Log ic , Warsaw, I929. 61 notations JL1*. 3^-PPP JL2*. ^P^-PQ JL3*. ^^PQ=>aQR-»PR However, as early as i n 1913,H.M. Sheffer discovered that the c l a s s i c a l sentential l o g i c needs, i n fact, one and only one primitive operator. One such operator, denoted as "|" and c a l l e d "Sheffer's stroke", i s defined i n the following ways-* D e f i n i t i o n of Sheffer's stroke. PIQ i s true i f f P and Q are not both true. Thus the two primitive operators of H i l b e r t ' s and Ackermann's axiomatic system f o r the c l a s s i c a l sentential calculus can be de-fined by Sheffer's stroke as followss Df a. -P =df PIP Df b. Pr>Q =df P |(Q|Q) Then, i n 1916 J.G.P.Nicod found that the c l a s s i c a l s entential l o g i c can be axiomatized with just one axiom i f we frame i t i n terms of Sheffer's stroke. The single axiom of Nicod's i s the following onei^ N i l (PI(QIR))l(((T|T)|T)I((S|Q) |((P|S) i(P|S)))) However, i n Nicod's axiomatization we need the following rule of strong modus ponens i f NI i s the only axioms P , P|(R|Q) / .*. Q 5. Sheffer, H.Ms "A set of f i v e independent postulates f o r Boo-lean algebra", Transaction of the American Mathematical So-c i e t y , 1913, pp.481-488. 6. Nicod, J.G.Ps "A reduction i n the number of the primitive pro-positions of l o g i c " , Proceedings of the Cambridge Philosophi- c a l Society. 1916, pp.32-40. 62 Again i t was Lukasiewicz who discovered that Nicod's system can be improved. This time i t turns out that not the number of the axioms, but the length of the single axiom can be shortened. In Nicod's single axiom f o r sentential l o g i c there are f i v e senten-t i a l l e t t e r s involved; but Lukasiewicz found that we need a s i n -gle axiom with only four sentential l e t t e r s . Lukasiewicz's shor-n ter axiom i s t h i s . L i t (PI(Q|R))|((SI (SIS))l((S|Q) I ((PIS) I (PIS)))) Or, i n compromised Polish notationt LI* i IIP IQRIIS I SSIISQU So f a r , i t seems that at l a s t we have the simplest system f o r the c l a s s i c a l sentential calculus, at lea s t i n the sense of the leas t number of primitive operators, the number of axioms, and the number of sentential l e t t e r s involved i n the single 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 here. 7. Cf. Borkowski, Lt Jan Lukasiewicz* Selected Works. 63 P A R T I I . A S Y N T A C T I C AND F O R M A L L Y S E M A N T I C A P P R O A C H TO T H R E E - V A L U E D C O N F I R M A T I O N T H E O R I E S , AND A T H R E E - V A L U E D S E M A N T I C S 64 1. A BRIEF INTRODUCTION In t h i s Part we w i l l s t a r t to construct several three-valued "formal theories of confirmation" that are intended to solve the paradoxes of confirmation. However, i n the end, i n each case, the formal theory of con-firmation that i s supposed to solve the paradoxes of confirmation turns out to contain some version of the paradoxes of confirma-t i o n . This becomes clear when, i n each case, we work out a com-plete semantics f o r the underlying q u a n t i f i c a t i o n a l l o g i c . Thus we l e a r n again the old lesson that syntactic, or f o r -mally semantic, adequacy of a confirmation theory may turn out to be semantically inadequate. One related lesson i s t h i s : on the one hand s y n t a c t i c a l or formally semantical p o s s i b i l i t y i s broader than semantical possi-b i l i t y : on the other hand semantical consideration i s more deci -sive than s y n t a c t i c a l or formally semantical consideration, at le a s t t h i s i s so i n the case of constructing an adequate theory of confirmation. So, the r e s u l t s of our endeavour i n t h i s Part are l a r g e l y ne-gative at the end, except that we have worked out a complete con-firmation semantics f o r three-valued q u a n t i f i c a t i o n a l l o g i c s with, or without, i d e n t i t y . This newly worked-out three-valued seman-t i c s would have i t s own independent value. 65 2. A DEFINITION OF "CONFIRMATION LOGIC" A " c o n f i r m a t i o n l o g i c " can be b r i e f l y d e f i n e d as "the u n d e r -l y i n g l o g i c o f an adequate c o n f i r m a t i o n t h e o r y " . B u t , t h e n , what c o n d i t i o n s must a c o n f i r m a t i o n t h e o r y f u l f i l l i n o r d e r to be e n t i t l e d "adequate"? S i n c e t h e r e a r e , I t h i n k , no i n f a l l i b l e c r i t e r i a o f such k i n d o f adequacy, l e t us l a y down t e n -t a t i v e l y , and o n l y t e n t a t i v e l y , the f o l l o w i n g c o n d i t i o n s , which are s u b j e c t to f u t u r e r e v i s i o n i n case we f i n d t h a t they are not j u s -t i f i a b l e t C o n d i t i o n 1 . I t must be i n a c c o r d w i t h our c o n f i r m a t i o n i n -t u i t i o n s such t h a t these i n t u i t i o n s are s y s t e m a t i z a b l e , c o n s i s t e n t and, above a l l , j u s t i f i a b l e . C o n d i t i o n 2. I t must be a b l e to s o l v e the paradoxes o f c o n -f i r m a t i o n and g i v e the s o l u t i o n a r e a s o n a b l e e x p l a n a t i o n . C o n d i t i o n 3. I t must be a b l e to be framed i n terms o f s y n -t a x o r f o r m a l s e m a n t i c s . C o n d i t i o n 4 . I t can p r o v i d e a d e f i n i t i o n o f "degree o f c o n -f i r m a t i o n " a n d / o r " p r o b a b i l i t y " so t h a t the s t a n d a r d axiom system f o r p r o b a b i l i t y i s d e r i v a b l e , f o r i t seems c l e a r t h a t p r o b a b i l i t y t h e o r y i s an e x t e n s i o n o f c o n f i r m a t i o n t h e o r y i n the same way t h a t the concept o f "degree o f c o n f i r m a t i o n " presupposes the concept o f " c o n f i r m a t i o n " . S i n c e i t seems t h a t c l a s s i c a l l o g i c cannot f u l f i l l the second o f above c o n d i t i o n s from what we have rev iewed about a l l proposed t h e o r i e s o f c o n f i r m a t i o n i n P a r t I , i t may be thus not i n our s e n -se a c o n f i r m a t i o n l o g i c . However, i t i s an open q u e s t i o n whether t h e r e i s such a t h i n g . In what f o l l o w s we w i l l s t a r t t e n t a t i v e l y to s e a r c h f o r i n t h r e e - v a l u e d l o g i c s , h o p e f u l l y , a c o n f i r m a t i o n l o g i c . 66 However, to search f o r or to construct a l l at 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 follows by looking 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 conditions. Thus the following two d e f i n i t i o n s : D e f i n i t i o n of a minimal confirmation theory. A confirmation theory i s a minimal confirmation theory i f f i t has the following properties * i ) I t employs (some form of) Hempel*s seven basic concepts; i i ) I t s a t i s f i e s some version of Hempel *s equivalence condition; i i i ) I t solves the paradoxes of confirmation. D e f i n i t i o n of a minimal confirmation l o g i c . A l o g i c i s a minimal confirmation l o g i c i f f i t i s the underlying l o g i c f o r a minimal confirmation theory. So defined, a minimal confirmation l o g i c i s not necessarily a confirmation l o g i c and a minimal confirmation theory i s not necessarily an adequate confirmation theory. However, i t i s the f i r s t step towards achieving a confirmation l o g i c . Since i t i s a le s s e r task and, hence, easier to f u l f i l l , we w i l l s t a r t our search f o r a confirmation l o g i c by looking, f i r s t , f o r a minimal confirmation l o g i c . 67 3. AN EXAMPLE OF MINIMAL CONFIRMATION LOGIC We give an example of a minimal confirmation l o g i c below. The minimal confirmation l o g i c 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 to be denoted as QMC3 » whose sentential l o g i c w i l l be denoted as MC3. The three truth-values of MC3 (and QMC3) are denoted as. t, f, n: and they are respectively to be interpreted as "truth" , " f a l s i t y " and "neither truth nor f a l s i t y " . MC3 has three primitive o p e r a t i o n s i ~ , v , and —»; and they are interpreted respectively as "the (external) negation" (or "complementation"), "disjunction" and "implication". Their truth rules are shown by the following truth tables i p V Q P —> Q t f n t f n ~P t t t t f n f t f n t t t t t n n n n t t MC3 has two other sentential connectives! & , and f-» ; and they are interpreted respectively as "conjunction" and "equivalence". They have the following truth tables i * \ P t & f Q n P <-> Q t f n t t f n t f n f f f f f t n n n f n n n t The conjunction and the equivalence of MC3 (and QMC3) are superfluous, f o r they can be defined by i t s three primitive op-erations as shown belowi 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 that although the conjunction i n the definiendum of Def 2 i s not a primitive operation, i t can be converted into a composition of primitive operations with the help of Def 1.) The 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 MC3 plus the e x i s t e n t i a l and the universal quantifications, which w i l l be respectively the generalizations of the d i s j u n c t i o n and the conjunction of MC3. We w i l l then have our f i r s t minimal confirmation theory denoted as MGT, i f we add to QMC3 the Equivalence Condition and Hempel's seven basic concepts of a confirmation theory, v i z . , "observation report", "hypothesis", "development of a hypothesis", "direct confirmation", "confirmation", "discon-firmation" and " n e u t r a l i t y " . The following 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 " d i s -conf irmation" w i l l be understood n a t u r a l l y as the "external negation" of QMC3• ( i i ) The "entailment" r e l a t i o n involved i n the d e f i n i -t i o n of "direct confirmation" and "confirmation" of MCT w i l l be the following " q u a s i - c l a s s i c a l entailment r e l a t i o n " denoted as " which i s quite p a r a l l e l to the " c l a s s i c a l entailment r e l a t i o n " : 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 r e l a t i o n . "P IJ-Q" i s true i f f "Q" must be true i n case "P" i s true. (So, 69 g i v e n t h a t "P" i s t r u e but M Q " i s a n y t h i n g o t h e r than t r u e , we haves "P |J-Q" i s f a l s e ; o therwise "P |HQ" i s t r u e . ) Now we make the f o l l o w i n g c l a i m s Theorem I . QMC3 i s a m i n i m a l c o n f i r m a t i o n l o g i c . P r o o f . T h i s i s due to the f a c t t h a t t h e c o n f i r m a t i o n t h e o r y MOT whose u n d e r l y i n g l o g i c i s QMC3 s o l v e s the paradoxes o f c o n f i r -m a t i o n as shown by the f o l l o w i n g lemmass Lemma ( 1 ) . Ba & Ra {- ( R a - » B a ) (where Bi (I) i s b l a c k , Ri Q i s a r a v e n , as a r a v e n . ) Lemma (2). ~ B a & Ra KCRa -^Ba) Lemma ( 3 - D . B a & ~ R a )(• (Ra - » B a ) Lemma (3-2). Ba&~Ra / r <'"'(Ra-•Ba) Lemma ( 4 . 1 ) . ~ B a & ~ R a ^ ( R a - ^ B a ) Lemma (4.2) ~ B a & - ~ R a ^ - ( R a - ^ B a ) When the above lemmas are a p p l i e d to the Raven H y p o t h e s i s t h a t a l l r a v e n s are b l a c k , t h e y w i l l y i e l d the f o l l o w i n g d e s i r e d r e -s u l t s when the development o f the Raven H y p o t h e s i s i s c o n s i d e r e d of an o b j e c t "a"s (1) The Raven H y p o t h e s i s i s c o n f i r m e d by and o n l y by b l a c k r a v e n s . (2) Any 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 the Raven H y p o t h e s i s . (3) Any b l a c k n o n - r a v e n i s n e u t r a l w i t h r e s p e c t t o the Raven H y p o t h e s i s ; and (4) so i s any n o n - b l a c k n o n - r a v e n . As to the d e r i v a t i o n s and n o n - d e r i v a b i l i t i e s o f the above l em-mas, we have the f o l l o w i n g t r u t h t a b l e s and p a r t i a l t r u t h t a b l e s to demonstrate them, assuming t h a t we have the Correspondence T h e -70 orem o f MC3» Lemma (1) Lemma (2) Ba Ra Ba&Ra lr (Ra - » B a ) ~Ba&Ra II- ~ ( R a - * B a ) t t t re t f f ft f t t f . f t t f f t f t t n n t n f f t t n f t f t f t t t t f f f f t t t f t f t f n f t n t n t t n n t n t n t t t t n n f f t t t f t f t n n n t t n f t Lemma (3.1) Lemma (3»2) Ba Ra Ba& ~Ra \[ (Ra - » Ba) Ba & ~Ra ||- —(Ra —> Ba) t f t t © n t t (f) f t t n Lemma (4.1) Lemma (4.2) Ba Ra ~ B a & ~ R a ||-(Ra—>Ba) ~Ba & -Ra |f- ~(Ra -^>Ba) n f t t t (D n t t t (f) f t f n Thus the paradoxes of c o n f i r m a t i o n are avo ided i n the min ima l c o n f i r m a t i o n t h e o r y MCT and, hence, QMC3 i s a min imal c o n f i r m a t i o n l o g i c as we c l a i m e d . QED. 4. THE AXIOMATIZATION OF MC3 In the fo l lowing we give an axiom system fo r MC3 for future reference * I. Formation r u l e s . 1. P r imi t i ve operators : ~ - , v, — 2 . A u x i l i a r y symbols* (, ). 3. Sentence l e t t e r s : P , Q , R , . . . . 4. Formation ru les of wffs (or sentences) : a) A sentence l e t t e r i s a wff . b) I f P,Q are wffs , so are ~P, P v Q , and P->Q. 5. Axioms: Ax 1) . P 3 ( Q 3 P ) Ax 2 ) . (PDQ) 0 ( ( Q OR) =>(P3R)) Ax 3 ) . (-Q3 - P)3 (P3Q) Ax 4). ( ( P 3 - P ) ^ P ) ^ P (where " P D Q " i s a shorthand fo r " ( P - » Q ) V Q " ' and " - P " i s fo r " P - » ~ P " . ) I I . Transformation r u l e s . 1. D e f i n i t i o n s : Def 1. P&Q =df ((P-*~P) v ( Q - » ~ Q ) ) - > - ( ( P - » — P ) v ( Q - » ~ Q ) ) Def 2 . P <-» Q =df (P-*Q) & (Q-*P) 2 . Rule of in fe rence : modus ponens, i . e . , P , 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 of uniformly rep lac ing any sentence l e t t e r fey any wff i n a theorem i s i t s e l f a theorem. 72 5. IN SEARCH OF MORE MINIMAL CONFIRMATION LOGICS L e t ' s c a l l the s e n t e n t i a l l o g i c of a min imal c o n f i r m a t i o n l o g i c , which i s a lways a q u a n t i f i c a t i o n a l l o g i c , i t s " s e n t e n t i a l m in ima l c o n f i r m a t i o n l o g i c " . Thus MC3 i s the s e n t e n t i a l min imal c o n f i r m a t i o n l o g i c of QMC3 . MC3 i s not the on l y min imal c o n f i r m a t i o n l o g i c , f o r we can have i n f i n i t e l y many such s e n t e n t i a l min imal c o n f i r m a t i o n l o g i c s i f we f o l l o w the p a t t e r n o f the c o n s t r u c t i o n o f MC3 i n m-valued l o g i c s f o r any m ^ 3 . However, i n what f o l l o w s we w i l l c o n c e n t r a t e on l y on t h r e e -v a l u e d min imal c o n f i r m a t i o n l o g i c s f o r two r e a s o n s : ( i ) F i r s t , t h r e e - v a l u e d l o g i c i s the s i m p l e s t k i nd among the m-valued l o g i c s , when m^-3. ( i i ) Among any c h o i c e s i t seems t h a t a t h r e e - v a l u e d l o g i c i s the on l y p roper c h o i c e f o r a p l a u s i b l e and adequate t h e o r y of c o n f i r m a t i o n , f o r I b e l i e v e t h a t the th ree t r u t h - v a l u e s of a p roper t h r e e - v a l u e d l o g i c seem, i n a way, to co r re spond to the th ree c o n f i r m a t i o n a l s t a t e s : c o n f i r m a t i o n , d i s c o n f i r m a t i o n , and n e u t r a l i t y . In what f o l l o w s we are go ing to c o n s t r u c t more t h r e e - v a l u e d s e n t e n t i a l min ima l c o n f i r m a t i o n l o g i c s i n a more sy s temat i c way. However, as we have mentioned t h a t t h e r e are i n f i n i t e l y many many-valued min imal c o n f i r m a t i o n l o g i c s , we cannot s tudy a l l o f them. Even i f we l i m i t o u r s e l v e s o n l y to the task of c o n -s t r u c t i n g t h r e e - v a l u e d min ima l c o n f i r m a t i o n l o g i c s the number of p o s s i b l e c a n d i d a t e s c o u l d be a c e l e s t i a l o n e , f o r there are ( 3 3 - D - 3 9 - ( 3 9 - l ) - ( 3 9 - 2 ) - ( 3 9 - 3 ) p o s s i b l e 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 s , i f we assume tha t 73 each such l o g i c contains a negation, a disjunction, a conjunc-t i o n , a conditional (or implication) and a bi-c o n d i t i o n a l (or equivalence). So, at t h i s point i t i s not a bad p o l i c y f o r us to lay down some c r i t e r i a that are i n t u i t i v e l y plausible 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 to pick out those more promising three-valued minimal confirmation l o g i c s . To begin with we have the following few c r i t e r i a : C r i t e r i o n I. The three truth-values w i l l be denoted as: t, f, and n; and they w i l l be interpreted as "truth" (or "confirma-t i o n " ) , " f a l s i t y " (or "disconfirmation") and "neither truth nor f a l s i t y " (or " n e u t r a l i t y " ) . C r i t e r i o n I I . Any sentential minimal confirmation l o g i c must contain at l e a s t a negation, a disjunction, a conjunction, a conditional (or implication) and a bi - c o n d i t i o n a l (or equiva-lence) . C r i t e r i o n I I I . Since any minimal confirmation l o g i c must be able to solve the paradoxes of confirmation when i t i s em-ployed as the underlying l o g i c of a confirmation theory, i t s negation i f i t has only one must be the external negation (denoted as "~") i n order to be able to express the concept of "complement" of a class or property such as non-black, and non-raven, etc. O f f i c i a l l y the external negation obeys, to re-peat, the following truth table: P II ~P t f f t n t C r i t e r i o n IV. Any three-valued minimal confirmation l o g i c 74 must c o n t a i n the " s t a n d a r d d i s j u n c t i o n " and the " s t a n d a r d c o n -j u n c t i o n " as i t s d i s j u n c t i o n and c o n j u n c t i o n . T h e i r t r u t h t a b l e s , to r e p e a t , a re g i v e n i n belows P V Q P & Q PX t f n t f n t t t t t f n f t f n f f f n t n n n f n The i n t e n t i o n of t h e i r a d o p t i o n i s t h i s : They g i ve us a n a -t u r a l g e n e r a l i z a t i o n f o r the e x i s t e n t i a l and 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 can then be 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 gene r -a l i z a t i o n s of the s tandard d i s j u n c t i o n and o f the s tandard c o n -j u n c t i o n . C r i t e r i o n V. Le t us denote the " c o n d i t i o n a l " (or " i m p l i c a -t i o n " ) of any t h r e e - v a l u e d min imal c o n f i r m a t i o n l o g i c as Then, i t s b i - c o n d i t i o n a l (or e q u i v a l e n c e ) w i l l be d e f i n e d n a t u -r a l l y as f o l l o w s : Def ( i ) . P-3»Q =df (P - i *Q)&(Q -^>P). With the above f i v e c r i t e r i a as the min imal requ i rement s f o r any t h r e e - v a l u e d min imal c o n f i r m a t i o n l o g i c we can prove the f o l -l ow ing theorem: Theorem II. There are 960 t h r e e - v a l u e d min imal c o n f i r m a -t i o n l o g i c s . P r o o f . (A) F i r s t , to show tha t Theorem II i s t r ue i s to c o n s t r u c t 960 c o n d i t i o n a l s such t h a t each o f them p l u s the e x t e r n a l nega -t i o n , the s tandard d i s j u n c t i o n , the s tandard c o n j u n c t i o n and the b i - c o n d i t i o n a l forms a s e n t e n t i a l min imal c o n f i r m a t i o n l o g i c . 75 (B) Secondly, to show that the above f i v e operations form a sentential minimal confirmation l o g i c i s to show that they con-t a i n the following few theses, whieh the avoidance of the paradoxes of confirmation are mainly due tot Thesis (l ) i Br & Rr \\-Rr -i*Br , where Bi (J) i s black, R: (l) i s a raven, r: a raven. Thesis ( 2 ) : ~Br & Rr (Rr #-^*Br) Thesis ( 3.1)* Br & ~Rr>Rr-i*Br Thesis ( 3 . 2 ) : B r & - R r J J M R r ^ B r ) Thesis ( 4.1): ~Br &-Rr^Rr -i»Br Thesis ( 4 . 2 ) : ~Br & ~ R r ^ ~ ( R r -i*Br) (G) Since each conditional of the minimal confirmation l o g i c s that we are constructing has to observe a l l of the above theses, we can construct each of the conditionals i n l i g h t of the theses and argue i n the following ways: a) F i r s t , Thesis (1) forces us to adopt the following p a r t i a l truth table f o r a l l of the conditionals that we are constructing: \Br Rr^ b) Then, Thesis ( 2 ) forces us to adopt the following p a r t i a l truth table f o r the conditionals: R r \ f n t y y where y€ {f,n\ . c) Thirdly, the pai r of theses Thesis (3.1) and Thesis ( 3 . 2 ) forces us to adopt either one of the following truth tables: Br t t * f t y n y t where y€ {f ,n}. d) Lastly, Theses (4.1) and ( 4 . 2 ) force us to adopt one of the following twelve possible p a r t i a l t ruth tables: fn f n f n fn fn f n fn f n fn f n f n fn f n ty XX yt X X nn tx nn yt nf tx nf yt fn tx f n yt f f tx f f yt t t yx t t ty where xe {t,f,n} and y e { 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 that there are 2 v 2 - 2 - 2 « ( 2 * 3 * 3 + 2 ' 3 ' 3 + 3 + 2 + 3 + 2 + 3 + 2 + 3 + 2 + 2 + 2 ) = 960 possible combinations f o r a l l x's and y*s. In other words we have exactly 9 6 0 conditionals (or implications) out of the possible combinations. Hence, there are 96O three-valued mini-mal confirmation l o g i c s . QED. 77 6. NEO-CLASSICAL MINIMAL CONFIRMATION LOGICS We have 960 minimal conf i rmat ion l o g i c s now, but the number i s s t i l l too large fo r us to have a d e t a i l e d study of them. More important ly , not every one of them i s as promising as the others . So, we are going to impose some more use fu l r e s t r i c t i o n s on them i n order to e l iminate the l e s s promising ones. Thus the fo l lowing two new r e s t r i c t i o n s : C r i t e r i o n VI. As f a r as the t ru th -va lues t and f are con-cerned, each cond i t iona l of any of the 960 minimal conf i rmat ion l o g i c s must comply with those of the c l a s s i c a l c o n d i t i o n a l . C r i t e r i o n VII. " P - i » P " must ho ld , fo r t h i s i s one of the most obvious theorem that we can expect from any l o g i c . Any minimal conf i rmat ion l o g i c that obeys the above two new r e s t r i c t i o n s 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 conf i rmat ion l o g i c " , s ince each t ruth table of any sentent ia l connective of the c l a s s i c a l l og i c and that of any n e o - c l a s s i c a l minimal c o n f i r -mation log i c are i d e n t i c a l as f a r as the t ru th -va lues t ' s and f s are concerned. We admit that the in tent ion of imposing the above two new r e s t r i c t i o n s i s not r e a l l y c o n f i r m a t i o n - t h e o r e t i c a l o r iented . In t h i s sense we need not to impose them. However, the in tent ion serves one of our i n t e l l e c t u a l c u r i o s i t y : Is there any important d i f fe rence between the c l a s s i c a l l o g i c and the n e o - c l a s s i c a l minimal conf i rmat ion l o g i c ? Before we can answer th i s question, l e t us t ry to ca lcu la te the number of the n e o - c l a s s i c a l minimal conf i rmat ion l o g i c s . It turns out that we have the fo l lowing r e s u l t : 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. After the imposing of the two new r e s t r i c t i o n s any condi-t i o n a l P -^ *Q of each of the ne o - c l a s s i c a l minimal confirmation l o g i c s can take only one of the following two forms: form 1 form 2 t f n t f n t t f y t f y f t t X t t y n y y t y t t where x€{t,f,n} and y€ {f,n}. So, there are 2-3*2- 2 + 2-2-2 = 32 n e o - c l a s s i c a l minimal confirmation l o g i c s . QED. 79 7. QUASI CONFIRMATION LOGICS AND QUASI CONFIRMATION THEORIES We have only 32 neo-classical minimal confirmation l o g i c s f o r the possible condidates of a confirmation l o g i c . However, i t i s s t i l l the case that not each of them i s as promising as the others, fo r not each of them can avoid Goodman's paradox:'*' and t h i s i s be-cause to avoid Goodman's paradox the following two equivalences 2 must not hold i n l i g h t of Leblanc's analysis: • • • Cond 1. ( D s ( ( D & R ) v H ^ R ) ) ) w ( D i - R ) ( Cond 2. ((D-i*~D)-M(D& R) v (-D & ~R)) ) A (D -i*-~R), where R: a ce r t a i n object i s a marble, D: an object i s drawn on a c e r t a i n day. After checking through each of the 32 n e o - c l a s s i c a l minimal confirmation l o g i c s we f i n d that Cond 1 and Cond 2 do not hold i n only 20 of the 32 n e o - c l a s s i c a l minimal confirmation l o g i c s . These 20 n e o - c l a s s i c a l minimal confirmation l o g i c s w i l l be fu r -ther c a l l e d "quasi confirmation l o g i c s " and denoted as QC3(i ) t where l ^ i ^ 2 0 , and each QC3(i) has the following respective con-d i t i o n a l s , denoted as f or each i = 1 , 2 , 3 , . . . ,20: 1> 2 t f n 3 , t f n s . 5 , 6^ t f n 7. t f n t f n 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 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 t f n t t t n f t t f n 8, t f n t t t f f t 9, 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 12, t f n t t f f f t -12, t f n t t n n t t 14 7-t f n t t f n t t t f n 15, 16 t f n t t f f t t 17, t f f t t n n t t 18 20, t f f t t f f t t t f n t t n f t t t f f t t f n t t t f f t t n f t t 1. Goodman's paradox w i l l be discussed i n Part IV below. 2. Leblanc, Hi "That posi t i v e instances are no help", pp.453-462. 80 So , i f we employ any one 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 as the u n d e r l y i n g l o g i c o f a new c o n f i r m a t i o n t h e o r y to be c a l l e d "quas i c o n f i r m a t i o n t h e o r y " , we c a n , h o p e f u l l y , not o n l y a v o i d the paradoxes o f c o n f i r m a t i o n but a l s o Goodman's p a r a d o x . 8. SOME PROPERTIES OF THE QUASI CONFIRMATION LOGICS The 20 quasi confirmation l o g i c s have some outstanding properties: (A) F i r s t , as we know, each of the 20 quasi confirmation l o g i c as the underlying l o g i c of a quasi confirmation theory can avoid the paradoxes of confirmation and (we s h a l l l a t e r show) Goodman's paradox. (3) The second outstanding property of each of the 20 quasi confirmation l o g i c s i s the following one: Theorem IV. Each of the 20 quasi confirmation l o g i c s i s "tr u t h - f u n c t i o n a l l y incomplete"; i n other words, there are truth-functions that cannot be expressed i n each of the 20 quasi confirmation l o g i c s . Proof. To give one example of truth-functions that cannot be expressed i n each of the 20 quasi confirmation l o g i c s , just consider the following truth t a b l e i p g(P) t n f n n t We claim now that there i s no truth function i n any of the 20 quasi confirmation l o g i c s that can express the truth-values of g(P). This i s becausei ( i ) a l l sen t e n t i a l operations of each of the 20 quasi confirmation l o g i c s are "normal" 1 or "q u a s i - c l a s s i c a l " , i . e . , i f the truth-values of the domain 1. Cf Rescher, Nicholas: Many-valued l o g i c , 19^9, PP.55-57. 82 of a truth function are the c l a s s i c a l truth-values t or f, then the truth values of the range of the truth function w i l l also be of the c l a s s i c a l truth-valuesj ( i i ) the truth-values of the composition of any number of normal operations are again normalj ( i i i ) but the t r u t h function g(P) i s not normal. Hence, we have that g(P) cannot be defined i n each of the 20 quasi con-firmation l o g i c s . QED. (C) Although each of the 20 quasi confirmation 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 t i s not a defect as an under-l y i n g l o g i c of a confirmation theory. Nevertheless, each of the 20 quasi confirmation l o g i c s has the following nice pro-perty: Theorem V. Each of the 20 quasi confirmation l o g i c s i s "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 the c l a s s i c a l truth-values t and f are concerned"s i n other words any three-valued normal operations can be expressed (or are definable) i n each of them. In p a r t i c u l a r , the following two t r u t h tables of a l l possible unary and binary normal operations can be expressed (and are definable) i n each of the 20 quasi confirmation l o g i c s : h(P,Q) k (P) t f n y X X y X X y y y X X y where x€ [t,f} and yg/t,f,n} • Note that Theorem V i s a consequence of the following more general theorem: Theorem VI. Each of the 20 quasi confirmation l o g i c s has an "extended (disjunctive) normal form theorem", which says that each sentence of any of the 20 quasi confirmation l o g i c s can be expressed as a conjunction of disjunctions, where each disjunct of every d i s j u n c t i o n i s one of the following "molecules" i such that these molecules have the following t r u t h tables and they can be defined i n each of the 20 quasi confirmation l o g i c s as shown belowi T p , F ( i ) p (where l ^ i ^ 2 0 ) , N p and mP P T p F ( i ) p N p mP n t X t f f f t f f f t f f n Def a. Def c. Def b. Def d. mP =df P&-P 8 4 9 . SOME MORE PROPERTIES OF THE QUASI CONFIRMATION LOGICS Before we can give some more i n t e r e s t i n g propreties about the 20 quasi confirmation l o g i c s , we need the following few d e f i n i t i o n s : D e f i n i t i o n of Isomorphic Logic. Two l o g i c s are "isomorphic l o g i c s " , i f a l l primitive operators of one lo g i c are definable by the primitive operators of the other l o g i c and vic e versa. D e f i n i t i o n of Equivalent Logic. Two l o g i c s LI and L2 are "equivalent l o g i c s " , i f the following two conditions are s a t i s -f i e d : ( i ) They are isomorphic l o g i c s ; ( i i ) they have the same "semantical entailment r e l a t i o n " , i . e . , we have: P j £ Q i f f P ^ Q. D e f i n i t i o n of Twin Logic. Two l o g i c s are "twin l o g i c s " , i f they are isomorphic l o g i c s , but not equivalent l o g i c s . Now we can claim the following few theorems about the 20 quasi confirmation l o g i c s again: Theorem VII. Each of the 20 quasi confirmation l o g i c s i s another's isomorphic l o g i c . Theorem VIII. A l l 20 quasi confirmations l o g i c s are twin l o g i c s to one another. Theorem IX. Lukasiewicz's three-valued sentential l o g i c L3 and any of the 20 (sentential) quasi confirmation l o g i c are twin logics,but L3 and any of the 20 (sentential) quasi confirmational l o g i c s are not equivalent l o g i c s . 85 We know that any of the 20 quasi confirmation l o g i c s are trut h - f u n c t i o n a l l y incomplete. However, i f we add to any of them the following operation denoted as "n" and interpreted as "the neutralized operation" which has the truth table given belowi p nP t n f n n n then each of the 20 quasi confirmation l o g i c s i s truth-function-a l l y complete. In other words, we make the following claim: Theorem X. Each of the 20 quasi confirmation l o g i c s a f t e r adding the neutralized operation to 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 that i f we add Slupecki's T-operation to Lukasiewicz's three-valued sentential l o g i c L3, then we w i l l have a truth-func-t i o n a l l y complete sentential l o g i c denoted as Now ( i ) Slu-pecki's T-operation i s nothing but the n-operation except that the former i s uninterpreted while the l a t t e r i s interpreted and denoted d i f f e r e n t l y , otherwise they obey exactly the same truth table; and ( i i ) L3 and any of the 20 (sentential) quasi confirmation l o g i c s are twin l o g i c s as indicated by Theorem IX. Hence, i f L^ i s truth-f u n c t i o n a l l y complete, so must be each of the 20 (sentential) quasi confirmation l o g i c s a f t e r adding to the n-operation. QED. 1. Cf., Rescher, N» Many-Valued Logic, 1969, p.335. 86 1 0 . THE AXIOMATIZATION OF A TRUTH-FUNCTIONALLY COMPLETE QUASI (SENTENTIAL) CONFIRMATION LOGIC S i n c e each of the 20 q u a s i ( s e n t e n t i a l ) c o n f i r m a t i o n l o g i c s a f t e r a d d i n g to i t the n - o p e r a t i o n i s t r u t h - f u n c t i o n a l l y com-p l e t e , we can p i c k out one o f them f o r s p e c i a l s t u d y f o r the s i m p l e r e a s o n t h a t a l l o t h e r q u a s i ( s e n t e n t i a l ) c o n f i r m a t i o n l o g i c s can have s i m i l a r p r o p e r t i e s . The one t h a t we w i l l p i c k out f o r s p e c i a l s tudy i s QC3(1) w i t h the n - o p e r a t i o n . T h i s s e n t e n t i a l l o g i c w i l l be denoted as CQSC3. S i n c e i t i s t r u t h - f u n c t i o n a l l y c o m p l e t e , we w i l l a l s o 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 two n e g a t i o n s , denoted r e s p e c t i v e l y as " - " and ' " " and i n t e r p r e t e d as "not" and "the c o n t r a d i c t o r y of" and read as "the i n t e r n a l n e g a t i o n " and "the i n t r i n s i c n e -g a t i o n " , which have r e s p e c t i v e l y the f o l l o w i n g t r u t h t a b l e s * p - P 'P t f f f t t n n f CQSC3 can be a x i o m a t i z e d as f o l l o w s * I . F o r m a t i o n 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 , n . 2. A u x i l i a r y symbols* (, ) . 3- Sentence l e t t e r s : P , Q , R , . . . . 4. F o r m a t i o n r u l e s of wf f s (or s e n t e n c e s ) . ( i ) A sentence l e t t e r i s a w f f . ( i i ) I f P , Q are w f f s , so are - P , P - * Q , P v Q, and nP . 5. Axioms* Ax 1. P ^ ( Q O P ) 87 Ax 2. ( P 3 Q ) 3 ( ( Q 5 R ) o ( P P R ) ) Ax 3- ( - Q o - P ) r ? (Pt>Q) Ax 4. ( ( P 3 - P ) 3 P ) 3 P Ax 5. nPo -nP Ax 6. -nPr>nP (Note that " O " i s a shorthand f o r "(P-*Q) v Q" and i s not part of the p r imi t i ve bas is of the language.) II. Transformation r u l e s . 1. D e f i n i t i o n s . Df 1. P & Q =df - ( - P v - Q ) Df 2. P<-»Q =df (P->Q) & ( Q - » P ) Df 3. - P =df P-*-P Df 4. 'P =df * P 2. Rule of in fe rence : modus ponens, i . e . , P , P-*Q f- Q 3. Rule of uniform s u b s t i t u t i o n : the r e s u l t of uniformly r e -p lac ing any sentence l e t t e r by any wff i n a theorem i s i t -s e l f a theorem. Note that the j u s t i f i c a t i o n of the above ax iomatizat ion f o r CQSG3 i s due to the fac t that i t and are twin l o g i c s . Note a lso that the q u a n t i f i c a t i o n a l genera l i za t ion 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 conf i rmation l o g i c " ) 88 11. CONFIRMATIONAL VALUATION AND A COMPLETE SEMANTICS FOR ANY THREE-VALUED QUANTIFICATIONAL LOGIC WITH IDENTITY Up to t h i s point we have not considered the valuation of the atomic and the general sentences of any quasi confirmation l o g i c with, or without, i d e n t i t y . In t h i s section we w i l l give a f u l l y developed semantics, which i s complete, f o r any three-valued 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 point of confirma-t i o n a l valuation. Truth tables as a matrix language t e l l us only the valuations of molecular sentences. So, how do we evaluate the atomic sentences and, i n general, the atomic as well as the general sentences of any of the quasi confirmation l o g i c s ? Before we can do t h i s , l e t us give b r i e f l y the grammar of CQQC3 and some relevant p r e l i m i -nary concepts. 1 I. The grammar of CQQC3 with i d e n t i t y . The expressions of the language of CQQC3 with i d e n t i t y are any f i n i t e length of strings of symbols of the following two kinds* (1) Variables. The i n d i v i d u a l variables are the lower case i t a l i c l e t t e r s "x" through "z" with, or without, numeral subscripts. (2) Constants. They f a l l into two major kinds* ( i ) The l o g i c a l constants, ( i i ) The non-logical constants. These include* a) Predicate l e t t e r s . b) Individual constants. A series of fundamental notions are, then, defined as follows* A predicate l e t t e r of degree n (or an n-ary, or n-place pre- dicate l e t t e r ) i s a predicate l e t t e r having as superscripts a nu-l.Here i n t h i s respect I t r y to follow Mates' approach expressed i n h i s Elementary Logic as c l o s e l y as possible, although there are i n e v i t a b l y some fundamental differences. 89 meral f o r the positive integer n, which w i l l usually be omitted, being understood. An i n d i v i d u a l symbol i s either an i n d i v i d u a l variable or an i n d i v i d u a l constant. An atomic formula i s either a sentence l e t t e r which i s de-fined as a 0-place predicate l e t t e r or an i-place predicate l e t -t e r followed by a s t r i n g of i i n d i v i d u a l symbols, where i i s any po s i t i v e integer. A formula i s either an atomic formula or else defined recur-s i v e l y by the usual formation r u l e s . Furthermore, an occurrence of an i n d i v i d u a l variable v i n a formula A i s bound. i f i t i s within an occurrence i n A of a f o r -mula of the form (3v)B or of the form(\/v)B; otherwise i t i s a free occurrence. A sentence (or wff) 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 variable occurs f r e e . F i n a l l y , a formula that i s not an atomic formula i s c a l l e d general, i f i t begins with a universal or e x i s t e n t i a l q u a n t i f i e r ; otherwise i t i s c a l l e d molecular. And, a sentence i s a sentence of the sentential calculus or sentential l o g i c (say, of CQQC3), i f i t contains no i n d i v i d u a l symbol. I I . Some additi o n a l terminology and semantical concepts. We need the following additional terminology and semantical concepts before we can discuss about confirmational valuations! (a) For any formula A, i n d i v i d u a l variable v, and i n d i v i d u a l symbol w, A v / / w i s the r e s u l t of replacing a l l free occurrences of v i n A by occurrences of 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 can 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 (and p r e c i s e l y l a t e r ) i n two steps* (1) "c" has a d e n o t a t i o n , namely, c , i f f c e x i s t s , (where "c" i s an i n d i v i d u a l c o n s t a n t 1 ) : and ( 2 ) i f c does not e x i s t , 2 t h e n " A ( c ) " has the t r u t h - v a l u e n , (where "A(c )" i s an atomic sentence c o n t a i n i n g "c" as one o f i t s i n d i v i d u a l c o n s t a n t s ) . S i n c e what we are g o i n g to have i s an a b s o l u t e s e m a n t i c s , any g i v e n domain D o f 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 i s always assumed to be the a c t u a l w o r l d ^ or i t s p r o p e r subset which i n c l u d e s the empty s e t . Thus the v a l u a t i o n o f any atomic sentence whose i n d i v i d u a l c o n -s t a n t s a l l have d e n o t a t i o n s c a n be c a r r i e d out i n the s i m i l a r way to the c l a s s i c a l v a l u a t i o n . Then, to p r e d i c a t e l e t t e r s o f degree 1 i t a s s i g n s " p r o p e r t i e s " ^ ( o r , more p r e c i s e l y , s e t s o f e n t i t i e s o r n o n - e n t i t i e s ) . To p r e d i -c a t e l e t t e r s o f degree 2 i t a s s i g n s b i n a r y " r e l a t i o n s " of e n t i t i e s o r n o n - e n t i t i e s , ^ and so o n . 1 . What an i n d i v i d u a l c o n s t a n t denoted i s c a l l e d " i n d i v i d u a l " , "ob-j e c t " o r " e n t i t y " . 2 . S i n c e a p r e c i s e d e f i n i t i o n o f c o n f i r m a t i o n r e q u i r e s r e f e r e n c e to some d e f i n i t e "language o f s c i e n c e " L as Hempel s a y s , we can assume t h a t any i n d i v i d u a l c o n s t a n t "c" appears i n L has a meaning and , hence, whether "c" e x i s t s ( i n the a c t u a l w o r l d ) o r not can be d e -c i d e d w i t h the h e l p o f the meaning o f "c" i n L . 3 . A second way to deve lop our 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 an i n t e r p r e t a t i o n i s a w o r l d o f p o s s i b l e e n t i t i e s and t h a t we a l s o have a p r e d i c a t e "A" meaning " . . . 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 not the case t h a t A c , we have case ( 2 ) above; o t h e r w i s e , we have case ( 1 ) . 4. 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 . 5. I f any se t c o n t a i n s some n o n - e n t i t i e s , the se t w i l l be c o n s i d e r e d as no 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 " . 6 . A t h i r d way to d e v e l o p our 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 never to l e t any n o n - e n t i t i e s e n t e r our b a s i c v o c a b u l a r i e s . Thus , any n o n - e n t i t y i s not i n any g i v e n domain nor i n any r e l a t i o n . And when we encoun-t e r some (complex) terms 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 by our t h e o r y o f d e f i n i t e d e s c r i p t i o n s . Thus , when no unique e n t i t y c a n s a t i s f y the d e s c r i p t i v e phrase "(the x)Dx", we have t h a t the t r u t h ^ v a l u e o f "Dc" i s n (where "(the x )" i s our 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 to sentence l e t t e r s i t assigns truth-values or one of the confirmation-statuses, v i z . , confirmation, disconfirmation or neu-t r a l i t y . ^ (c) Let Int and Int* he two interpretations of CQQC3 with or 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 constant. Then, Int i s a k-variant of Int* exactly when Int and Int* are the same or d i f f e r only i n what they assign to k. (d) A sentence i s v a l i d (or l o g i c a l l y true), i f i t i s true un-der every interpretations otherwise i t i s i n v a l i d . (e) A sentence A i s a consequence of a set of sentences S, which can also 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 of S are true and A i s not true. (f) A set of sentences S i s consistent (or 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 of S are true. F i n a l l y , we can give the confirmational valuation i n our o f f i -c i a l sense below. I I I . Confirmational valuation. At l a s t we can give the c o n f i r -mational valuation p r e c i s e l y as follows: (1) Let us st a r t from the simplest case. Consider the 1-place atomic sentence: Ga. Case 1. Suppose that ( i ) under an i n t e r p r e t a t i o n Int the i n -divi d u a l constant "a" denotes no e n t i t y i n a domain D or ( i i ) the predicate l e t t e r "G" refe r s to no property under the Int. Then, we make the assignment: V(Ga)=n, where " V ( )" means "the valuation of ( )" or "the truth-value of ( )". Case 2. Suppose that the "a" under the Int denotes some en-7. This i s another fundamental difference of the confirmational va-lua t i o n and the c l a s s i c a l valuation. 8 . Note that here we do not adopt the second or the t h i r d way of de-veloping our confirmational valuation. 92 t i t y , say, a i n domain D, and also suppose that the "GH under the same Int refe r s to some property, say, G. Then, there are two sub-cases to be further considered: Subcase 2.1. Suppose that the a has the property G under the Int. Then, we make the assignment: V(Ga)= t under the Int. Subcase 2 .2 . Suppose that the a f a i l s to possess the proper-ty G under the Int. Then, we have: V(Ga) = f under the Int. (2) Now consider the general case of a sentence of j-place pre-cate l e t t e r , where j > l , such as t h i s one: Ga^a2...a... Case 1. Suppose that ( i ) under i n t e r p r e t a t i o n Int one or more or a l l of the i n d i v i d u a l constants "a^","a 2",...,"aj" f a i l to denote any e n t i t y i n a given domain D, or ( i i ) the j-place predicate l e t t e r "G" refers to no property or no r e l a t i o n under the Int. Then, we have: VfGa^ag...a^)= n under the Int. Case 2. Suppose that each of the j i n d i v i d u a l constants de-notes some entity, say, a-^.a^,. •. »a.. i n a domain D under an i n t e r -pretation Int, and also that the j-place predicate NG M refers to some property or r e l a t i o n , say G under the same Int. Then, we have: ( i ) V(Ga^a2 • • .a^ ) = t u n c * e r "the Int, i f <a^,a 2,... » aj> n a s the pro-perty or r e l a t i o n G under the Int: otherwise ( i i ) V(Ga 1a 2.•.a^) =f under the Int. (3) The valuations of ~ A , A v B, A & B, A-»B, A+*B, etc. can be car r i e d out according to t h e i r truth tables. (So can be any sen-tence involving some new sentential connectives whose truth tables are once given and added to CQQC3-) (4) Suppose that A=(\/v)B. Then, ( i ) V ( A)=t under interpre-t a t i o n Int exactly when V(B v / / k) = t on every k-variant of the Int; ( i i ) V ( A)=f under the Int, exactly when V ( B V / / k ) = f f o r at le a s t 93 one k - v a r i a n t o f t h e I n t ; ( i i i ) i n a l l o t h e r c a s e s , V ( A ) = n u n d e r t h e I n t . (5) S u p p o s e t h a t A = ( 3v )B . T h e n , ( i ) V ( A ) = t u n d e r a n i n t e r -p r e t a t i o n I n t e x a c t l y w h e n V ( B V / ^ C ) = t f o r a t l e a s t one k - v a r i a n t o f t h e I n t ; ( i i ) V ( A ) = f u n d e r t h e I n t e x a c t l y w h e n V ( B V / / k ) = f f o r e v e r y k - v a r i a n t o f t h e I n t ; ( i i i ) i n a l l o t h e r c a s e s , V ( A ) = n u n d e r t h e I n t . (6) F i n a l l y , how do we e v a l u a t e a n i d e n t i t y s e n t e n c e s u c h a s t h i s o n e : d = e ? ( w h e r e " d " a n d B e " a r e t w o i n d i v i d u a l c o n s t a n t s . ) We e v a l u a t e i t i n t h e f o l l o w i n g way* ( i ) I n t h e c a s e w h e n w h a t " d " o r " e " o r "both d e n o t e a r e n o t i n a d o m a i n D u n d e r a n i n t e r p r e -t a t i o n I n t , we h a v e : V ( d = e ) = n u n d e r t h e I n t ; ( i i ) i n t h e c a s e w h e r e b o t h " d " a n d " e " d e n o t e some e n t i t i e s i n D u n d e r t h e I n t , we h a v e t h a t V ( d = e ) = t u n d e r t h e I n t i f t h e y d e n o t e t h e same e n t i t y i n D u n d e r t h e I n t ; o t h e r w i s e ( i i i ) we h a v e : V ( d = e ) = f u n d e r t h e I n t . (7) Now we c a n i n t r o d u c e a u s e f u l t e r m . U n d e r 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 , s e n t e n c e s s u c h a s "(3x ) G x " , " ( V x ) H x " , " K a ' \ " a = a " a n d " a = b H w i l l h a v e t h e t r u t h - v a l u e n i f t h e g i v e n d o m a i n i s t h e e m p t y s e t ( a n d , h e n c e , " a " a n d " b " s t a n d f o r n o e n t i t i e s . ) S u c h a k i n d o f s e n t e n c e s w i l l b e c a l l e d " a n e m p t y s e n t e n c e " ( r e l a t i v e t o t h e i n t e r p r e t a t i o n w h e n t h e d o m a i n i s t h e e m p t y s e t ) . S o , a t l a s t we h a v e p r o v i d e d a f u l l y d e v e l o p e d o r c o m p l e t e s e -m a n t i c s f o r o u r t h r e e - v a l u e d l o g i c s i n c l u d i n g CQQC3 ( a s w e l l a s f o r a n y 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 w i t h i d e n t i t y i f i t w i l l a d o p t 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 a s i t s s e m a n t i c s . ) 94 12. THE FAILURE OF THE 20 QUASI-CONFIRMATION LOGICS TO SA-TISFY A MONOTONICITY CONDITION1 It appears that there are 20 semantic and syntact ic s o l u -t ions of the paradoxes of conf i rmation from the semantics pro -vided i n the l a s t s e c t i o n . However, these 20 solut ions seem to c o n f l i c t with a monotonicity c o n d i t i o n . The i n t u i t i v e idea of what might be c a l l e d monotonicity i s that i t should be the case that i f to observation reports that confirm a hypothesis are added some observation reports that cannot d isconf i rm that hypothesis the r e s u l t i n g observation r e -ports would s t i l l confirm the hypothesis . However, t h i s i s not the case f o r any of the 20 q u a s i - c o n -f i rmat ion l o g i c s that are claimed to solve the paradoxes of conf i rmat ion . Thus consider an observation report 0 that con-f i rms the raven hypothesis! H: Ct/x) (Rx-*Bx) ; fo r example, l e t 0 be the conjunct ion (1) (Ra 1 & B a ^ & . . . & (Ra R& Ba R ) . The development of the raven hypothesis r e l a t i v e to t h i s obser-va t ion i s D, the conjunct ion (2) ( R a 1 - » B a 1 ) & . . . & (Ra n~*Ba n) . Now, l e t 0^ be the claimed observation report based on an object whose name i s "b" such that "Bb" i s true and "Kb" i s n e u t r a l ; 0^, 1. This sec t ion was r e - w r i t t e n a f t e r my ora l defence when a s e r i -ous mistake was discovered by Prof . Richard E. Robinson. 95 then, i s ( 3 ) Bb & ~Rb . The development of H rel a t i v e to 0 ^ i s the conditional (k) Rb -»Bb , where "-*" i s any conditional of the 2 0 quasi-confirmation l o g i c s . Not only does 0^ neither semantically e n t a i l s nor semantically e n t a i l s the negation of D^, but the observation report 0* ob-tained by conjoining 0 and 0^ (5) (Ra 1 & Ba 1) & ... & (Ra n& Ba n) & (Rb &Bb) neither semantically e n t a i l s the development D* ( 6 ) (Ra x -tBa^^) & ... & ( R a n ~ * B a n ) & (Rb-*Bb) nor semantically e n t a i l s the negation of D*. Thus, 0 * neither confirms nor disconfirms the raven hypo-thesis 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 under the semantics given i n the l a s t section? If "Rb" i s neutral, either what "b" names i s not a bona f i d e object or the predicate "R" i s interpreted by something that i s not a bona f i d e property according to our confirmational seman-t i c s . But i f "b" names something that i s not a bona f i d e object, then "Bb" w i l l not be true; and i f the in 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 according to our conf irmational semantics, f o r any i such that 1 4 i 4 n . Thus, according to our confirmational semantics given i n the 96 l a s t section the problem outlined above would not happen. Un-fortunately, the above semantics i s not the only one possible. There i s a r e a l problem of finding a logic that both solves the paradoxes of confirmation and s a t i s f i e s the monotonicity condi-t i o n i n t u i t i v e l y presented above. Note that i t might appear that Hempel's solution or disso-l u t i o n of the paradoxes of confirmation runs afoul of the mono-t o n i c i t y condition as well. Thus, l e t 0 and D be as above, and l e t 0_2 be the observation report based on an object whose name i s "c" such that "Rc" i s true and no truth-value i s assigned to "Be". Then, p_2 i s simply "Rc" and the development of the raven hypothesis r e l a t i v e to i s : (7) R C D B C . However, the observation report 0 # obtained by conjoining 0 and p_2 i s : (8) (Ra 1.Ba 1)" ... • (Ra n.Ba n)'Rc which neither semantically e n t a i l s the development D# (9) (Ra 1 3 B a 1 ) (Ra n=>Ba n). (Rc P B C ) nor semantically e n t a i l s the negation of D#. Thus, 0 # neither confirms nor disconfirms the raven hypothe-s i s . However, t h i s i s not a problem f o r Hempel because i f "Be" i s f a l s e , then the hypothesis w i l l be disconfirmed, and so i t should be the case that 0 # neither confirms nor disconfirms the 2 . Technically 0 and D restated i n c l a s s i c a l l o g i c notation. 97 raven hypothesis. This i s quite d i f f e r e n t from the case f o r the 20 solutions discussed above. In short, the 20 solutions of the paradoxes of confirmation discussed above are to be rejected, but t h i s r e j e c t i o n need not apply to Hempel's theory. 98 P A R T I I I . A C O N F I R M A T I O N L O G I C AND A P L A U S I B L E S O L U T I O N OF T H E P A R A D O X E S OF C O N F I R M A T I O N 99 1. A BRIEF INTRODUCTION In t h i s Part we s t a r t afresh by paying more attention to se-mantical problems to construct a " q u a s i - c l a s s i c a l three-valued l o g i c " , which i s very close to the c l a s s i c a l l o g i c . Then, we strengthen the q u a s i - c l a s s i c a l three-valued l o g i c to a "confirmation l o g i c " by introducing into i t some new sen-t e n t i a l connectives. Among them there are an i n t e r n a l negation which i s d i f f e r e n t from the external negation, an implication which i s d i f f e r e n t from the conditional, 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 confirmation l o g i c as the underlying l o g i c of a new confirmation theory c a l l e d the " i n t e r n a l c o n f i r -mation theory" and go on to solve the paradoxes of confirmation i n the following way« On the one hand the paradoxes of confirma-t i o n are derivable as the natural outcome of the Raven Hypothe-s i s , which i s commonly understood as a "conditional law"; on the other hand the paradoxes of confirmation are avoided i f the Raven Hypothesis i s framed as a de-neutralized Implicational Raven Hy-pothesis, which i s claimed to represent an "implicational law". Only i n the l a t t e r sense of the de-neutralized Implicational Ra-ven Hypothesis can we say that we have a plausible solution of the paradoxes of confirmation. 100 2. A THREE-VALUED QUASI-CLASSICAL LOGIC Consider the following three-valued matrix language* ( i ) I t s three truth-values are* t (or 1), f (or -1), and n (or 0). And they are interpreted respectively as "truth" (or "confirmation"), " f a l s i t y " (or "disconfirmation") and "neither truth nor f a l s i t y " (or " n e u t r a l i t y " ) . ( i i ) The matrix language has the following f i v e sentential connectives (or operations)* ~ (the external negation, or complementation), v (disjunction), & (conjunction), -» (conditional), and (bi-conditional) * and they are respectively to be read and interpreted as* non, or, and, i f , then i f and only i f (which i s abbreviated as* i f f ) . ( i i i ) The f i v e sentential connectives obey the following truth rules described by the truth tables as shown below* p V Q P & Q P - 4 Q P *-* Q t f n t f n t f n t f n ~P t t t t f n t f n t f n f t f n f f f t t t f t t t t n n n f n t t t n t t t Now we w i l l have a decent three-valued sentential l o g i c de-p noted as L~ as well as i t s q u a n t i f i c a t i o n a l extension (with or 101 2 w i t h o u t i d e n t i t y ) denoted as QLj, i f we add to the above m a t r i x language a f u l l y - d e v e l o p e d s e m a n t i c s , e s p e c i a l l y the v a l u a t i o n s o f i t s atomic sentences as w e l l as the v a l u a t i o n s o f i t s q u a n t i -f i e d s e n t e n c e s . Here we w i l l s i m p l y add to i t the f u l l y - d e v e l o p e d semant ics o f c o n f i r m a t i o n a l v a l u a t i o n s g i v e n i n s e c t i o n 11 o f P a r t I I . Thus we c l a i m t h a t we have a decent t h r e e - v a l u e d s e n t e n t i a l 2 2 l o g i c L - and a decent 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 Q L ~ . 102 3. AN AXIOMATIZATION OF I2 For the sake of l a t e r references and discussions we w i l l 2 give here, f o r the time being, also an axiom system f o r i t as follows! I. Formation r u l e s . 1. Primitive operators: ~ , v, &. 2 . A u x i l i a r y symbols i ( , ) . 3. Sentence l e t t e r s : P,Q,R,... . 4. Formation rules of wffs (or sentences). a) A sentence l e t t e r i s a wff. b) I f A, B are wffs, so are ~A, AvB, and A&B. c) These are the only rules of wffs. Axioms. Ax 1.1 (A v A)-»A Ax 1 . 2 A - » ( A V B ) Ax 1.3 (A v B ) - * (B vA) Ax 1.4 (A-»B)-*((CvA)-»(CvB)) Ax 2.1 A — » ( A & A ) Ax 2 . 2 ( 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 of inference: modus ponens, i . e . , A , A-»B I- B 3. Three laws of interchange. a) Law of double-negation (abbr. as DN). Suppose that a wff A occurs i n a theorem. Then, A can be replaced at any of i t s occurrences by A, and vice versa. b) Law of transposition (abbr. as Transp). Any occurrence of a wff A&B (respectively, AvB) i n a theorem can be replaced by B& A (respectively, B v A ) , and vice versa. c) Law of iderapotence (abbr. as Id). Any occurrence of a wff A i n a theorem can be replaced by A v A (or, by A&A), and vice versa. 4. Rule of uniform substitution: the r e s u l t of uniformly replacing 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 2 Note that there are a few important differences of the axiom system of the three-valued sentential l o g i c L^ given i n the l a s t section and the axiom system f o r the c l a s s i c a l two-valued sentential l o g i c C2 as i t i s given by Hi l b e r t and Ackermanni 1 (1) Their axiom system f o r C2 needs only two pri m i t i v e oper-ators, v i z . the c l a s s i c a l negation —r and the c l a s s i c a l disjunc-2 t i o n v; while i n our axiom system of L^ we need three primitive operators. (2) Their axiom system has four axioms, v i z . HA 1. (A v A) OA (where "A 3 B " i s defined as " T A V B " . ) HA 2. A o ( A v B) HA 3- (A v B) O (B v A) HA 4. ( A v B ) o ( ( C v A ) o ( C v B ) ) 2 But i n our axiom system f o r L^ we need something more i f the three laws of interchange are omitted f o r the conjunctor of 2 Lj i s a primitive operator which i s not definable by other p r i -2 mitive operators of L^. (3) In C2 the conjunctor can be defined by i t s negation and disjunctor as follows i p . Q =df - 7 ( 7 P Y - 7 Q ) > where "•" i s the "conjunctor" of C2, such that the following two conditions are s a t i s f i e d * ( i ) ( p . Q ) = ^ ( ^ P v - 7 Q ) , and 1. Cf. H i l b e r t D. and W. Ackermanm P r i n c i p l e s of Mathematical Logic, t r . by L.M.Hammond, G.G.Leckie and F . Stemhardt, 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, "the valuation of A".) In other words, we have the following theorem* Theorem 1. In c l a s s i c a l s entential l o g i c C2, (1) since V(P.Q) = Vt>(-rPv->Q)), i t follows that ( i ) (P.Q)s-n>Pv-*Q) and, hence, i t i s legitimate to make the following d e f i n i t i o n ! ( i i ) P.Q =df ->(-rPv-Q): and, dually, (2) since V(P v Q) = V(->(--P. -»Q)), i t follows that ( i i i ) (Pv Q)= -»(-rP.~'Q) and, hence, i t i s also legitimate 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 we can only have the following weaker, and yet very important, theoremt 2 Theorem 2. In the three-valued sentential l o g i c L^, (1) i t i s not true that V(P&Q) = V H - P v - Q ) ) . Hence, ( i ) i t i s not legitimate to make the d e f i n i t i o n * P&Q =df ~(~Pv~Q), although 2 ( i i ) i t i s the case that i n we always have* P&Q-f*~(~P v~Q); also, dually, (2) i t i s f a l s e that ( i i i ) V(PvQ) = V(~ (~P &~Q)), i t follows that (iv) i t i s not legitimate to make the d e f i n i t i o n * 2 PvQ =df ~(~P &~Q), although (v) i t i s always true i n to have* P v Q<-»~(~P&~Q). (The importance of the above two theorems w i l l become c l e a r when we l a t e r discuss the question* what i s a "deviant logic"?) We l i s t here some of the more important theorems f o r the pur-pose of l a t e r reference* Theorem 3 (The completeness theorem). I f A|f-B, then Al-B, 2 f o r any two wffs A and B of L^.where "|(-" and " (-" are respective-l y the "semantical entailment r e l a t i o n " and 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). If A(-B , then A||-B , f o r 2 any two wffs A and B of L y Theorem 5 (The deduction theorem). I f A|-B , then i t follows that }-(A-*B), where A i s any f i n i t e set of wffs and B i s any wff of Ly 2 Theorem 6 (The converse deduction theorem). In , i f |-(A->B), then A |-B , f o r any two wffs 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 and C2 2 that j u s t i f i e s c a l l i n g a "three-valued q u a s i - c l a s s i c a l senten-t i a l l o g i c " t Theorem 7. ( i ) A|-B holds i n C2 i f f A |-B holds i n : and ( 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, also j u s t i f i e s the use of the same notation f o r the semantical entailment relations and the 2 sy n t a c t i c a l consequence relations respectively of C2 and of L~. 107 5. A QUASI-HEMPELEAN CONFIRMATION THEORY We w i l l have a quasi-Hempelean external confirmation theory, i f the following three conditions are s a t i s f i e d i (I) Its underlying l o g i c 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 -t i o n a l l o g i c QL^ (without i d e n t i t y ) . (II) I t has also Hempel's Equivalence Condition 1 provided that the "equivalence" here simply means the " b i - c o n d i t i o n a l " of QL|. (III) I t has also Hempel*s seven basic concepts of a c o n f i r -mation theory, namely, ( i ) observation report, ( i i ) hypothesis, ( i i i ) development of a hypothesis, (iv) d i r e c t confirmation, (v) 2 confirmation, (vi) disconfirmation, and ( v i i ) n e u t r a l i t y , pro-vided that the "entailment" involved i n the d e f i n i t i o n s of the l a t t e r four concepts w i l l be understood as the semantical e n t a i l -ment r e l a t i o n " \[" and the "disconf irmation" of our new c o n f i r -mation theory w i l l be defined i n terms of the external negation 2 of QL^. Thus i n our new confirmation theory "B confirms H" i s "B lhH d" and "B disconfirms H" i s "B ||-~Hd", where H d i s "the de-velopment of H with respect to the i n d i v i d u a l s whose names occur e s s e n t i a l l y i n B".^ Since the new confirmation theory i s so close to Hempel*s theory of confirmation and since i t s disconfirmation i s defined 2 i n terms of the external negation of QL^ , we w i l l c a l l i t "the quasi-Hempelean external confirmation theory" or, simply, "the external confirmation theory". 1. Cf. Hempelt Aspects of S c i e n t i f i c Explanation, p.31. 2. Ibid, p.37 or, better, section 5, Part I. 3. Ibid, p.36 or, better, section 5, Part I. 1 6. PARADOXES 0? CONFIRMATION REGAINED In the newly constructed quasi-Henpe lean external confirma t i o n theory a l l "paradoxes of confirmation" can be re-derived, 2 mainly because i n QL^ we have the following 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 e n t i t y . These truths can be v e r i f i e d by t h e i r respective truth tables with the help of the meaning of the semantical e n t a i l -ment r e l a t i o n . With the help of our d e f i n i t i o n s of confirma-t i o n and disconfirmation i t follows from the above r e s u l t s that s ( A ) Any black raven confirms the Raven Hypothesis that a l l ravens are black or, i n symbols, (Vx) (Rx-»Bx). ( B ) Any non-black non-raven also confirms the Raven Hypo-the s i s . (C) Any black non-raven confirms the Raven Hypothesis, i n p a r t i c u l a r , (C.l) Any non-raven confirms the Raven Hypothesis, and (C-2) Anything black also confirms the Raven Hypothesis. (D) The Raven Hypothesis i s disconfirmed by and only by a non-black raven. I t i s indeed very odd that we have the r e s u l t s (B), (C), 109 (C.l) and (C.2) they are "the paradoxes of confirmation" regained i n our new quasi-Hempelean external confirmation theory. 110 7. AN ANALYSIS OF THE FAILURE The three-valued l o g i c s that we have constructed up to t h i s point a l l f a i l as confirmation l o g i c s , f o r the paradoxes of con-firmation are re-derivable i n the confirmation theories which employ these l o g i c s . The reason they f a i l i s that they repeat the same pattern of mistake. The pattern of t h i s mistake i s that a l l of them con-t a i n " q u a s i - c l a s s i c a l conditionals", by which we mean any condi-t i o n a l s whose truth tables are of the following formi P _k^ Q PX t f n t t f u f t t V n X y z where u,v,x,y,z * f t , f t n \ . In other words, a q u a s i - c l a s s i c a l con-d i t i o n a l of a three-valued l o g i c i s a binary operation whose truth table i s i d e n t i c a l to the one of the c l a s s i c a l conditional as f a r as the truth-values t (truth) and f ( f a l s i t y ) for P and Q are con-cerned . Any confirmation theory whose underlying l o g i c has such a q u a s i - c l a s s i c a l conditional w i l l c e r t a i n l y contain the paradoxes of confirmation with the minimal help of Hempel 1s concepts of (i) the development of a hypothesis, ( i i ) confirmation, ( i i i ) a q u a s i - c l a s s i c a l negation,and (iv) a q u a s i - c l a s s i c a l conjunc-tion , which have the following truth tables t P and Q n e g a t i o n of P P \ t f n t t f n f f f f f t n n f n X where x e { t , f , n } . I l l To see that the paradoxes of confirmation can be derived i n such a q u a s i - c l a s s i c a l confirmation theory i s to see that i n such a l o g i c we have» (1) f - i ^ f t and (2) f i t t and so, by d e f i n i t i o n of confirmation (with the help of the concept of development of a hypothesis) that we havei (A) Any non-black non-raven confirms the Raven Hypothesis; and (B) any non-raven which i s black also confirms the Raven Hypothesis. Therefore , we must not repeat t h i s same pattern. However, to r e a l i z e that we should not repeat the same pat-tern of mistake does not mean that we should surpress the quasi-c l a s s i c a l conditional, otherwise we would commit another mistake of surpressing i n our new theory the expression of the paradoxes of confirmation, instead of solving them. Here i t seems that we are trapped i n a predicament! E i t h e r ( i ) we surpress any q u a s i - c l a s s i c a l conditional from our new l o g i c and, hence we surpress the e x p r e s s i b i l i t y of the paradoxes of confirmation , or else ( i i ) we introduce a q u a s i - c l a s s i c a l conditional into our new l o g i c and derive the paradoxes of con-firmation i n our new theory of confirmation. In the l a t t e r case we have the paradoxes of confirmation and i n the former case we do not even have a chance to solve the paradoxes of confirmation. 112 8. CONDITIONAL LAW VS. IMPLICATIONAL LAW It seems that the only way to get out the predicament des-cribed i n the l a s t section i s t h i s i ( i ) F i r s t , we admit that we need a q u a s i - c l a s s i c a l condi-t i o n a l and, hence, we have also had to admit that the paradoxes of confirmation are not solvable 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 conditional as a univer-s a l ( q u a s i - c l a s s i c a l ) conditional law. ( i i ) But that does not mean we cannot frame the Raven Hypo-thesis i n a d i f f e r e n t way, say, as a universal implicational law such that the paradoxes of confirmation are solvable. The Raven Hypothesis as a conditional law i s framed, say, i n QL-j i n the following way* Hi ( V X ) ( R X ^ B X ) , which w i l l be c a l l e d "the Conditional Raven Hypothesis" from now on. Suppose that we have also an "Implicational Raven Hypothe-s i s " (or "Causal Raven Hypothesis") which i s symbolized as H*i (Vx)(Rx ==>Bx), and read s t i l l as " a l l ravens are black" and yet understood i n the following sensei $. H^t That x i s a raven implies that x i s black, (or, i n a sense, also as H;>« That x i s a raven causally implies that x i s black.) 113 Now we ask: can we f i n d such a new connective " =£" to be un-d e r s t o o d as "implication" i n the sense of H]_ (and, perhaps, a l s o * i n the sense o f H 2 ) such that the paradoxes of confirmation are not derivable i n the proposed new form of "Implicational Raven Hy-pothesis"? If there i s such an implication i n a three-valued l o g i c , then we w i l l have a (causally) implicational solution of the paradoxes of confirmation, though we do not have any ( q u a s i - c l a s s i c a l l y ) con-d i t i o n a l solution. However, t h i s i s good enough f o r us to get out of the predi-cament of solving or not solving the paradoxes of confirmation. Anyway, t h i s i s almost a l a s t chance f o r us to solve the pa-radoxes of confirmation. So, we w i l l s t a r t to search f o r such an implication i n the next section. 114 9. IN SEARCH OF AN IMPLICATION I A few conditions must be observed i f there i s such an im p l i -cations (A) Condition 1. The implication should not be quasi-c l a s s i c a l . That i s , i t should not have the following p a r t i a l t r uth table J £^4 t f n P=*-Q t f n t f t t Otherwise, as we argue i n section 7 the paradoxes of confirma-t i o n w i l l be re-derivable. (B) Condition 2. The implication should be "semi-classical", by which we mean i t should contain the following p a r t i a l truth tables P =#Q t f n t f n t f Otherwise, a black raven won't be able to confirm the Implica-t i o n a l Raven Hypothesis and a non-black raven won't be able to disconfirm i t ei t h e r . This 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 . So, i t i s reasonable to have Condition 2. By Condition 1 and Condition 2 there are s t i l l the following p o s s i b i l i t i e s f o r a binary operation denoted as possible implication! to be the * Q n t f n t f y l x l x2 y2 y3 y^ y5 115 where xl,x2 €ff,n} and y l , y 2 , y 3 , y 4 , y 5 e{ttt,nj. . So, we have i = 2 ' 2 - 3 - 3 -3-3-3 = 972 , i . e . , we have 972 possible candidates f o r such an implication. (C) Condition 3« The possible implication cannot have the following "counter-classical" p a r t i a l truth table e i t h e r i i . t f n t f f f n Otherwise, ( i ) a black non-raven or ( i i ) a non-black non-raven would disconfirm the Implicational Raven Hypothesis. This would be as co u n t e r - i n t u t i t i v e as both of them confirming the Implicational Raven Hypothesis. Thus we have only the following p o s s i b i l i t i e s f or the possi-ble implication: P \ P t f n t t f y l f n n y2 n y3y4y5 where yl,y2,y3,y4,y5e ft,f,n} . So f a r we have i = 3^ = 243, i . e . , we have 243 possible candidates f o r the possible implication. (D) Condition 4. The possible implication cannot have the following 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 truth value n applies i n the case when the ent i t y 110 does not exist then a non-entity such as a centaur, etc., would confirm the Implicational Raven Hypothesis, f o r we would have 2 the following r e l a t i o n held i n QLy ~ R c & ~ B c |[-(Rc=^Bc) where "c" i s "a centaur (or, a unicorn, e t c . ) " . And i t i s ab-surd f o r any non-entity "to confirm the Implicational Raven Hypo-th e s i s . We think that any non-entity i s neutral with respect to the Implicational Raven Hypothesis. This i n t u i t i o n j u s t i f i e s f o r us Condition 4 as well as the following conditions (E) Condition 5- The possible implication cannot have the following p a r t i a l truth tables P t f n t f n f Otherwise, as we have said that i t would enable any non-entity to disconfirmthe Implicational Raven Hypothesis, and that i s co u n t e r - i n t u i t i v e . Condition k plus Condition 5 give us, with the previous r e s u l t , the following possible t r u t h table f o r the possible implications i Q t f n P t f n t f y l n n y2 y3 y k n where y l , y 2 , y 3 , y k 6}t,f,n} . Thus f a r we have i = 3 , i . e . , we have 81 more possible candidates for the possible implication. 117 However, the assignment of V(y5) = n would create f o r us the following d i f f i c u l t y : Any non-entity w i l l disconfirm the Implicational Raven Hypothesis, although i t won't confirm the Implicational Raven Hypothesis, i f the underlying l o g i c of our 2 new confirmation theory i s QL^ plus t h i s implication. The only way to overcome t h i s new d i f f i c u l t y i s : 2 2 (i)Besides adding the would-be implication to QL^ (and L^) 2 we also have to add the following " i n t e r n a l negation" to QL-^ (and ): p -P t f f t n n where "-" i s to be read as "not" and interpreted as "opposition". ( i i ) Also change the "disconfirmation* 1 of our new confirmation theory to "with respect to the i n t e r n a l negation". Thus i n our new confirmation theory "B disconfirms H" i s defined 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 disconfirmation any non-entity i s neither con-firming nor disconfirming the Implicational Raven Hypothesis. In other words, under the new d e f i n i t i o n of disconfirmation any non-entity i s simply neutral to the Implicational Raven Hy-pothesis. I f eventually we can f i n d the implication we want, then 2 plus the implication and the i n t e r n a l negation w i l l be de-2 noted as C3 and QC3, and thus w i l l be QL^ plus the implication and the i n t e r n a l negation. The new confirmation theory, whose underlying 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 disconfirmation i s defined i n terms of the in t e r n a l negation. 118 10. IN SEARCH OF AN IMPLICATION II Up to t h i s point we s t i l l have 81 p o s s i b i l i t i e s f o r the possible implication. This number i s derived from the insight provided from the f i v e conditions on implication to interpret the Implicational Raven Hypothesis l a i d down i n the previous section. We would not have any more insight of t h i s type, f o r we have exhausted a l l possible truth-value combinations on which we could ground the Implicational Raven Hypothesis. Remember that we are i n search of a possible truth rule f o r sentences of the following general form: (1) P=»Q, or, i n a way, of the following form: (2) Ga=^Hb where P,Q,G,H,a and b are a l l un-interpreted; and we are not i n search of the truth rule of the following form: (3) Rc =*Bc , where R and B are a l l interpreted as follows: R: (l) i s a raven, Bi ® i s black. Thus the possible combinations of truth-values of (2) of the following form .Hb Ga =*Hb t f n t y i f n y3 y^+ (where y l , y 2 , y 3 , y 4 6 ( t , f , n \ ) are ruled out from the possible 119 considerations, because the imposed interpreted r e s t r i c t i o n s of (3) that G = R, H = B, a = c=b, and there are indeed ravens ( i . e . R) and blackness ( i . e . B) i n the world. Thus the c o n f i r -mational valuation rules out the possible truth-value combina-tions of ( i ) V(Ga)=t and V(Hb)=n, ( i i ) V(.Ga) = f and V(Hb)=n, ( i i i ) V(Hb)=t and V(Ga)=n, and (iv) V(Hb)=f and V(Ga)=n. In other words, the formally possible truth-value combinations of ( i ) , ( i i ) , ( i i i ) and (iv) are impossible now from the seman-t i c point of view of confirmational valuation when G and H are respectively interpreted as "... i s a raven" ( i . e . R) and "... i s black" ( i . e . B). So, from now on I w i l l concentrate my attention on form (3 ) , not (1) nor (2 ) , f o r i n s p i r a t i o n about the insight of the possi-ble truth-value combinations of the truth rule of the possible implication. Once we have t h i s understanding, we can have the following more in s i g h t s : (F) Condition 6. (the Neutralized Condition). The t r u t h -values of y l , y2, y3 and yk of the l a s t p a r t i a l truth table should be: n. In other words, we should have the following p a r t i a l truth table f o r the possible implication! The reason f o r t h i s i s : since the formal sentence (2 ) , v i z . Ga=^Hb, once interpreted as sentence ( 3 )» v i z Rc=»Bc, we must have the above truth-value assignments due to the following two considerations! 120 ( i ) Since both R and B are now interpreted (respectively as "... i s a raven" and "... i s black") and there are such proper-t i e s i n the actual world, the only chance f o r both Rc and Be to have the truth-value n (as given) according to the confirmational valuation i s that "c" denotes nothing i n the actual world. ( i i ) I f "c" denotes nothing i n the actual world, i t i s out-side the "natural range of relevance" according to von Wright. In such case, ( i . e . , when "Ga=*Hb" i s interpreted as "Rc=*Bc") we should better assign Rc =*Bc the truth-value n, i n order that we have a chance of solving the paradoxes of confirmation a l a von Wright's treatment of the paradoxes v i a i t s "natural range of relevance of app l i c a t i o n " . If the above reasoning i s correct, then the implication that we are looking f o r should have the following truth table 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 refuses to interpret the c l a s s i c a l horseshoe operator 3 as an "implication",because he thinks the horse-shoe operator does not give us any reasonable, say, causal connection between the antecedent and the consequence. For example, i n c l a s s i c a l l o g i c we can i n f e r from (1) The sky i s blue and the sky i s not blue to the conclusion: (2) UBC i s i n Vancouver. This inference 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 following theoremt (3) P and not-P D Q. C.I. Lewis refuses to interpret the horseshoe operator as "implication" f o r the reason that there i s no " i n t r i n s i c " , say, causal, connection, i n our present example, between (1) and (2 ) . So, he c a l l s (3) a "paradox of implication" i f we interpret the horseshoe operator as an "implication". He l i s t s at l e a s t 13 such paradoxes of i m p l i c a t i o n * 1 1. P 3 ( Q 3 P ) 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. Cf. Lewis, C.I. and C.H. Langfordt Symbolic L o g i c , p . 8 6 f f . Dover, I932, 1959. 122 7. -?(PoQ)o-Q 8. - r(PoQ) => (P3 - » Q ) 9. - » ( P 3 Q ) 5 (~»P 3 Q ) 10. -r(POQ)^ ( T P 3 - » Q ) 11. -7 (P^Q) r>(Q =>P) 12. P o ( Q v - 7 Q ) 13. (P.^P)QQ The above sentences representing the paradoxes of implica-p t i o n ( 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 plus , i f we substitute a l l occurrences of the horseshoe by and under-stand the negation, disjunction and conjunction of the c l a s s i c a l l o g i c as the external negation (or any negations to be introduced 2 into such as the i n t e r n a l and the i n t r i n s i c negations), d i s -2 junction and conjunction of L^. This would constitute our f i r s t plausible j u s t i f i c a t i o n of the truth rule of the implication. We understand that 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 bleak, f o r the avoidance of the paradoxes of implication could be just a favor of luck. 123 12. JUSTIFICATION OF THE TRUTH RULE OF THE IMPLICATION II Since the j u s t i f i c a t i o n of the truth rule of the implica-t i o n given i n the previous section i s not rigorous, f o r a long time I was content simply to c a l l i t "the f a t c o n d i t i o n a l " or "the f a t arrow". However, I r e a l i z e d that I could not give i t s truth rule a rigorous j u s t i f i c a t i o n i f I could not f i n d out, i n the f i r s t place, what C.I. Lewis means by "implication", f o r he i s the f i r s t l o g i c i a n who c r i t i c i z e s and r e j e c t s the i n t e r p r e t a t i o n of the c l a s s i c a l "horseshoe" operator as an "implication". Lewis has 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 i mplication". I t i s t h i s : 1 C.I. Lewis' d e f i n i t i o n of s t r i c t implication. According to Lewis, P s t r i c t l y implies Q i f f " i t i s f a l s e that i t i s pos-s i b l e that P should be true and Q f a l s e . " Does the " 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 s t r i c t implication? It turns out not to. One counter-example i s : l e t us have the following two sentences, HI1 A l l brothers are male, H2: A l l s i s t e r s are female. Then, we have that ~H1 s t r i c t l y implies ~H2, but the truth-value of ~H1=±~H2 i s n. Hence, the f a t arrow i s not a s t r i c t i mplication. However, l e t us write " |(P=>Q)H as "P |f=»Q". Then, i t seems cl e a r that i f we have that P|f=* Q then we have that P s t r i c t l y im-p l i e s Q, although not conversely. 1. Lewis and Langford: Symbolic Logic, p.124 and p.244. 12k So, ||=* i s stronger than ||-. We w i l l come back to the problem of t h e i r strengths i n a moment. Quine also has a d e f i n i t i o n of "implication", which he thinks i s a l o g i c a l r e l a t i o n between two sentences. His d e f i n i t i o n of 2 implication i s t h i s t Quine's d e f i n i t i o n of implication. Sentence SI implies sentence S2 i f f no i n t e r p r e t a t i o n makes SI true and S2 f a l s e , hence i f and only 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 material conditional whose antecedent i s SI and whose consequence i s S2. In a word, implication 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 and of confirmation l o g i c are "implication" according to Quine. J Thus, i n confirmation l o g i c \[ i s the implication as well as the semantical entailment. Since ||=> i s stronger than |(-, the former can be c a l l e d "the strong implication" or "the strong (semantical) entailment". But what does that mean i n l o g i c though? In logic " |[-" as a semantical entailment r e l a t i o n i s truth-preserving i n any l o g i c a l inference. Is ||=* also truth-preserving? The answer i s no. Thus, fo r instance, the following inference, P . P->Q |f- Q i s a v a l i d argument; but, correspondingly, we do not havej P , P^Q Q However, i f a l l premises of an argument are necessarily true, and the argument i s v a l i d , then ||=> w i l l preserve the property, i . e . the conclusion w i l l be also a necessary truth. Thus " ||=»" i s necessary-truth-preserving (or v a l i d i t y - p r e s e r v i n g ) . Since truth consists of necessary truth and contingent truth, necessary-truth preserving i s , of course, stronger than truth pre-2. Quine» Methods of Logic, revised ed., p . 3 ^ . 3 . There may be a problem of "use" and "mention" neglected here. 125 serving. Thus "strong implication" c l e a r l y means that i t s necessary-truth-preserving i s stronger than the truth-preserving of \\~ , i . e . the semantical entailment r e l a t i o n as "implication". S i m i l a r l y , i s stronger than the conditional —* i n C3« So, i t can be c a l l e d "the strong c o n d i t i o n a l " . But what does that r e a l l y mean? The truth-value of P => Q can be described as i s equivalent to P-»Q provided that i n the f i r s t place P i s presupposed to be true. Then, what i s the stronger part i s , of course, the clause that P i s f i r s t l y presupposed to be true. Hence, "strong conditional" can also be c a l l e d "presuppositional conditional" and "P=*Q" can be read as " i f P then Q, where P i s pre-supposed to be true" or, more b r i e f l y , "P only i f presupposed by Q". Below we l i s t some i n t e r e s t i n g properties about ||=> » II"anc* ~* * Theorem 8. In L^, C3 and QC3 |(=> i s stronger than \\- , which i s i n turn stronger than —> . Theorem 9« I f P ||=>Q then P||-i3, but not conversely; or i n other words, i f |f-(P=»Q) then ||-(P-*Q). but not conversely. The following example shows that the converse part of the pre-ceding theorem does not holdi We have that P||-P, but not that P |f=>P. To see t h i s , just l e t V(P)=n (or f ) . (Note that f o r lack of a shorter term, "P=»Q" w i l l be s t i l l read as "P implies Q", while we reserve "entailment" f o r \[ and "strong entailment" or "strong implication" f o r ||=» , although i n English the best reading that I can think of for "P=*Q" i s i P i s followed by Q ( f a c t u a l l y or l o g i c a l l y ) . ) 4. This has been much f u l l y described i n next section on p.128. 126 13. A LONG WAY TO REACH THE TRUTH RULE OF THE IMPLICATION In the previous four sections I have t r i e d to described how I reached the truth rule f o r implication as well as how i t s truth rule i s j u s t i f i e d . I t seems somewhat complicated; but, act u a l l y , the way I reached the truth rule f o r implication i s even more complicated. Since i t also provides a way to j u s t i f y the truth rule f o r implication, I would l i k e to describe i t below. (I) F i r s t , my i n i t i a l insight i n constructing a three-valued confirmation l o g i c was to i d e n t i f y , or at l e a s t associate, the three truth-values t (truth), f ( f a l s i t y ) and n (neither truth nor f a l s i t y ) with the three confirmation-states c (confirmation), d (disconfirmation) and n ( n e u t r a l i t y ) . I believed that 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 to pursue the matter i n t h i s l i g h t . (II) Second, I believe that the paradoxes of confirmation are counter-intuitive, i . e . genuinely puzzling. Further, I thought that the only 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 , since i t seems that Hempel's con-firmation theory proper i s flawless. This conviction re-enforces point ( I ) . (III) I, then, constructed, i n a s y n t a c t i c a l (and formally semantical)approach, a minimal confirmation theory, whose under-l y i n g l o g i c 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 of confirmation. Unfortunately, a f t e r r e f l e c t i o n the avoidance of the para-doxes of confirmation i s only apparent! they are avoided by "sup-presing", i . e . , neglecting t h e i r t o t a l meaning and complete i n -terpretation from a f u l l y developed semantical perspective. Some important lessons have been learnedi 12? Lesson 1. A purely s y n t a c t i c a l (and formally semantical) approach to confirmation theory i s i n s u f f i c i e n t . Lesson 2. Any underlying l o g i c of a confirmation theory must have a fully-developed semantics. Lesson 3» The sentential l o g i c of the underlying 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 so i s i t s q u a n t i f i c a t i o n a l one. Lesson 4. The paradoxes of confirmation should not be solved by suppressing 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 . Lesson 5« The paradoxes of confirmation are natural conse-quences of the Conditional Raven Hypothesis otherwise,they would be "solved" by suppression but they should not be the consequ-ence of the Implicational Raven Hypothesis. (IV) So f a r , i t s t i l l remained unclear what implication i s . (V) However, before we can answer what "implication" i s , we would better construct a three-valued " q u a s i - c l a s s i c a l " l o g i c which i s so close to the c l a s s i c a l l o g i c that the paradoxes of confirma-t i o n are natural consequence of the " q u a s i - c l a s s i c a l confirmation theory" with such an underlying l o g i c . (VI) So, we have L 2 and 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 con-2 cept of "implication" can be added to L^ i n our present approach. (VIII) I also found that i f we add the i n t e r n a l negation (-) 2 1 to Ly then the following concept of "presupposition" i s definablei D e f i n i t i o n of Presupposition, p presupposes Q i f the follow-ing condition holdst (P ||-Q) & (-P l|-Q)f where "&" i s the mata-language analogue of the "conjunction" of L 2 and " fr" i s the semantical entailment r e l a t i o n . 1. Cf. Haack, Susans Deviant Logic, p . l 4 l . The idea i s to capture Frege's d e f i n i t i o n or presupposition, which i s : P presupposes Q * df P i s neither true nor f a l s e unless Q i s true. 128 (IX) Implication, denoted as •=», can have the following truth rule i n order to capture von Wright's idea of "natural range of re-levance" i ( i ) In the f i r s t place, P=*Q presupposes P. Hence, i f V(P) t then V(P Q) = n; and ( i i ) i f V(P) = t, then V(P =^Q) = V(P->Q), where "-•" i s the 2 conditional of L^. It follows from the above truth rule that the new implication has the truth table: It turns out that the above truth table i s i d e n t i c a l to the one reached at section 10 although they are reached by quite d i f -ferent routes. The above route seems to give us a presuppositional j u s t i f i c a t i o n of the truth rule of the implication. Note that I did not i n fact reach the above truth rule f o r the implication as smoothly as I described i t here. In fa c t , before I reached i t I s t i l l made many mistakes and t r i e d many other routes. The f a i l u r e s have been p a r t i a l l y recorded as Condition 1 to Condi-t i o n 5 of section 9 and, espec i a l l y , as Condition 6 of section 10 . t f n t f n n n n n n n 129 14. A BRIEF VIEW OF C3 AND QC3 Before the constructing a new confirmation theory to solve the paradoxes of confirmation, we'd better stop at t h i s point to have a b r i e f view of i t s underlying l o g i c QC3. C3 i s plus the i n t e r n a l negation - and the implication . Thus C3 contains the following sentential connectivess ~ (the external negation), v (disjunction), & (conjunction), —» (conditional), <-> (bi-conditional), — (the i n t e r n a l negation), =4. (the impl 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. C3 i s t r u t h - f u n c t i o n a l l y complete. Proof. We know from E.L.Post's work that any three-valued sentential 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 contains the d i s j u n c t i o n of C3 plus the following unary operation "o" c a l l e d "(Postian) c y c l i c operation", which has the following truth t a b l e t 1 p OP t f f n n t Since oP can be defined i n G3 as follows: 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 following truth tables p oP (~P& ((P=»P)&(~P =*~p)))v ' ~_ (p V -P) t f f t f t t t n f t n f t f? f t t f t f n t f n f n f n t f n t f n f f t t f n t t nn n n n n tn t tn i i i t n n nn 1. Cf. Rescher*N» Many-valued Logic, p.53. (I969) 130 we have that C3 i s truth f u n c t i o n a l l y complete. QED. Since we have implication, i t would also he natural f o r us to define a "bi-implication" i n C3 as follows: 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 bi - i m p l i c a t i o n w i l l have the truth table: p1 P<s=» Q t f n t t f n f f n n n n n n Note that our bi - i m p l i c a t i o n turns out to 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 that we have had. Note also that the bi- 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 (truth) value". In view of t h i s we introduce into C3 al-so the following "equivalence" which has the truth table: t f n P == Q t f n t f f f t f f f t Since C3 i s truth - f u n c t i o n a l l y complete, i t i s better for us to introduce into i t the following " i n t r i n s i c negation", denoted as and read as "a contradiction of" and interpreted as "contra-d i c t i o n " , whose truth table i s given as follows: t f n f t f Thus i n C3 "P& 'P" i s a contradiction; but neither "P&~P" nor "P&-P" are. (On the other hand, the law of excluded-middle i n C3 131 can only be expressed i n terms of the external negation as: ||-Pv~P: and the law of double-negation can be expressed only i n terms of the i n t e r n a l negation as: ||-P=—P. f Then, QC3, which w i l l be our confirmation l o g i c f o r the new confirmation theory that w i l l solve the paradoxes of confirmation, i s C3 plus both the e x i s t e n t i a l and the universal quantifications denoted as (3 ) and (\/ ) which are respectively the generaliza-tions of the disjunction and the conjunction of G3. After adding the confirmational valuation described i n section 12 of Part II to QG3 (with or without i d e n t i t y ) , we claim: Theorem 11. QC3 (with or without i d e n t i t y ) i s a "decent l o g i c " , meaning: ( i ) i t has a fully-developed 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 that the under-l y i n g l o g i c (and semantics) of i t s meta-language, meta-meta-language, etc., can be QC3 i t s e l f . Thus from now on the l o g i c a l words and semantical concepts that appear i n our discussions (except otherwise stated) should be under-stood as those of QC3 (instead of those of QC2), f o r the obvious reason that we can use the vocabularies of QC3 to discuss QC2 but not conversely f o r example, we do not have the counterpart of "implication" of QC3 i n QC2. 2. See the truth tables for the external negation ~ and the i n t e r -nal negation - given respectively on p.100 and p.11? f o r checking. 132 15. ELEMENTARY INTERNAL CONFIRMATION THEORY We introduce the following two d e f i n i t i o n s about confirma-t i o n theoriesi D e f i n i t i o n of an elementary confirmation theory. A con-firmation theory i s an "elementary confirmation theory", i f a l l of i t s hypotheses are atomic or molecular sentences. D e f i n i t i o n of a general confirmation theory. A confirmation theory i s a "general confirmation theory", i f i t has general senten-ces as well 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 basic concepts borrowed from Hempel's theory of confirmation, v i z . , "observation report", "hypothesis" (which i s confined only to atomic or molecular sen-tences), "development of a hypothesis", "direct confirmation" , "confirmation", "disconfirmation" (which i s defined i n terms of the i n t e r n a l negation, thus "B disconf irms H" i s "B|(--Hd" ) 1 and "neut r a l i t y " , and also the following "equivalence condition": The Equivalence Condition. Whatever confirms (disconfirms, or i s neutral with respect to) one of two l o g i c a l l y equivalent sentences also confirms (disconfirms, or i s neutral with respect to) the other. The f i r s t important r e s u l t of t h i s elementary i n t e r n a l con-firmation theory i s t h i s : Theorem 12. Suppose that a hypothesis i s either an atomic or a molecular sentence. Then, ( i ) i n Hempel's confirmation theory, 1. Note that "Hd" i s "the development of H w.r.t. the set of i n d i -viduals e s s e n t i a l l y mentioned i n B". 133 given that the only sentence l e t t e r s that occur i n an observation report B are those that occur i n the hypothesis H, we have the following r e s u l t : (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 disconfirmed by B i f f H i s F (which i s "False" i n the sense of QC2); (1 . 3 ) H i s never neutral with respect to B. On the other hand, (II) i n the newly constructed elementary i n t e r n a l confirmation theory, given that the only sentence l e t t e r s that occur i n an observation report B are those that occurinthe hypothesis H, we have the following neat r e s u l t : (11.1) H i s confirmed by B i f f H i s true (which i s t of QC3); (11.2) H i s disconfirmed by B i f f H i s f a l s e (which i s f of QC3); (11.3) H i s neutral with respect to B i f f H i s neither true nor f a l s e ( i . e . , H i s n of QC3). Examples of (II.3) of Theorem 12 are: HI: This centaur i s black; H2: That unicorn i s blue; H3: Satan i s clever; H4: The ether wind i s cooling. (And, perhaps, also H5« Pegasus = Pegasus, i f we add the i d e n t i t y , v i z . "=", to the vocabulary to C3.) Note that Theorem 12 has j u s t i f i e d our intended i d e n t i f i c a -t i o n that the three truth-values t, f, and n are respectively i d e n t i f i a b l e to the three confirmation-statuses c (confirmation), 134 d (disconfirmation) and n ( n e u t r a l i t y ) . The r e s u l t of Theorem 12 also 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 confirmation theory "non-Hempelean", f o r the re s u l t s of (I) and (II) are quite d i f f e r e n t . Note also that only the equivalence (not the bi - c o n d i t i o n a l , nor the bi-implication) of QC3 observes the Equivalence Condition given i n t h i s section. In other words, we have* Theorem 13. I f ||-(Hl«H2)t then we have the following cor-responding property: HI i s confirmed (disconfirmed, or neutral with respect to an observation report) i f f H2 i s confirmed (dis-conf irmed, or neutral with respect to the same observation report)? on the other hand we cannot have the above corresponding property even i f we have: ||-(HI <-* H2 ), or we have: |f- (HI 4=*H2 ). This Theorem thus j u s t i f i e s the introduction of the equiva-lence, viz.,"=", into the vocabulary of C3 and QC3. 135 16. THE GENERAL INTERNAL CONFIRMATION THEORY The elementary i n t e r n a l confirmation theory can be e a s i l y ex-tended now to a general i n t e r n a l confirmation theory, which i s also non-Hempelean, by making the following two major thingst (I) Add any general sentences, besides atomic and molecular ones, as possible hypotheses. (II) On top of the elementary i n t e r n a l confirmation theory we also add the following few concepts! D e f i n i t i o n s of posi t i v e , negative and neutral instances. Sup-pose that G V / / c, which i s the development of the hypothesis ()/v)Gv with respect to the reference class {cJ, i s confirmed by (discon-firmed by, or neutral to) some observation report. Then, G V / /° i s a posi t i v e (negative, or neutral) instance of hypothesis (Vv)Gv. D e f i n i t i o n of "strong confirmation". A universal hypothesis i s strongly confirmed i f a l l of i t s instances are posit i v e ones. D e f i n i t i o n of "pure n e u t r a l i t y " . A universal hypothesis i s purely neutral i f a l l of i t s instances are neutral ones. D e f i n i t i o n of "weak confirmation" (or "mixed n e u t r a l i t y " ) . A universal hypothesis i s weakly confirmed (or mixed neutral) i f ( i ) some of i t s instances are po s i t i v e and ( i i ) some other instances are neutral ones, while ( i i i ) none of i t s instances are negative. The l a s t three concepts are something new; and they are non-Hempelean. They w i l l play some important role l a t e r when we come to the solut i o n of the paradoxes of confirmation. Thus, at l a s t , we have a non-Hempelean general i n t e r n a l con-firmation theory, which w i l l be c a l l e d "the i n t e r n a l confirmation theory" from now on f o r the sake of s i m p l i c i t y . 136 17- A QUASI-SOLUTION OF THE PARADOXES OF CONFIRMATION If we frame the Raven Hypothesis that a l l ravens are black i n terms of the implication of QC3 as the following Implicational Raven Hypothesis, IRH: (Vx)(Rx =^Bx) where RJ 0 i s a raven, B« ® i s black, to be read s t i l l as " a l l ravens are black", then we claim the following r e s u l t i Theorem 14. If we re-frame the Raven Hypothesis as an im-p l i c a t i o n a l law as (Vx) (Rx =^ Bx), then the paradoxes of c o n f i r -mation are not derivable i n the i n t e r n a l confirmation theory. Proof. (1) Suppose that a i s a black raven. Then, we have: Ra & Ba \\-(Ra =*3a), i . e . , any black raven confirms the Implicational Raven Hypothesis. (2) Suppose that b i s a not-black raven. Then, we have: Rb & -Bb ||--(Rb =*3b), i . e . , any not-black raven disconfirms the Implicational Raven Hypothesis. (3.1) Suppose that c i s a black peony. Since i t i s not a raven, we have that -Rc; and since i t i s black, we have that 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 black not-raven i s neutral with respect to the Impli-cational Raven Hypothesis. (3.2) Suppose that d i s a white swan. Since i t i s not a raven and i t i s not black, we have: -Rd&-Bd. So, together we have: i . e . , any not-black not-raven i s neutral to the Implicational Raven Hypothesis. (3.3) Suppose that "e" i s "a blue unicorn". Then, according to the confirmational valuation ( i n the sense of absolute seman-t i c s ) we have: V(Re)= n=V(Be), f o r there are no unicorns ( i n the actual world). Hence, we have: ~-Re &~BeJ~ So, together we have: i . e . , any non-black non-raven i s neutral to the Implicational To sum up, we have: Theorem 15. In the i n t e r n a l confirmation theory we have the following r e s u l t s about the Implicational Raven Hypothesis: (1) Only black ravens confirm i t : (2) Only a not-black raven disconfirms i t ; and (3) A l l other cases are neutral to i t . -Rd& -BdJ^(Rd =*Bd) and -Rd & -Bd Jjf- (Rd =»Bd), ~ R e & ~ B e J p ( R e =*Be) and Raven Hypothesis. QED. 1. Note that 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 th i s point when I was i n c l i n e d to claim the paradoxes of confirmation had been solved, we f i n d the following two c r i t i c i s m s about the "solution" of the paradoxes of confirmation offered i n the previous section 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 truth-values of the Implicational Raven Hypothesis i s neither true nor f a l s e i n the actual world;so long as there exists 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 black, the truth-value of the Implicational Raven Hypothesis can-not be true i n the actual world. This i s odd. In contrast the truth of the Raven Hypothesis i n Hempel's theory of confirmation i s not excluded by such considerations. Here i t seems that there i s an exchange: On the one hand the paradoxes of confirmation are avoided inthe i n t e r n a l confirmation theory, when we frame the Raven Hypothesis as the Implicational Raven Hypothesis? on the other hand the truth-value of the Impli-cational Raven Hypothesis i s neither truth nor f a l s i t y , instead of t r u t h . In contrast i n Hempel's confirmation theory, although we have the paradoxes of confirmation, the truth-value of the Raven Hypo-thesis i s , simply, Truth (we may suppose). 2 ) In general, i n the i n t e r n a l confirmation theory the t r u t h -value of any universal implicational hypothesis cannot be truth so long there exists one i n d i v i d u a l other than those mentioned i n the antecedent of the implication. This i s more than odd. Any such theory of confirmation i s 139 seriously inadequate, f o r the theory cannot r e s u l t i n anything true, i f an empirical hypothesis i s framed i n terms of the im-p l i c a t i o n of QG3, though i t may r e s u l t i n some hypotheses "being confirmed. This c r i t e r i o n of inadequacy i s due to the nature of science that science i s obviously i n search of "truth", not, or not just, "neither truth nor f a l s i t y " . I suffered a shocking experience when I discovered how v u l -nerable I was to the above vigorous c r i t i c i s m s , e s p e c i a l l y point 2). For a long time I consoled myself that, perhaps, science i s also i n search of "non-falsity" and, hence, that the in t e r n a l confirmation theory i s not t o t a l l y inadequate and s t i l l has i t s own l i m i t e d value. 140 19. A PLAUSIBLE SOLUTION OF THE PARADOXES OF CONFIRMATION The above two vigorous c r i t i c i s m s of the "solution" of the pa-radoxes of confirmation i n p a r t i c u l a r and the inadequacy of the i n -ternal confirmation theory i n general, as stated i n the l a s t sec-tio n , posed a serious problem for me fo r a long time. However, they can be overcome. The key to a plausible solution which answers the two above-mentioned c r i t i c i s m s i s t h i s i Although the truth-value of the Im-p l i c a t i o n a l Raven Hypothesis, IRHi ( V X ) ( R X - * B X ) , i s neither true nor f a l s e , i t s confirmation-status i s "weakly con-firmed", which i s stronger and better than "purely n e u t r a l " . 1 Thus i f we can exclude those neutral instances from the IRH, the refined IRH, to be c a l l e d "the de-neutralized Implicational Raven Hypothe-s i s " (IRH^), w i l l be true, and i t s confirmation-status i s now strong 2 l y confirmed. In order to achieve t h i s we w i l l r e s t r i c t the domain of any hy-pothesis to i t s "natural range of relevance", following von Wrightj and i n order to see how t h i s is done we w i l l consider the simplest form of a universal implicational hypothesis i Hi (VX) (FX = * GX), and denote i t s "de-neutralized hypothesis" as "H^". The "natural u range of relevance" of W i s then defined as follows i f xiFx & (Gx v -Gx)> . 1. The d e f i n i t i o n s of "weak confirmation" and "pure n e u t r a l i t y " are given i n section 16 of t h i s Part, p.135. 2. See i t s d e f i n i t i o n on p.135-141 In general, suppose a hypothesis H i s of the following forms Hs H(F1,F2 F i , G1,G2 Gj ), where F l , F 2 , . . . f F i are antecedents of any implication i n H while Gl,G2,...,Gj are consequents of any implication i n H, and i , j ^ . l . Then, the "natural range of relevance" of }r associated with hypo-thesis H i s defined as f o l l o w s : {x: (F1&F2&...&Fi)&(Glv -G1)&(G2v -G2)&...&(Gjv -Gj)} . Thus the natural range of relevance of the de-neutralized Im-p l i c a t i o n a l Raven Hypothesis IRH associated with the IRH i s : {x: Rx& (Bx v -Bx)} . Hence, IRH^ i s not only strongly confirmed (with respect to i t s reference class of individuals) but also now true. Thus we have answered the two c r i t i c i s m s raised e a r l i e r : 1) In p a r t i c u l a r , the de-neutralized Implicational Raven Hypo-thesis IRH can be true. 2) In general, the i n t e r n a l confirmation theory can also pro-vide truths (besides neither truth nor f a l s i t y ) i f any empirical hypothesis i s framed i n terms of the implication of QG3. So, i t follows that ( i ) we have solved the paradoxes of c o n f i r -mation i f the Raven Hypothesis i s understood as the IRH and ( i i ) the i n t e r n a l confirmation theory remains adequate under the vigo-rous c r i t i c i s m s once we have introduced the # - r e s t r i c t i o n of a hy-pothesis into i t . A few important side r e s u l t s can also be mentioned here: 142 ( I ) Our strategy f o r a plausible solution of the paradoxes of confirmation v i a the concept of "the de-neutralized hypothesis" associated with a hypothesis (say, e.g., the Raven Hypothesis) can-not be applied to Hempel's theory of confirmation. This i s mainly because no neutral instance about an object whose complete i n f o r -mation r e l a t i v e to the Raven Hypothesis (or to any other hypothe-s i s ) can be knocked out i n Hempel's theory of confirmation. So, any such move to avoid the paradoxes of confirmation i s destined 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 "taking (some) relevant instances as i r r e l e v a n t ones". (II) In the i n t e r n a l confirmation theory there i s always a me-chanic way to t e l l whether or not an object c i s i n the natural range of relevance of a hypothesis H. The mechanic way i s t h i s . Suppose that we have a hypothesis H and suppose that we also have a complete information of an object c r e l e a t i v e to a l l predicates appearing i n H. Then, we have: c i s i n the natural range of relevance of H i f f observation report B(c) i s not neutral to H. Since B(c) i s never neutral 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 universal relevance 1, we can never have the above r e s u l t i n Hempel's theory of confirmation. Hence, there i s no way to lay down the natural range of relevance of a hy-pothesis i n Hempel's theory. This poses a great d i f f i c u l t y f o r von Wright i n his solution of the paradoxes of confirmation v i a the concept of "natural range of relevance". (III) Another c r u c i a l thing to note i s that i n von Wright's solution of the paradoxes of confirmation v i a the concept of "na-1. Cf. section 7, Part I, p.20ff. 143 turnal range of relevance" two l o g i c a l l y equivalent hypotheses w i l l i n general have two d i f f e r e n t "natural ranges of relevance"; but t h i s i s not the case i n our approach. Thus, e.g., i n von Wright's approach the following two l o g i c a l l y equivalent hypotheses: Hit (x)(Rxr>Bx) H2: (x) (-Bx =>-Rx) w i l l obviously have two d i f f e r e n t "natural ranges of relevance". But i n our approach the following two counterpart-hypotheses, H3« ((Vx)(Rx=>Bx))#, H4: ((l/x)(-Bx=V-Rx))#, are not l o g i c a l l y equivalent nor have they the same "natural range of relevance". In f a c t , what i s l o g i c a l l y equivalent to H3 i s the hypothesis: H5: ((Vx)((Pxv -Px) & (Rx=>Bx))) #, where "P" i s any predicate. But, then, H3 and H5 have the same "natural range of relevance". Thus the Equivalence Condition i s s t i l l preserved i n the i n t e r -nal confirmation theory even under the # - r e s t r i c t i o n ; and t h i s i s why we also f e e l that von Wright's si m i l a r s o l u t i o n of the para-doxes of confirmation 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 equiva-lence condition i s not preserved under the transformation of "the natural range of relevance" of a hypothesis i n his approach. (IV) F i n a l l y , also note the following important property: Theorem 17. The "confirmation" of a de-neutralized implica-t i o n a l hypothesis ((Vx)(Fx=»Gx))^ i s automatically a "selective confirmation" i n the in t e r n a l confirmation theory i f we define the "contrary" of the given hypothesis as " ((Vx) (Fx=*—Gx) ) ^ M . Proof. This i s due to the fac t that the confirmation of a de-neutralized implicational hypothesis such as, H #« ((\/x)(Fx=*Gx))#, i s confirmed i f and only i f ( i ) i s confirmed and ( i i ) the "con-tr a r y " CH# of CH #t ((Vx)(Fx=*-Gx))#, i s disconfirmed i n the i n t e r n a l confirmation theory. QED. 145 PART IV. ADEQUACY CONDITIONS FOR CONFIRMATION AND THE GOODMAN PARADOX 146 1. ADEQUACY CONDITIONS FOR CONFIRMATION LAID DOWN BY HEMPEL According to Hempel any confirmation theory must observe the following conditions i n order to be adequate; 1 (8.1) Entailment Condition. Any sentence which i s entailed by an observation report i s confirmed by i t . (8.2) Consequence Condition. If an observation report con-firms every one of a class K of sentences, then i t also confirms any sentence which i s a l o g i c a l consequence of K. (The following two more conditions are consequences of the above condition.) (8.21) Special Consequence Condition. If an observation re-port confirms a hypothesis H, then i t also confirms every conse-quence of H. (8.22) Equivalence Condition. If an observation report con-firms a hypothesis H, then i t also confirms every hypothesis which i s l o g i c a l l y equivalent with H. (8.3) Consistency Condition. Every l o g i c a l l y consistent ob-servation report i s l o g i c a l l y compatible with the class of a l l the hypotheses which i t confirms. (This implies that (8.31) Unless an observation report i s s e l f - c o n t r a d i c t o r y , i t does not confirm any hypotheses which contradict each other, and that (8.32) Unless an observation report i s s e l f - c o n t r a d i c t o r y , i t does not confirm any hypotheses which contradict each other.) (8.33) Conjunction Condition. If an observation report con-1. Hempelt Aspects of S c i e n t i f i c Explanation, pp .31-3 k . 147 firms each of two hypotheses, then i t also confirms t h e i r conjunc-t i o n . These conditions must be met f o r any adequate theory of con-firmation according to Hempel; i n other words, they are necessary conditions f o r any adequate theory of confirmation. In t h i s Part we w i l l t r y to examine t h e i r necessity, and come to a conclusion which i s much weaker than Hempel's view when he l a i d down the conditions. However, before we proceed to examine them, l e t us examine a rejected "Converse Consequence Condition", which i s of some i n t e r -est and has been proposed by some authors. 148 2. THE CONVERSE CONSEQUENCE CONDITION AND ITS REJECTION The Converse Consequence Condition can be stated as f o l l o w s : 1 (8.4) Converse Consequence Condition. If an observation re-port confirms a hypothesis H, then i t also confirms any sentence K which has H as i t s consequence. Hempel r e j e c t s the above Converse Consequence Condition, f o r the following two reasons: (I) The Converse Consequence Condition i s incompatible with the Consequence Condition. To show t h i s , l e t us assume that Hem-pel's theory of confirmation (or any theory of confirmation) i s adequate and, hence, consistent. Then, l e t us make the following two notational abbreviations* 1. B* r i s a raven. 2 . H: Hook's law (or any physical law or statement). It follows by Hempel's d e f i n i t i o n : 3. B confirms B. Since i n c l a s s i c a l l o g i c the following statement i s always true, 4. ( B » H )O B , by the Converse Consequence Condition we have: 5. B confirms (B-H). Likewise, i t i s always true i n c l a s s i c a l l o g i c that we have* 6. (B-H) "D H . Hence, by the Consequence Condition we have* 1. Hempel* Aspects of S c i e n t i f i c Explanation, p. 3 2 . 1 4 9 ?. B confirms H. The above r e s u l t means that any observation report B could confirm any hypothesis H, even i f H i s a f a l s e or a contradic-tory statement. This i s absurd. Thus the Consequence Condition and the Converse Consequence Condition are incompatible. Hence, they cannot both be v a l i d . Hempel thinks that here the Converse Consequence Condition has to go f o r the following reasont (II) A hypothesis i s confirmed by an observation report should not imply that a stronger hypothesis i s also confirmed by the same observation report. Thus the hypothesis that r i s a raven i s con-firmed by the observation report that r i s a raven; but the obser-vation report should not without any other good reason also con-firm the stronger hypothesis that r i s a raven and Hook's law. The above reason seems to be sound. So, we agree with Hempel to r e j e c t the Converse Consequence Condition. 150 3. CARNAP AND THE CONSEQUENCE CONDITION Carnap thinks that the consequence condition i s i n v a l i d . He gives the following counter-example t 1 The i n i t i a l evidence i s that ten chess players 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 junior players and others are senior players. Their d i s t r i b u t i o n i s as follows t Moreover, we know that only one w i l l win and that each has an equal chance of winning. Let H = 'a woman wins', and l e t K = 'a stranger wins'. Then the degree of c o n f i r -mation of H and the degree of confirmation of K on the i n i -t i a l evidence i s . 5 . Now l e t us suppose that we are given the additional information that a senior player has won. Then the degree of confirmation of H given 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 confirmation of K given t h i s new evidence i s .6; so t h i s new evidence confirms K. The degree of confirmation of Hv K on the i n i t i a l evidence was, however, .7 ; but, given t h i s new information, i t i s only .6. This new evidence, therefore, disconf irms HvK even though i t confirms H and confirms K, and either of the l a t t e r e n t a i l s the former. The above "counter-example" shows that the consequence con-d i t i o n i s indeed not v a l i d i n Carnap's theory of confirmation, where "degree of confirmation"(as, i n a sense, p r o b a b i l i t y ) i s c l e a r l y a quantitative concept of confirmation: and t h i s i s quite 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 confirmation. So, i t could happen that the consequence condition holds i n Hem-pel's theory of confirmation, although i t does not hold i n Car-nap's theory of confirmation. Thus the following question a-Ql: Do. Carnap and Hempel talk about the same thing when they 1. Cf. B.A. Brody: "Confirmation and explanation", the Journal of Philosophy, LXV, 10; p.288; or Carnap's o r i g i n a l version i n his Logical Foundations of P r o b a b i l i t y . 2nd ed., p.383. 2. Carnap: Logical Foundations of Probabilit y, p .25 f f . Juniors Seniors M, W, W M, M New Yorkers Strangers M, M W, W, W r i s e : 151 talk about "confirmation"? Q2» Is i t possible that the consequence condition holds i n the context of Hempel's q u a l i t a t i v e theory of confirmation but not i n the context of Carnap*s quantitative theory of con' firmation? We w i l l t r y to answer the above two questions and related ones i n the next few sections. 152 4. EXPLICATION OF THE CONCEPT OF CONFIRMATION Carnap thinks that there are two explications of the concept of confirmation! 1 (I) Confirmation as firmness of hypothesis. Here "firm-ness" means "of high p r o b a b i l i t y " . (II) Confirmation as increase i n firmness of hypothesis. As an in t e r p r e t a t i o n of Hempel's notion of confirmation the f i r s t e x p l i c a t i o n i s e a s i l y seen to be untenable or inapplicable. To see t h i s , l e t us make the following two assumptions! HIs A l l flowers are purple. Bis a^ i s a purple flower. Then i t i s obvious that the observation report B l confirms the hypothesis HI i n Hempel's theory. Thus were the f i r s t e x p l i c a -t i o n of confirmation applicable to Hempel's theory of confirma-t i o n , the hypothesis HI would be firm. But hypothesis HI i s , i n fact, f a l s e . Hence, i t cannot be firm, nor can i t have high p r o b a b i l i t y f o r most flowers are not purple. Carnap's second ex p l i c a t i o n of confirmation as increase i n firmness of hypothesis seems to be applicable or compatible to Hempel's theory provided that we make the following two c l a r i f i -cations s i ) Some hypotheses, when they are well confirmed, cannot increase i n firmness. For example, the hypothesis H2s There are flowers i s well confirmed by the observation report, 1. Carnaps Logical Foundations of Probability, p.xv f f . 153 B2: a 2 i s a flower. How could such a hypothesis as H2, which i s confirmed now by the observation report B2 and hence has p r o b a b i l i t y one, be made "firmer" i . e . , to have a higher p r o b a b i l i t y than the present one 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 under-standing what Carnap means by "increase i n firmness of a hypo-th e s i s " as "increase 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 "increase of the p r o b a b i l i t y of the hypothesis without any upper l i m i t " . i i ) In Hempel's theory of q u a l i t a t i v e confirmation 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 . In consequence when we say that Hempel's concept of confirmation can be interpreted as "increase i n firmness of a hypothesis", 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 observation report increases or decreases the firmness of a hypothesis i n Hempel's theory of q u a l i t a t i v e confirmation? Here what we can say i s that we can see no d i f f i c u l t y f o r Hempel to introduce a concept of a p r i o r i p r o b a b i l i t y into his 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 that Hempel's concept of confirmation can be interpreted as, or i s compatible with, Carnap's notion of "increase i n firmness of a hypothesis". 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 formulating adequate c r i t e r i a of q u a l i t a t i v e confirmation had best be tackled, a f t e r a l l , by means of the quantitative concept of confirma-t i o n . This has been suggested e s p e c i a l l y by Carnap, who holds that "any adequate explicatum f o r the c l a s s i f i c a -tory concept of confirmation must be i n accord with at 2. Hempeli Aspects of S c i e n t i f i c Explanation, p.50. 154 l e a s t one adequate explicatum f o r the quantitative concept of confirmation. ... In other words: on some suitable 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 pro-b a b i l i t y of H on B should exceed the a p r i o r i proba-b i l i t y of H whenever B q u a l i t a t i v e l y confirms H. With the above two c l a r i f i c a t i o n s , we say that Carnap*s second ex p l i c a t i o n of the concept of confirmation as increase i n firmness of hypothesis i s applicable or, or at l e a s t compi-table with, Hempel's theory of confirmation. 155 5. CARNAP AND THE CONJUNCTION CONDITION Carnap t r i e s to rej e c t not only the Consequence Condition but also the Conjunction Condition of Hempel's theory of q u a l i t a t i v e confirmation. His "counter-example" f o r the l a t t e r i s t h i s * 1 Let the p r i o r evidence e contain the following i n f o r -mation. Ten chess players p a r t i c i p a t e i n a chess tourna-ment i n New York C i t y . Some of them are l o c a l people, some from out of town: some are junior players, some are seniors; some are men (M) and some are women (W). The d i s t r i b u t i o n i s known to be as follows: Local players Out-of-towners Juniors M, W, W M, M Seniors M, M W, W, W Furthermore, the evidence e i s assumed to be such that on i t s basis each of the ten players has an equal chance of becoming the winner, hence the chance f o r any player to win i s l / l O . It i s also assumed that i n each case of e v i -dence that c e r t a i n players have been eliminated, the re-maining players have equal chance of winning. Let h be the hypothesis that a man wins. Let i, be the evidence that a l o c a l player wins. Let j be the evidence that a junior wins. ~~ Using the background information given above i n the t a -ble, we can obtain the following values: c(h,e) = 1/2; c(h, e.i) = 3/5; c(h, e.j) = 3/5; c(h, e . i . j ) = 1/3. Thus, i, and j are each p o s i t i v e l y relevant to hypothesis h, while the conjunction i . j i s negatively relevant to h. In other words, i . confirms h and j confirms h but i . j discon-firms h. — So, the Conjunction Condition i s not v a l i d according to Car-nap. I t seems that we can t r y to save Hempel's Conjunction Condi-1. Carnap: Logical Foundations of Pro b a b i l i t y , pp.382-383. Also Marsha Hanen: "Confirmation, explanation and acceptance",p.126. In f a c t the version that we have here i s adopted from Hanen's, given i n the a r t i c l e . Note that "c(h,e)" i n the counter-example means "the degree of confirmation c of the hypothesis h, given evidence e". 156 t i o n f o r the same reason that we have t r i e d to save the Consequ-ence Condition, i . e . , Carnap's concept of quantitative confirma-t i o n i n h i s theory of confirmation i s quite d i f f e r e n t from Hem-pel's concept of q u a l i t a t i v e confirmation. Hempel's concept of " q u a l i t a t i v e confirmation" i s the r e l a t i o n between an observa-t i o n report and a hypothesis, while Carnap's "quantitative con-firmation" employing "degree of confirmation" as " p r o b a b i l i t y " i s the r e l a t i o n of t o t a l evidence plus p r i o r p r o b a b i l i t y with a hypothesis. Obviously they are two quite d i f f e r e n t r e l a t i o n s . So, i t could turn out to be the case that i n Carnap's theory of p r o b a b i l i t y as quantitative confirmation the Conjunction Condi-t i o n (as well as the Consequence Condition) does not hold, while i n Hempel's theory of confirmation the Conjunction Condition (as well as the Consequence Condition) holds. We w i l l come back to the problem of the v a l i d i t y of the Con-junction Condition (as well as that of the Consequence Condition) l a t e r i n section 11, where we w i l l have more decisive thing to say. 15? 6. GOODMAN'S PARADOX AND THE CONJUNCTION CONDITION Besides Carnap's "counter-example" to rej e c t the conjunction condition, Goodman also offers another reason to show the incre-d i b i l i t y of the conjunction condition. Goodman's counter-example, which i s adapted from Goodman's to Hempel*s o r n i t h o l o g i c a l para-digm here, i s t h i s one. Consider the following 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 defined as follows: An object i s said to be b l i t e i f i t has been ex-amined before time t ^ (which i s any fix e d future time, say, the year A.D. 2000) and i s black or has not been examined before time t^ and i s white. According to Hempel's theory of confirmation both HI and H2 are well confirmed, i . e . , they are both confirmed by each and every actual observation report available to us i n the world. Now Goodman asks: What should we predict from the two well-confirmed hypotheses HI and H2? Should we say: (1) Any raven examined a f t e r t ^ i s black? Or should we say* (2) Any raven examined a f t e r t^ i s b l i t e ? If the prediction i s the affirmation of (1), then we w i l l have: (3) Any raven examined a f t e r i s black. 1. Cf. Goodman: Fact. F i c t i o n , and Forecast. pp . 7 3 - 7 k ; and also Hempel: Aspects of S c i e n t i f i c Explanation, p . 5 ° . 158 On the other hand i f the prediction i s the affirmation of (2), then we w i l l have: ( k) Any raven examined a f t e r ^ i s white, for a f t e r time ± 1 any b l i t e raven i s , by d e f i n i t i o n , white. But (3) and ( k ) are two incompatible predictions. This (and other sim i l a r examples) w i l l be c a l l e d "Goodman's paradox" or "the Good-2 man paradox". One immediate consequence, some people^ w i l l say, i s that the conjunction condition should be rejected, otherwise we would have two incompatible predictions from the above example of Goodman's paradox. To t h i s , there are a few d i f f e r e n t responses: (i) One response i s to agree to re j e c t the conjunction condi-t i o n . ( i i ) Another d i f f e r e n t response i s to solve Goodman's paradox. Were Goodman's paradox solvable, the conjunction condition could be retained. We are going to review some proposed solutions i n the next few sections. ( 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 discussion of the conjunction condition, by the way, i s Hempel's response. Facing Goodman's paradox Hempel has t h i s to say: ... confirmation whether i n i t s q u a l i t a t i v e or i n i t s quantitative form cannot be adequately defined by s y n t a c t i c a l means alone. That has been made clea r e s p e c i a l l y by Goodman, who has shown that some hypothe-2. C f . , s a y , Skyrms: Choice & Chance, p . 6 l f f . 3. Cf. Marsha Hanen: An Examination of Adequacy Conditions f o r Con-firmation. Ph.D. di s s e r t a t i o n , Brandeis Univ., 19?0. k . Hempel: Aspects of S c i e n t i f i c Explanation, p.50. 1 5 9 ses of the form * (x)(PxoQx)' can obtain no confirma-t i o n at a l l even from evidence sentence of the form • Pa.Qa'. Thus Hempel admits that his sy n t a c t i c a l approach to c o n f i r -mation theory i s a f a i l u r e i n the l i g h t of Goodman's paradox. 160 7. CARNAP'S SOLUTION OF THE GOODMAN PARADOX Carnap's solution of the Goodman paradox i s that any projec-t i b l e predicate must be n o n - t e m p o r a l T h u s " b l i t e " i s not pro-j e c t i b l e , f o r i t obviously involves some time element i n i t s de-f i n i t i o n . Unfortunately, both "black" and "white", as Goodman pointed out, can be defined i n terms of " b l i t e " and "whack" (which i s to be defined below) and, hence, both of them also involve time ele-ment as can be shown by the following d e f i n i t i o n s : Df 1. An object i s said to be black i f i t has been examined before time t ^ (which i s any fixed future time, say, the year A.D. 2000) and i s b l i t e or has not been so examined and i s whack. Df 2 . An object i s said to be 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 . where " b l i t e " and "whack" are understood as follows: Df 3. An object i s said to be b l i t e i f i t has been examined before time t ^ and i s black or i s not so examined and i s white. Df 4. An object i s said to be whack i f i t has been examined before time t ^ and i s white or not so examined and i s black. Thus from the view point of " b l i t e " and "whack" the predicates "black" and "white" also involve time element i n a symmetric way. Hence, Carnap's proposal of the solution of the Goodman para-dox does not work. 1. Carnap: "On the application of inductive l o g i c " , Philosophy and Phenomenological Research, 19^7, pp.133-147. 161 8. SALMON'S SOLUTION OF THE GOODMAN PARADOX Wesley Salmon's proposed solution of the Goodman paraodx i s t h i s : Any p r o j e c t i b l e predicate must be ostensively d e f i n a b l e . 1 Salmon thinks that " b l i t e " and "whack" are not ostensively de-fin a b l e while "black" and "white" are. Accordingly " b l i t e " and "whack" are not p r o j e c t i b l e , but "black" and "white" are. Hence, the Goodman paradox i s solved. 2 Some people r e j e c t t h i s as an ad hoc solution f o r two main reasons: ( i ) The idea of "ostensive d e f i n i t i o n " i s ambiguous. To us who are "black"-and-"white" language speakers,a "black"(or "white") object can have the "ostensive d e f i n i t i o n " simply by pointing to a "black" (or, respectively, "white") thing. To us who are accus-tomed to such a "black"-and-"white" language, i t seems that there i s no way f o r " b l i t e " and "whack" to have such an "ostensive de-f i n i t i o n " . But "blite"-and-"whack" language speakers could object i n the following ways " b l i t e " (or "whack") could also have i t s "ostensive d e f i n i t i o n " i n the following ways F i r s t they point to a blaek. thing and say that an object i s " b l i t e " i f i t looks l i k e that thing and examined, say, before the year A.D. 2000; and, then, they point to a white thing and say that a " b l i t e " object should look l i k e that white thing i f the " b l i t e " object i s examined i n or a f t e r the year A.D. 2000. Thus "blite"-and-"whack" language speakers could also have "ostensive d e f i n i t i o n s " f o r the predicates "blite","whack", etc. 1. Salmons "On v i n d i c a t i n g induction", Philosophy of Science, 1963. 2. Cf. Hanens An Examination of Adequacy Conditions f o r Confirma- ti o n , Ph.D. d i s s e r t a t i o n , 1970, pp.96-110. 162 ( i i ) " B l i t e " and "whack" are also "ostensively definable" i n another sense so long as we allow discrete elements or we allow a c e r t a i n a r t i f i c i a l observation instrument which involves a cer-t a i n discrete time element. So, f o r instance, a " b l i t e " colour observation instrument w i l l be a c e r t a i n meter whose pointer rests on a black thing when an object ha& been examined before time t ^ and whose pointer rests on a white thing when the object has not been so examined before time t ^ . 3 Thus i t appears that Salmon's proposed solution may suffer from a c e r t a i n ambiguity i n "ostensive d e f i n i t i o n " . 3. Goodman: " P o s i t i o n a l i t y and picture", Philosophical Review, I960, pp.523-525. 163 9. AN ATTEMPTED SOLUTION OF THE GOODMAN PARADOX In t h i s section I w i l l attempt a semantical (and syntactical) solu t i o n of the Goodman paradox. F i r s t , l e t us generalize the two predicates " b l i t e " and "whack" into two fa m i l i e s of predicates as follows: t i Df 5. An object 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 black. t. Df 6. An object i s whack i f i t has been examined before time t^ and i s white or not so examined and i s black. Now l e t t^= t^, tg» ...» t n , .... Then, we have the following two families of i n f i n i t e l y many predicates* t, t 2 t b l i t e , b l i t e , b l i t e , "^ 1 ^2 "^ n whack , whack , ..., whack Then, we ask: Can the predicates "black" and "white" be defined t- t. i n terms of any pair of the predicates " b l i t e " and "whack "? And the answer i s : Yes, they can be so defined by any pair of t i t. " b l i t e " and "whack ", where t- i s any fixed time. Thus the pre-h dicates "black" and "white" can be defined by the pa 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 fix e d time l a t e r than now, and l e t us J consider the following two hypotheses: HI: A l l ravens are black. t i H2: A l l ravens are b l i t e J . It appears to be obvious that both hypotheses are well confirmed. 164 But they have incompatible predictions: HI predicts that any raven examined l a t e r than t . w i l l be black, while H2 predicts that any t . raven examined l a t e r than t . w i l l be b l i t e J , v i z . , white. J This i s the Goodman paradox. A question i s raised now: How to make choice between HI and H2? To answer t h i s question, l e t us f i r s t consider the following family of hypotheses: H 2 . 1 : A l l ravens are b l i t e , t 2 H 2 . 2 : A l l ravens are b l i t e , • • t H 2.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 n , where t 1 < t 2 <... < t n _ ^ < t n (which i s supposed to be the time now) and " <" i s "the e a r l i e r than" r e l a t i o n . Then, l e t us assume that we just examined a raven and found i t to be black. Then, we have: (i) Hypotheses 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 respect to the l a s t examined black raven.) In the above sense of ( i ) and ( i i ) we w i l l c a l l the hypothesis H2 "not a h i s t o r i c a l l y well-established hypothesis", while H i s a " h i s t o r i c a l l y well-established hypothesis", f o r no matter which t. t. p a i r of b l i t e and whack i s employed to define "black" HI i s always well confirmed. Thus i t appears that HI says -more than £ 2 although both of them are well confirmed. 165 So, we propose the following 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 well-established while the other i s not, we should prefer the f i r s t hypothesis to the second i n confirmation theory f o r the simple reason that the l a t t e r says l e s s h i s t o r i c a l l y than the former. Once we have the above Thesis, we should prefer HI to H2, and the Goodman paradox w i l l have a semantical (syntactical) solution becuase the above Thesis can be described i n terms of semantics (and syntax). Thus we have a semantical (and synta c t i c a l ) solution of the Goodman paradox. 166 10. THE GOODMAN PARADOX REGAINED By a symmetric argument to the previous attempted solution of the Goodman paradox we f i n d that the Goodman paradox can be regained as followsi F i r s t , generalize the two predicates "black" and "white" into two families of predicates as follows! t i Df 7. An object i s black i f i t has been examined before time t^ and i s b l i t e or not so examined and i s whackt (where " b l i t e " and "whack" are understood, to repeat, as follows: Df 8. An object i s b l i t e i f i t has been examined before the year A.D. 2000 and i s black or not so examined and i s white; Df 9. An object 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 black. ) . * i Df 10. An object 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 following two families of i n f i n i t e l y many predicates: t, t 2 t black , black , black , ...; t l t2 *n white , white , white , .... Then, we ask: Can the two predicates " b l i t e " and "whack" be t. t. defined i n terms of any pair of the predicates "black " and "white "? And the answer i s : Yes, they can be defined i n the following waysi Df 11. An object i s b l i t e i f i t has been examined before time t i * i t. and i s black or i s not so examined and i s white 167 Df 12. An object i s whack i f i t has been examined before t. t. time t^ and i s white 1 or i s not so examined and i s black 1 . Now l e t time t . be any f ixed time l a t e r than theyear A. D. 2000 and l e t us consider the following two hypotheses: t . H3« A l l ravens are black J . H4: A l l ravens are b l i t e . It appears to be obvious that both hypotheses are well con-firmed. But they have incompatible predictions: H3 predicts that any raven examined l a t e r than A.D. 2000 w i l l be black, while H4 predicts that any raven examined l a t e r than A.D. 2000 w i l l be white. According to the attempted solu t i o n described i n the previous section, we should form the following family of hypotheses: H3.1: A l l ravens are black . *2 H3 .2 : A l l ravens are black t n - l H3.n-1: A l l ravens are black t H3.n: A l l ravens are black , where t 1 < t 2 < . . . ' t n _ i < * n (which i s supposed to be the time now) and " <" i s "the e a r l i e r than" r e l a t i o n . Then, we assume that we just examined a raven and found i t to be black. Then, we have: (i) Hypotheses H3.1 , H3.2, H3.n-1, H3.n are a l l discon-firmed; 168 ( i i ) Both H3 and H k are confirmed (with respect to the l a s t examined raven.) In the above sense of ( i ) and ( i i ) H3 i s not " h i s t o r i c a l l y well-established hypothesis", while H4 i s . So, by the Preference Thesis given i n the previous section we should prefer H k to H3. And t h i s conclusion i s exactly oppo-s i t e to the preference we made i n the previous section by the same evidence of a just examined black raven. So, the attempted solut i o n of the l a s t section cannot be r i g h t , otherwise we would have a contradiction. 169 11. A TENTATIVE CONCLUSION OF THE VALIDITY OF THE ADEQUACY CONDITIONS FOR CONFIRMATION IN LIGHT OF MARSHA HANEN'S STUDY Hanen has a very thorough study of the adequacy conditions 1 2 f o r confirmation. Her conclusion is« It thus seems that none of the most obvious appearing adequacy conditions for confirmation i s v a l i d f or the increase of firmness notion of confirmation with a possible exception 3 an exception may be the Equivalence Condition, which i s v a l i d f o r Carnap's explicatum. F i r s t note that "the most obvious appearing adequacy condi-tions f o r confirmation" mentioned i n the above quote include a l l adequacy conditions l a i d down by Hempel given i n section one of th i s Part, and that "Carnap*s explicatum", just mentioned,is ex-a c t l y *the increase of firmness notion of confirmation" referred to i n the preceding quote. I agree with Hanen's conclusion with one possible exception, and the exception i s the Entailment Condition which says that any sentence which i s entailed by an observation report i s confirmed by i t . In f a c t , my view about the Entailment Condition does not d i f f e r too much from what Hanen says about i t provided that two possible confusions are c l a r i f i e d , f o r Hanen sayst ... consider Hempel's Entailment Condition.... I t i s 1. Cf. Haneni An Examination of Adequacy Conditions for c o n f i r -mation, Ph.D. d i s s e r t a t i o n , Brandeis Univ., 1970, also "Con-firmation and adequacy conditions", Philosophy of Science. 1971, pp.361-368, and "Confirmation, explanation and accep-tance" i n Analysis and Metaphysics, ed. by Keith Lehrer, 1975 pp.93-128. 2. Haneni "Confirmation, explanation and acceptance", p.102. 3. Ibid, p.126, footnote 29 . 4. Ibid, p.102. 170 usual to regard t h i s condition as v a l i d f o r the firmness concept of confirmation. But i f we say that, f o r evidence to count as confirming a given hypothesis there must be some as yet undetermined cases, then t h i s condition 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 fo r g_ , since , i f £ i s true, £ w i l l be true as well, and there w i l l be nothing l e f t to determine. In t h i s s i t u a -t i o n , g. functions as a hypothesis that i s exhausted i n the sense that a l l of i t s instances have, as i t were, a l -ready been examined. I t looks as though we can say that p_ v e r i f i e s i n which case i t does not merely confirm i t . Since i n Hempel's theory of confirmation, as well as i n our i n t e r n a l confirmation theory, the Goodman condition-* on a univer-s a l conditional hypothesis that i s said to be "confirmed" i s not imposed, Hanen's point about "undetermined cases" mentioned i n the above quote does not a r i s e . This i s the f i r s t possible con-confusion that we need to c l a r i f y . Hanen also saysi^ But i f we take an increase of firmness notion of con-firmation, 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 condition holds only i n a r e s t r i c t e d form. The hypothesis must not be l o g i c a l l y true f o r then, even though i t i s entailed by any observation statement, i t i s incapa-ble of receiving support i n the increase of firmness sense, i t s degree of confirmation already being 1. S i m i l a r l y , i f the degree of confirmation of the hypothesis (say, (3x)Fx) i s already 1 on the basis of p r i o r evidence (say, Fa), then new evidence (say, Fb) cannot increase that degree of con-firmation and so cannot be said to be conf irming i n the i n -crease of firmness sense. Note that we have already r e a l i z e d the above awkward s i t u a t i o n « that may a r i s e i n Carnap's "increase i n firmness" notion of c o n f i r -5. The Goodman condition of a universal conditional hypothesis to be confirmed i s that ( i ) i t be "supported" ( i . e . "there be some posi t i v e instances"), ( i i ) "unviolated" ( i . e . "there be no ne-gative instances") and ( i i i ) "unexhausted" ( i . e . "there be some as yet undetermined instances"). Cf. Goodmani Fact, F i c t i o n , and Forecast. Chapter IV. 6. Hanem "Confirmation, explanation and acceptance", p.102. 171 mation i n section four of t h i s Part, and there we have suggested a small modification of Carnap's explicatum from "increase i n firmness of a hypothesis" to "increase i n firmness of a hypothe-s i s up to p r o b a b i l i t y one" i n order to avoid the above awkward s i t u a t i o n and to save Carnap's explicatum. The modification of Carnap*s explicatum also 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 possible confusion that may ari s e from the remark that my view about the Entailment 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 Condition while Hanen t r i e s to r e j e c t i t . 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 con-firmation. Thus, the disagreement about the v a l i d i t y of the Entailment Condition between Hanen and myself could be only apparent, f o r she uses Carnap's explicatum of "increase i n firmness" of con-firmation while I t r y to save both Carnap's explicatum and the Entailment Condition by modifying Carnap's explicatum i n order to avoid the awkward s i t u a t i o n pointed out by Hanen and also d i s -cussed by us i n section four of t h i s Part. In the following I w i l l t r y to give reasons why we r e j e c t the Conjunction Condition, the Consequence Condition, the Special Con-sequence Condition, and the Consistency Condition. (I) The r e j e c t i o n of the Conjunction Condition. In section six of t h i s Part we have already described a "counter-example" made by Carnap to r e j e c t the Conjunction Condition. However, we t r i e d to defend Hempel's Conjunction Condition by saying that Carnap's 7. Ibid, p.102. 172 counter-example i s given i n terms of " d e g r e e of conf irmation" which i s a concept of "quantitative conf irmation", while Hempel's Conjunc-t i o n Condition may he only an adequacy condition f o r " q u a l i t a t i v e confirmation". Since the q u a l i t a t i v e conception of confirmation and the quantitative conception of confirmation are quite d i f f e r -ent, i t may turn out that the Conjunction Condition holds i n Hem-pel's theory of confirmation while i t f a i l s to hold i n Carnap's theory of quantitative confirmation. Once we have Goodman's paradox, the Conjunction Condition i s not defensible i f the predicates " b l i t e " , "whack", etc., are i n -g troduced into Hempel's theory. To see t h i s , l e t us repeat the argument given i n section six of t h i s Part. F i r s t , consider the two hypotheses: HI: A l l ravens are black. H2: A l l ravens are b l i t e 2 0 0 0 . Then, suppose that we have the following observation report: B: r i s a black raven. So, i n Hempel*s theory of confirmation we have: (1) B confirms HI: (2) B confirms H2. Hence, by the Conjunction Condition we have: ( 3 ) B confirms H1&H2. 8 . The introduction of these new predicates, by the way, forces Hempel to say: I believe Carnap i s r i g h t i n his estimate that the concept of confirmation defined i n my essay"...is c l e a r l y too narrow." See Aspects of S c i e n t i f c Explanation, p.50. 173 But H1&H2 i s a hypothesis with a contradictory pred i c t i o n which says that any raven w i l l be both black and white a f t e r the year A.D. 2000. Since we do not want any adequate theory of con-firmation to have any contradictory prediction, the Conjunction Condition must be rejected. (II) The r e j e c t i o n of the Consequence Condition. Carnap has a counter-example to rej e c t the Consequence Condition as has been described i n section three of t h i s Part. We t r i e d to defend Hem-pel's Consequence Condition by saying that i t would hold i n Hempel's theory of q u a l i t a t i v e confirmation, while i t f a i l s to hold i n Car-nap's quantitative confirmation theory. Once we have the Goodman paradox, the Consequence Condition i s not defensible, f o r consi-der the two hypotheses HI and H2 and the observation report B des-cribed above. We have the following 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 of confirmation (without employing the Conjunction Con-di t i o n ) : (4) B confirms H1&H2. Since HI & H2 has some contradictory pred i c t i o n a f t e r the year A.D. 2000, we always have the following r e s u l t i (5) ( H l & H 2 ) o P i , where P i i s any predic t i o n of HI & H2 a f t e r the year A.D. 2000. So, by the Consequence Condition we havei (6) B confirms P i . Since the r e s u l t of (6) i s not acceptable i n any adequate the-ory of confirmation, we have to re j e c t the Consequence Condition (and, f o r the same reason, the Special Consequence Condition.) 174 (III) The r e j e c t i o n of the Consistency Condition. Consis-tency Condition saysi Every l o g i c a l l y consistent observation re-port i s l o g i c a l l y compatible with the class of a l l the hypotheses which i t confirms. In the above example of r e j e c t i n g the Conse-quence Condition once we have the r e s u l t of (4), the Consistency Condition i s at the same time rejected f o r the reasons ( i ) the observation report B i s l o g i c a l l y consistent, and ( i i ) HI and H2 are l o g i c a l l y incompatible--- since t h e i r predictions about the colors of ravens a f t e r the year A.D. 2000 are incompatible and, hence, ( i i i ) B and HI &H2 are l o g i c a l l y incompatible at lea s t B, HI and H2 altogether are incompatible about the color of ravens a f t e r the year A.D. 2000. Hence, the Consistency Con-d i t i o n must be rejected. Thus we have rejected Hempel*s Conjunction Condition, Con-sequence Condition (including the Special Consequence Condition) and the Consistency Condition as we claimed. It seems that i n the end Hempel himself has come to r e a l i z e t h e i r i m p l a u s i b i l i t y as well, f o r he says i n his l a t e s t Postscript (1964) on Confir-Q mation: 7 One and the same observable phenomenon may well be accounted f o r by each of two incompatible hypotheses, and the observation report describing i t s occurrence would then normally be regarded as confirmatory f o r either hypothesis. This point does seem to me to carry considerable weighttbut i f i t i s granted, then the con-sequence condition has to be given up along with the con- sistency condition. Otherwise, a report confirming each of two incompatible hypotheses would count as confirming any consequence of the two, and thus any hypothesis what-ever, (my i t a l i c s ) And, hence, Hempel concludes i n his Postscript (1964) on Confir-9. See Hempel: Aspects of S c i e n t i f i c Explanation, p.50. 175 mation: I believe Carnap i s r i g h t i n his estimate that the con-cept of confirmation defined 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." Accordingly, I think that 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 suf- f i c i e n t , but are not necessary f o r the confirmation of hy-pothesis H by an observation report B. (my i t a l i c s ) Thus we can agree with Hempel that those adequacy conditions l a i d down by him are not r e a l l y necessary, although they may be s u f f i c i e n t . In f a c t , the Conjunction Condition, the Consequence Condition (including the Special Consequence Condition) and the Consistency Condition are a l l unnecessary, i f what we have argued above i s r i g h t . (IV) The problem of the Equivalence Condition. Hanen i s very sympathetic with the Equivalence Condition, although she knows that Goodman's and Scheffler's theory of s e l e c t i v e confirmation does not observe the Equivalence Condition and, hence, the Equivalence Condi t i o n must be rejected i n Goodman's and Scheffler's theory of selec-t i v e confirmation. In her Ph.D. d i s s e r t a t i o n she seems to come to the following conclusion: On the primary l e v e l of confirmation 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 Confirmation we need the Equi-valence Condition; but on the secondary l e v e l of conf irmation which i s derived from the primary confirmation, we have to r e l i n q u i s h the Equivalence C o n d i t i o n . 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 Condition as an adequacy condition f o r confirmation on the one hand, 10. Ibid.. p.50. 11 . Hanen: An Examination of the Adequacy Conditions f o r Confirma-t i o n , pp.221-222. 12. Hanen, op. c i t . , p . H 8 f f . 176 but r e t a i n i t as a good condition f o r the theory of acceptance on the other. Thus, to sum up, according to Hanen we do not need any of the adequacy conditions laldjiown by Hempel fo r conf irmation;, although we need the Equivalence Condition i n theory of acceptance. But why should the Equivalence Condition be rejected i n the the-ory of confirmation while retained i n the theory of acceptance? Hanen's reasons are: F i r s t , i n any adequate theory of acceptance we need the follow-ing rule or condition: Equivalence Rule (for acceptance). Suppose that KI and K2 are two l o g i c a l l y equivalent hypotheses. Then, KI i s accepted i f f K2 i s accepted. This i s c l e a r and, I suppose, nobody would argue against i t . Second, "questions about confirmation are d i f f e r e n t from ques-13 tions about acceptance". Or more f u l l y i n Hanen's words: I am suggesting ... that confirmation and acceptance can be viewed as occurring on two l e v e l s .... The usual condi-tions of adequacy should not be r i g i d l y applied to c o n f i r -mation r e l a t i o n s . But they may be applied once we ask which among confirmed hypotheses should be accepted ... . But one does not want to make too much of the idea of two l e v e l s , f o r I do not wish to claim that they are separate l o -g i c a l l e v e l s or anything of the sort. Rather, the important consideration i s that questions about confirmation are d i f - ferent from questions 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 Equivalence Con-d i t i o n as an adequacy condition f o r confirmation and as an adequa-cy condition (or rule) f o r acceptance: l k Hempel argues on behalf of the Equivalence Condition that hypotheses function i n deductive explanations and predic-tions and i t would be very odd to suppose that a given hy-13. Hanen, op. c i t . . p.121. 14. Hanen, op. c i t . , p.121. 177 pothesis could serve as a premise i n a deductive argument where i t s l o g i c a l equivalent could not. And indeed that would be odd. But there i s an extra step needed the move from saying that H i s confirmed to saying that i t i s acceptable as a premise i n a p a r t i c u l a r deductive argument. If i t is_ so acceptable, then c e r t a i n l y so i s any hypothesis l o g i c a l l y equivalent to i t but i t surely does not follow that i f i t i s confirmed, then so i s any equivalent hypo-t h e s i s . Whether the l a t t e r claim i s true depends upon the p a r t i c u l a r theory of confirmation i n use and the d i f f i c u l -t i e s into which attempts to adhere s t r i c t l y to the usual adequacy conditions seem to lead us i s good reason to avoid imposing them a p r i o r i . Now to sum up without making any c r i t i c a l review of what Hanen says above, Hanen thinks that we do not need any of the adequacy conditions l a i d down by Hempel. One serious question i s immediately raised: " I f we no longer require of a theory of confirmation that i t meets the condition of adequacy, i s not everyone free to propose his own theory?" (For i f i t i s the case, then we would be l e f t without any means of choos-ing among them.)1-' To bring out the point, consider the following new type of selective confirmation: 1^ A universal conditional hypothesis (x)(FxaGx) 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 denial", v i z . (x)(-Fx oGx), i s disconfirmed (where both "con-firmed" and "disconfirmed" are i n the Hempelean sense of con-firmation and disconfirmation.) Under t h i s new type of sel e c t i v e confirmation we havet a non-black non-raven s e l e c t i v e l y confirms the raven hypothesis while a black raven i s neutral to i t a complete reversal of the s i t u a -t i o n of Goodman*s and ScheffIer•s theory of selec t i v e confirmation. So, which idea of selective confirmation i s legitimate? If both are legitimate, then the paradoxes of confirmation w i l l 15. Stewart, J.Pi "Comments on Marsha Hanen's "Confirmation, expla-nation and acceptance"", p . l . 16. Ibid.. p.3. 178 obviously come back. However, Hanen thinks that only Goodman's and Scheffler's idea of selective confirmation i s the legitimate 17 one, f o r she says: I f one i s n ' t wedded to the equivalence condition, 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 -tions or utterances, then i t seems natural to say that, a necessary condition f o r t r e a t i n g hypotheses as i n com- p e t i t i o n i s that t h e i r antecedent classes be the same,... In the case of a hypothesis and i t s antecedent denial, t h i s condition f a i l s . (my i t a l i c s ) I am quite sympathetic with the necessary condition, to be c a l l e d "Hanen's condition", given i n the above quote, (for that i s , i n a way, the l i n e of thought carried out i n the i n t e r n a l con-1 o firmation theory.) But, then, a second question a r i s e s : Is Hanen's condition a necessary condition f o r confirmation? or f o r acceptance? or what? If i t i s a necessary condition f o r confirmation, does Hanen con-t r a d i c t h erself to r e j e c t any adequacy conditions l a i d down by Hem-pel? Since Hanen says nothing about the above questions, I would venture the following answers, which are the most reasonable ones, that I can think of: ( i ) Hanen's condition i s an adequacy condition f o r comparative (not q u a l i t a t i v e ) confirmation, f o r as she says: I t i s "a necessary condition 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 i t a l i c s ) ( i i ) Since Hanen's condition i s not among Hempel's adequacy conditions f o r q u a l i t a t i v e confirmation, she i s not inconsistent i n r e j e c t i n g a l l the adequacy conditions l a i d down by Hempel. 17. Hanen, op. c i t . , p.104. 18. The concept of "confirmation" of the i n t e r n a l confirmation the-ory i s automatically a " s e l e c t i v e confirmation" i n the sense of Goodman and Scheffler. (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 think that Hanen's r e j e c t i o n of a l l adequacy condition l a i d down by Hempel i s also a plausible p o s i t i o n with two great merits* 1) I t frees us from imposing any "adequacy" condition a p r i o r i i n constructing a theory of q u a l i t a t i v e confirmation. 2) That, of course, does not mean any theory of confirmation i s as good as any other. S t i l l , Hanen's p o s i t i o n has the great merit of separating the problem of the construction of a confirma-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 or absurd theory of confirmation w i l l be rejected when i t s absurdity i s found l a t e r in i t s a pplications. Thus we should not impose a p r i o r i any "adequacy" condition to any confirmation theory, e s p e c i a l l y i n i t s construc-t i o n period. This i s es p e c i a l l y true when we know very l i t t l e about the p r a c t i c a l i t y of many competitive theories of confirma-t i o n . By practice, we w i l l eventually select the l e a s t odd one. However, I do not hesitate to add t h i s to Hanen's position* Any confirmation theory which v i o l a t e s the Equivalence Condition i s l i k e l y to have the following 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 formulations of an equivalent hypothesis may have d i f f e r e n t confirming instances, t h i s r e s u l t would create great inconvenience i n i t s a p p l i c a t i o n as pointed out by Hempel. 2) The theory of q u a l i t a t i v e confirmation may become incoherent or unnecessarily complicated when i t extends into a theory of quan-t i t a t i v e confirmation. For i n the standard theory of p r o b a b i l i t y we need the following p r i n c i p l e either postulated as an axiom or 20 derived 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 equivalent, then they have the same p r o b a b i l i t y . Thus any theory of q u a l i t a t i v e confirmation must observe the 20. Cf., E l l e r y E e l l s * Rational Decision and Causality, p.222. 180 above Equivalence P r i n c i p l e i n order to be extendable to a theory of quantitative confirmation that i s i d e n t i c a l to or i n accord with the standard theory of p r o b a b i l i t y , f o r the Equivalence Condition i n confirmation theory i s the counterpart condition or p r i n c i p l e to the Equivalence P r i n c i p l e i n the standard theory of p r o b a b i l i t y . In defence f o r a theory of q u a l i t a t i v e confirmation which v i o -l a t e s the Equivalence Condition, "... one could argue that there need be no connection between the p r o b a b i l i t y calculus which i s formal, mathe-matical, and quantitative 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 saysi "... t h i s i s a rather 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 goal between the two approaches to induction, 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 at a l l f o r the other. One wants to say that the statements of the p r o b a b i l i t y calculus accord with 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 confirmation theory 23 which v i o l a t e s the Equivalence Condition may say« J "... some p r o b a b i l i t y t h e o r i s t s for the s u b j e c t i v i s t or personalists persuasion d e l i b e r a t e l y v i o l a t e s the pro-b a b i l i t y calculus i n the sense that they do not require of an i n t e r p r e t a t i o n of 'probability* that i t render a l l the axioms of the calculus true." 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 confirmation evaluations 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 -culus) , i f these evaluations are to constitute a coherent set of b e l i e f s . Thus the axiom of the quantitative concept of confirmation are j u s t i f i e d , i n that they are necessary conditions 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 of Adequacy Conditions f o r Confirmation, p.195 22. Ibid., p.195. 23. Ibid., p.195. 24. Ibid.. p.58. 181 Since the r e s u l t obtained by Ramsey and De F i n e t t i about the s u b j e c t i v i s t ' s or personalist's approach to p r o b a b i l i t y theory mentioned i n the above quote i s conclusive and since Hanen thinks that Goodman's and Scheffler's theory of sel e c t i v e confirmation can be extended to a theory of quantitative confirmation, I sus-pect that i t s extension would be incoherent at some point since 2 5 the Equivalence Condition i s v i o l a t e d . J However, the point i s s t i l l t h i s t Not to k i l l a theory of qua-l i t a t i v e confirmation just because i t does not observe the Equiva-lence Condition, although the v i o l a t i o n of the Equivalence Condi-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 confirmation i n -coherent l a t e r when the theory i s extended into a quantitative con-firmation. Anyway, i t s l a t e r "incoherence" i s only a speculation, even i f i t i s highly p l a u s i b l e . S t i l l , we should k i l l a theory of q u a l i t a t i v e or quantitative confirmation only when we have i n f a c t found that i t i s incoherent. Note that my agreement with Hanen's po s i t i o n on the one hand and my appeal the a d m i s s i b i l i t y and workability of the Equivalence Condition as the counterpart of the equivalence P r i n c i p l e within the context of the standard p r o b a b i l i t y theory i s i n accord with Hempel's and Carnap's l a t e s t view, for Hempel has t h i s to say i n his "Postscript (1964) on Confirmation"! 2 6 Perhaps the problem of formulating adequate c r i t e r i a of q u a l i t a t i v e confirmation had best be tackled, a f t e r a l l , by means of the quantitative concept of confirma-25. Note that the theory of sel e c t i v e confirmation turns out to be "incoherent" i n i t s claim that i t can solve the paradoxes of confirmation , f o r I f i n d that new paradoxes of confirmation can be e a s i l y derived i n Goodman's and Scheffler's theory of s e l e c t i v e confirmation to be discussed i n section 3, Part V. 26. Hempel, op. c i t . , p.50. 182 t i o n . This has been suggested e s p e c i a l l y by Carnap, who holds that "any adequate explicatum f o r the c l a s s i f i c a - tory concept of confirmation must be i n accord with at le a s t one adequate explicatum f o r the quantitative con- cept of confirmation" .... (my i t a l i c s ) Thus, i n conclusion, we agree with Hanen to r e j e c t a l l adequa-cy conditions l a i d down by Hempel f o r q u a l i t a t i v e confirmation with the understanding that there i s l i k e l y a problem of pot e n t i a l i n -coherence when the q u a l i t a t i v e confirmation theory i s extended to a quantitative one i f the Equivalence Condition i s not observed i n the q u a l i t a t i v e confirmation theory. However, f o r i n t r i n s i c reasons (say, f o r the sake of consis-tence, coherence and greater e x p r e s s i b i l i t y ) the i n t e r n a l c o n f i r -mation theory needs the Equivalence Condition (since any l o g i c a l l y equivalent hypotheses of a given hypothesis are confirmed, or d i s -conf irmed, by the same observation reports i n the i n t e r n a l c o n f i r -mation theory) and, perhaps, also the Entailment Condition ( i f peo-pel would accept my modification of Carnap*s explicatum of c o n f i r -mation discussed e a r l i e r . ) PART V. TENTATIVE CONCLUSIONS 184 1. INTRODUCTION In t h i s l a s t Part we w i l l sum up and at the same time compare the differences of Hempel's theory of confirmation, Goodman's and Scheffler's theory of selective confirmation and the i n t e r n a l con-firmation theory as they represent, I think, the three most impor-tant approaches to confirmation theory. With what we have studied up to now and, espec i a l l y , with the help of some newly discovered paradoxes of confirmation derived i n the theory of selective confirmation as well as i n the int e r n a l confirmation theory, I come to the following most reasonable con-cl u s i o n that I can think oft the only plausible solution of the paradoxes of confirmation i s t h e i r d i s s o l u t i o n . 185 2. A SUMMARY OF HEMPEL'S THEORY OF CONFIRMATION Hempel's theory of confirmation has the following outstanding propertiesi 1. I t i s the f i r s t comprehensive theory of q u a l i t a t i v e c o n f i r -mation. 2. Its underlying l o g i c i s the c l a s s i c a l two-valued q u a n t i f i -c a t i o n a l l o g i c . 3. His theory of confirmation o f f e r s a S a t i s f a c t i o n C r i t e r i o n of Confirmation; i . e . , i n his theory of confirmation we have* Theorem 18. An observation report B ( c 1 , c 2 , ...,c^) confirms a hypothesis H i f f f c ^ . C g c ^ s a t i s f i e s H, (where c^ . C g , • • . » c ^ are the i n d i v i d u a l constants 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 observation report 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 loosely that c "confirms" the raven hypothesis, although what they mean i s that c s a t i s f i e s the raven hypothesis and, hence, the t o t a l information about c, v i z . Rc.Bc, confirms the raven hypothesis.) 4. A l l adequacy conditions of confirmation theory discussed i n the l a s t Part hold i n his theory of confirmation. Above a l l , the Equivalence Condition holds i n his theory of confirmation. 5. The paradoxes of confirmation are dissolved i n his theory of confirmation, f o r according to Hempelt ( i ) they are not genuine, ( i i ) they have no l o g i c a l and objective grounds and, hence, ( i i i ) they are psychological i l l u s i o n s . 186 Or i n Hempel's words: "The impression of a paradoxical s i t u a t i o n i s not objec-t i v e l y founded: i t i s a psychological i l l u s i o n . " (my i t a l i c s ) "... our factual knowledge that not a l l objects are black tends to create an impression of paradoxicality 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 turns out that the paradoxes of confirmation... are due to a misguided i n t u i t i o n i n the matter rather 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 are derived." (my i t a l i c s ) 6. What i s controversial i s that many people think that the pa-radoxes of confirmation are genuine and they can be l o g i c a l l y and objectiv e l y formulated and, hence, they are not psychological i l l u -sions as Hempel thinks and persuades us to believe. 7. One serious trouble of Hempel's theory of confirmation i s that Hempel's theory of confirmation allows the practice of "indoor p ornithology". Thus the following hypotheses are a l l confirmed i n Hempel's theory (simply because I have no o f f i c e at UBC): HI: A l l ravens i n my o f f i c e at UBC are black. H2: A l l ravens i n my o f f i c e at UBC are white. H3« A l l ravens i n my o f f i c e at UBC are both black and white. Perhaps the above type of hypotheses i s too t r i v i a l f o r o r n i -thologists, f o r who cares about the ravens i n my o f f i c e at UBC. So, consider the following bigger hypotheses: H4: A l l pandas i n Canada are black. H5* A l l pandas i n Canada are white. H6: A l l pandas i n Canada are both black and white. H7« Each panda i n the moon has two eyes. H8: Each panda i n the moon has three eyes. HQ* Each panda i n the moon has one thousand and one eyes. 1. Hempel: Aspects of S c i e n t i f i c Explanation, p.18 and p.20. 2. Goodman: Fact. F i c t i o n , and Forecast, pp.70-71. 18? Now each of the above bigger hypotheses H4- H9 i s confirmed (and, hence, presumably true) 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 objects i n Canada or i n the moon confirms each of hypotheses H4- H9. Here i t seems quite cl e a r now as the above examples shown that the practice of "indoor ornithology" may produce a kind of mytho-l o g i c a l or f i c t i o n a l ornithology. That Hempel's theory of c o n f i r -mation allows such a mythological or f i c t i o n a l practice indicates that there i s something wrong with Hempel's theory of q u a l i t a t i v e confirmation or that the underlying l o g i c ( i . e . c l a s s i c a l l o g i c ) of Hempel's theory of confirmation cannot deal 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 think that 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 topics of (theory of) comparative confirmation. For, then, with the establishment of the following hypotheses: H^Q: A l l ravens are black ^ 1 1 ! P a n c* a s a r e black H 1 2: Each panda has two eyes as truths or laws, the hypotheses H2, H3, H5, H6, H8 and H9 can be eliminated as l e s s good and impractical,simply because H 1 Q, H 1 1 and H 1 2 are true and more general on the one hand and simpler-^ i n form and, hence, more useful on the other hand. 3. Cf. Goodman: "safety, strength, s i m p l i c i t y " , Philosophy of S c i -ence, 1961, pp.150-151. 188 3. A SUMMARY OF GOODMAN'S AND SCHEFFLER'S THEORY O F SELECTIVE CONFIRMATION Goodman's and Scheffler's theory of select i v e confirmation has the following i n t e r e s t i n g , important as well as peculiar proper-t i e s » 1. I t i s based on top of Hempel's theory of confirmation. 2. The theory of selective confirmation appears not to allow the practice of indoor ornithology. 3. The theory of selective confirmation does not have a Sa t i s -f a c t i o n C r i t e r i o n of Confirmation. 4. The paradoxes of confirmation appear not; to be derivable i n the theory of selective confirmation. If so, i t would follow: ( i ) the paradoxes of confirmation are genuine, and ( i i ) they are l o g i c a l l y and objectively grounded and, hence, ( i i i ) they are not psychological i l l u s i o n s . 5. In the theory of select i v e confirmation the Equivalence Con-d i t i o n does not hold (as f a r as select i v e confirmation i s concerned). Hence, i n general, 5.1. the formulation of a hypothesis i s c r u c i a l to i t s selec-t i v e confirmation, and 5.2. there are hypotheses, e.g. (x)Gx, which cannot have se-l e c t i v e confirmation simply because they do not have contraries. 6. By d e f i n i t i o n of selective confirmation the contrary of a hypothesis must be disconfirmed f o r the hypothesis to be se l e c t i v e -l y confirmed. I t follows that a universal conditional hypothesis such as the following one, H : ( 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 following hy-pothesis, CHi -(x ) ( Px3-Gx), confirmed. That amounts t o i 1 we must have both H and the follow-ing hypothesis, CH*: (3x)(Fx.Gx), confirmed i n order to s e l e c t i v e l y confirm the universal conditional hypothesis H. This seems to be a very strong and undesirable as-sumption. At l e a s t , i t i s a stronger assumption than the assump-t i o n of adding an e x i s t e n t i a l clause to a universal conditional hypothesis made by the A r i s t o t a l i a n solution of the paradoxes of confirmation c r i t i c i z e d by Hempel. Also some c r i t i c s f e e l that property 5*1 and property 5.2 made above are undesirable, inconvenient and, perhaps, also wrong i n theory or, at l e a s t , i n s c i e n t i f i c practice; and, furthermore, pro-perty 5.2 seems also to suggest that the theory of selective con-firmation i s inadequate. Above a l l , the peculiar property 5 seems to be the most harmful one to the theory of selective confirmation, f o r without observing the equivalence condition the theory of selective confirmation i s l i k e l y to be i n c o n f l i c t with the standard p r o b a b i l i t y theory when i t i s extended into a theory of quantitative confirmation as we have t r i e d to indicate i n section 11 of the previous Part. 7. F i n a l l y , my l a t e s t discovery i s that Goodman's and Scheffler's 1. However, from the transformation of CH to CH* the equivalence condition has been employed, and we know that the equivalence condition i s rejected i n the theory selective confirmation. So, the c r i t i c i s m followed maybe not r e a l l y true. 2. Hempeli Aspects of S c i e n t i f i c Explanation, pp.16-17. 190 theory of sele c t i v e confirmation does not r e a l l y solve the para-doxes of confirmation. Thus, consider the observation report: (1) Ra.Ba, which s e l e c t i v e l y confirms the raven hypothesis: H: (x)(Rxr>Bx); while each of the observation reports, (2.1) -Rb.-Bb, (2.2) -Rc.Bc, i s " s e l e c t i v e l y neutral" to ( i . e . neither " s e l e c t i v e l y confirms" nor "disconfirms") the raven hypothesis H. But each of the f o l -lowing observation reports, (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 confirm the raven hypothesis H, f o r we have: (4.1) (Ra.Ba). (-Rb.-Bb) \[ (Ra a B a ) . (Rb t>Bb), (4.2) (Ra.Ba). (-Rc.Bc) ||-(RaoBa). (Rc oBc), (4.3) (Ra.Ba). (-Rb.-Bb). (-Rc.Bc) ||- (RaoBa). (Rb=>Bb). (Rc oBc ), i . e . , ( 3 . 1 ) , (3-2) and (3.3) each confirms the raven hypothesis H; and we also have: (5-1) (Ra.Ba). (-Rb.-Bb) ||- -((Ra=> -Ba). (Rb => -Bb)), (5.2) (Ra.Ba). (-Rc.Bc) fl- -((Rao -Ba). (Rc=> -Be )), (5-3) (Ra.Ba). (-Rb.-Bb).(-Rc .Be ) ||- -((Ra=?-Ba). (Rbo-Bb). (RcP-Bc)), i . e . , ( 3 .1 ) . (3-2) and (3.3) each disconfirms, respectively, i t s contrary. Hence, (3.1), (3*2) and (3.3) each s e l e c t i v e l y confirms the raven hypothesis H, these are the newly discovered paradoxes of confirmation derived i n the theory of selective confirmation. So, the theory of selective confirmation does not r e a l l y solve a l l paradoxes of confirmation. In other words i t i s an u n j u s t i f i e d claim to say that Goodman's and Scheffler's theory of selec t i v e confirmation has solved the paradoxes of confirmation. 192 4. A TENTATIVE CONCLUSION FOR THE INTERNAL CONFIRMATION THEORY In the development of the in t e r n a l confirmation theory when we understood the Raven Hypothesis (which says that a l l ravens are black) as the hypothesis: RH: (tyx) (Rx =»Bx), we almost had a solution of the paradoxes of confirmation, f o r we can have the following r e s u l t s about the RH i n the i n t e r n a l c o n f i r -mation theory: (i ) A raven that i s black and only a raven that i s black can confirm the RH; ( i i ) A raven that i s non-black w i l l disconfirm the RH; and ( i i i ) A l l other instances are neutral to the RH. So f a r so good. The only trouble with the above "solution" of the paradoxes of confirmation i s that the RH can never be true i n the actual world where, as i t turns out to be, there are non-black objects which w i l l make the RH never be true i n the in t e r n a l confirmation theory when we come to actual a p p l i c a t i o n . The non-truth, a l b e i t also non-falsity, of the RH i s t h e o r e t i c a l l y both odd and undesirable. So, we do not have a genuine solu t i o n of the paradoxes of con-firmation thus f a r . Then, I think that I can perhaps refi n e and improve the above "solution" of the paradoxes of confirmation by introducing the con-cept of "# - r e s t r i c t i o n " of a hypothesis into the i n t e r n a l confirma-t i o n theory i n order to get r i d of the neutral cases of the RH so that the refined RH can have the truth-value "truth". Thus i f we understand the Raven Hypothesis as: 193 RH#: ((Vx)(Rx =*Bx))#, we can s t i l l preserve the above three properties ( i ) , ( i i ) and ( i i i ) on the one hand and have RJr had the truth-value "truth" on the other hand. Thus i t seems that f i n a l l y we have a solut i o n of the paradoxes of confirmation. At t h i s point 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 "solution" of the paradoxes of confirmations (I) The f i r s t c r i t i c i s m i s t h i s . Suppose that we have the Con-sequence Condition i n the i n t e r n a l confirmation theory. Then, note that we have the following semantical 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 true that i n the actual world we have the obser-vation reports (2) (Ra& Ba) & (-Rb &-Bb) where "a" i s "a black raven" and "b" i s "a non-black non-raven", i t follows with the help of the Consequence Condition that the hy-pothesis, (3) ( ( V X ) ( R X = * B X ) ) # S E ((\/x)(-Bx=>-Rx))#, i s confirmed by (2) f o r (2) confirms (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 derive the paradoxes of confirmation i n the i n -ternal confirmation theory (with the help of the Equivalence Condi-tion) according 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 that the Consequence Con-1 9 4 d i t i o n does not hold i n the i n t e r n a l confirmation theory, 1 f o r i t has been rejected as an adequacy condition f o r confirmation i n sec-t i o n 1 1 , Part IV. Moreover, ( 3 ) i s not a l o g i c a l equivalence and, hence, we cannot employ the Equivalence Condition to i t i n order to derive the paradoxes of confirmation from i t . (II) The second c r i t i c i s m i s t h i s . In the i n t e r n a l confirma-t i o n theory the two hypothesesi ( 4 ) (3x)Rx& (3x)-Bx&((yx)(Rx=*Bx)) # and ( 5 ) (3x)Rx& (3x)-Bx& ((t/x)(-Bx=*-Rx)) # are l o g i c a l l y equivalent. Let us c a l l ( 4 ) "the strong Raven Hypothesis". Since ( 4 ) i s l o g i c a l l y equivalent to ( 5 ) , ( 5 ) i s also "the strong Raven Hypo-thesis", e s p e c i a l l y when we have the Equivalence Condition i n the i n t e r n a l confirmation theory. Consequently, what confirms (or disconfirms) ( 4 ) w i l l also con-firm (or, respectively, disconfirm) ( 5 ) » and vice versa. In p a r t i c u l a r , we have t h i s i the observation report ( 2 ) con-firms both ( 4 ) and ( 5 ) , and t h i s i s one version of the paradoxes of confirmation. This new version of the paradoxes of confirmation i s a f a t e f u l blow to my claim that the paradoxes of confirmation can be solved i n the i n t e r n a l confirmation theory. (III) Worse s t i l l , without employing the Equivalence Condition 1 . Note that the f i r s t c r i t i c i s m was made before I came to the re-j e c t i o n of the Consequence Condition as an adequacy condition fo r the i n t e r n a l confirmation theory. 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 conditions) some new forms of the paradoxes of confirmation can be derived as follows: Although i n the in t e r n a l confirmation theory the observation report, BI: Ra&Ba confirms the de-neutralized Implicational Raven Hypothesis RH ,and each of the following two observation reports, B2: -Rb&-Bb, B3: -Rc&Bc i s neutral to RH^, we have that each of the following three obser-vation reports, B4: (Ra& Ba) & (-Rb &-Bb), B5: (Ra & Ba) & (-Rc & Be ), B6: (Ra&Ba)& (-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 of RH with respect to the class of i n d i -viduals e s s e n t i a l l y mentioned i n each of the observation reports B4, B5 and B6 i n the in t e r n a l confirmation theory. In other words, i n the i n t e r n a l confirmation theory we have the following two important r e s u l t s : ( I I I . l ) Individually, the observation report that a raven i s black confirms RH , while the observation report that a non-raven i s non-black or the observation report that a non-raven i s black u (or both) i s neutral to RW . 196 (III.2) C o l l e c t i v e l y , as a statement of t o t a l evidence, a state-ment that a raven i s black conjoined to a statement that a non-raven i s non-black or to a statement that a non-raven i s black (or to both) confirms RH^. The r e s u l t ( I I I . l ) i s f i n e ; but the r e s u l t (III.2) i s some new version of the paradoxes of confirmation gained i n the int e r n a l con-firmation theory. The r e s u l t ( I I I . l ) gives us the impression that the paradoxes of confirmation are genuine and solvable; but i n f a c t that i s only a psychological i l l u s i o n f o r the paradoxes of confirmation are not r e a l l y solvable as the r e s u l t (III.2) has shown. The above surp r i s i n g r e s u l t s that some new paradoxes of c o n f i r -mation can be derived from RH as well as from the strong raven hy-pothesis seem to suggest the following conclusions! 1) Even i f we change the underlying l o g i c of Hempel's theory of confirmation from c l a s s i c a l l o g i c to QC3 t h i s change i s , f i r s t of a l l , r a d i c a l and quite undesirable i n the sense that l o g i c i s always said to be the l a s t thing to go f o r change new d i f f i c u l -t i e s as well as new and old paradoxes of confirmation w i l l occur or recur i n some unexpected ways. 2) Thus, f i r s t , when we understand the Raven Hypothesis as RH, although we can avoid the paradoxes of confirmation, the truth-value of RH unexpectedly turns out to be neutral ( i . e . neither true nor f a l s e ) . u 3) Then, we de-neutralize RH to have RH i n order to get r i d of u a l l neutral instances of RH, although RH77^ turns out to be true, some new form of the paradoxes of confirmation such as the above r e s u l t (III.2) now unexpectedly occurs. 197 4) A l l these, i n turn, seem to suggest that the paradoxes of confirmation 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 confirmation the-ory i s no exception, even i f one t r i e s to read and understand the raven hypothesis i n other plausible ways, as well as to make other plausible refinements. 5) So, Hempel i s perhaps r i g h t a f t e r a l l to conclude that we can only have a "di s s o l u t i o n " of the paradoxes of confirmation. At le a s t t h i s seems to be the most reasonable conclusion that 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 that there are some s l i g h t differences 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 confirmation 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 in t e r n a l confirmation the-ory i (A) In Hempel's theory of confirmation from the d i s s o l u t i o n of the paradoxes of confirmation we have: (i ) Ra.Ba, -Rb.-Bb or -Rc.Bc (or any number of t h e i r con-junction) confirms the raven hypothesis H; ( i i ) Rd.-Bd (or i t s conjunction with any other observation report) disconfirms the raven hypothesis H. (B) In the i n t e r n a l confirmation theory from the d i s s o l u t i o n of the paradoxes of confirmation what we have are: (i ) Ra&Ba as the t o t a l evidence confirms the raven hypothe-s i s RH #j ( i i ) -Rb&-Bb or -Rc & Be (or both) as the t o t a l evidence i s neutral to the raven hypothesis RH ; ( i i i ) Rd& -Bd (or i t s conjunction with any other observation report) disconfirms the raven hypothesis RH ; 1 9 8 (iv) Ra&Ba and i t s conjunction with either -Rb& -Bb or -Rc & Be (or both) as the t o t a l evidence also confirms the raven it hypothesis RH . The significance of ( A ) ( i ) means that the paradoxes of c o n f i r -mation can only be considered as psychological i l l u s i o n s i n Hempel's theory; however, the significance of ( B ) ( i i ) seems to suggest that i n a sense t h a t the paradoxes of confirmation are not psychological i l l u s i o n s i n the i n t e r n a l confirmation theory, although they turn out to be eventually i n the sense of (B)(iv) when we also come across a raven that i s black. However, the s i g n i f i c a n t differences just noted above could be only apparent once we r e a l i z e that to confirm the raven hypothesis RH77^ i n the i n t e r n a l confirmation theory at l e a s t one black raven must be e s s e n t i a l l y mentioned i n the t o t a l evidence or, i n other words, to make RH true there must exist at l e a s t one black raven i n the world. But there i s no such " r e s t r i c t i o n " being b u i l t into the raven hypothesis H i n the confirmation of H i n Hempel's theory of confirmation, nor i n the making of the truth of the raven hypo-thesis H i n c l a s s i c a l l o g i c . 7) F i n a l l y , note that the s t r i k i n g s i m i l a r i t y of the deriva-tions of the new paradoxes of confirmation that the t o t a l evidence, Ra and Ba conjoined to either -Rb and -Bb or -Rc and Be 199 (or both) confirms or s e l e c t i v e l y confirms the raven hypothesis that a l l ravens are black, respectively, i n the i n t e r n a l c o n f i r -mation theory and i n the theory of selective confirmation. The s t r i k i n g s i m i l a r i t y of the new paradoxes of confirmation derived i n the above mentioned two theories i s i n f a c t due to the same root: both theories employ ( i ) Hempel's concept of t o t a l evidence ( i . e . t o t a l observation report) and ( i i ) the concept of entailment of the development of a hypothesis i n defining the concepts of "confirma-t i o n " and " s e l e c t i v e confirmation". I t i s true that Hempel also employs the concept of t o t a l e v i -dence and the concept of entailment of the development of a hypo-thesis i n defining his concept of "confirmation". But he also ac-cepts gladly the paradoxes of confirmation, once he derives them. Thus conditions ( i ) and ( i i ) stated above seem to impose on the i n t e r n a l confirmation theory and the theory of selective confirma-t i o n the new paradoxes of confirmation. 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