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Conditional and biconditional reasoning : a developmental study using new norms Taylor, John Robert 1981

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CONDITIONAL AND BICONDITIONAL REASONING: A DEVELOPMENTAL STUDY USING NEW NORMS by JOHN ROBERT TAYLOR B.Ed., U n i v e r s i t y of Toronto, 1972 M.Sc., Monash U n i v e r s i t y , 1969 B.Sc. (Hons.), Monash U n i v e r s i t y , 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION i n THE FACULTY OF GRADUATE STUDIES Department of Mathematics and Science Education We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1981 © John Robert T a y l o r , 1981 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f L^u.^i/U^(Vo«*  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V ancouver, Canada V6T 1W5 Date ^0 HOV j [R% I r>r> C I ~> /Id \ DEDICATION To my parents JOHN and ELSIE TAYLOR ho by nature and nurture e me the freedom to choose ACKNOWLEDGEMENTS A f t e r years of immersion i n t h i s p r o j e c t , i t i s easy to lose s i g h t of the sources of encouragement and i n s p i r a t i o n . To my Committee members, Dr. J . Coombs; Dr. E. Hobbs; Dr. S. S. Lee; Dr. D. Owens; Dr. G. S p i t l e r ; Dr. J . Y u i l l e , I owe s p e c i a l thanks. Their confidence i n me gave that e s s e n t i a l margin of safety to ensure completion. I am p a r t i c u l a r l y indebted to Dr. S p i t l e r for her patience and wisdom, and to Dr. Coombs and Dr. Owens for t h e i r i n v a l u a b l e help w i t h the d i f f i c u l t task of proof-reading. Thanks are due a l s o to Pat, my f r i e n d s , and my colle a g u e s , for t h e i r empathy, i n t e r e s t , and camaraderie. i v ABSTRACT The present study attempted to answer the question of the extent to which i d i o s y n c r a t i c i n t e r p r e t a t i o n s of premises a f f e c t e d performance on s y l l o g i s t i c reasoning t a s k s . A model was developed which accounted for both l i n g u i s t i c f a c t o r s and l o g i c a l f a c t o r s . The co n s t r u c t s i n the model were incorporated i n t o a new set of norms for s y l l o g i s t i c reasoning. Two t e s t s were developed f o r t h i s study. The R e f e r e n t i a l Test measured i n t e r p r e t a t i o n s of given l o g i c a l expressions using p h y s i c a l r e f e r e n t s . The S y l l o g i s m Test contained standard s y l l o g i s m s in which a s p e c i f i c l o g i c a l expression was embedded as the f i r s t premise of each s y l l o g i s m . One v e r s i o n of each of these two t e s t s was constructed f or each of four l o g i c a l expressions. One l o g i c a l expression was a c o n d i t i o n a l p r o p o s i t i o n , and the other three were s y n t a c t i c a l l y d i f f e r e n t b i c o n d i t i o n a l p r o p o s i t i o n s . One ve r s i o n of each t e s t was administered to 916 students i n Grades 6, 8, 10 and 12, one p r o p o s i t i o n f or each student. The c r i t e r i o n measures for the R e f e r e n t i a l Tests revealed (a) many d i f f e r e n t i n t e r p r e t a t i o n s of the same p r o p o s i t i o n , and (b) c l e a r d i f f e r e n c e s among a l l four p r o p o s i t i o n s . Developmental trends were unclear. The c r i t e r i o n measures f o r the Syl l o g i s m Tests revealed (a) c l e a r developmental trends f o r the a b i l i t y to reason c o n s i s t e n t l y , independent of apparent i n t e r p r e t a t i o n s of premises, and (b) some d i f f e r e n c e s and some s i m i l a r i t i e s among the four V p r o p o s i t i o n s , almost a l l of which contravened norms of formal l o g i c . I n t r a s u b j e c t comparisons between Sy l l o g i s m Tests and R e f e r e n t i a l Tests (using the same l o g i c a l p r o p o s i t i o n f o r each student), demonstrated that for almost a l l students, s y l l o g i s t i c reasoning was governed by both the i n t e r p r e t a t i o n of the f i r s t premise and the a b i l i t y to generate that i n t e r p r e t a t i o n rather than a truncated v e r s i o n . These comparisons a l s o suggested that some apparently r e g r e s s i v e behaviour of Grade 12 students could be a t t r i b u t e d to developmental changes i n meanings of l o g i c a l p r o p o s i t i o n s which were not accompanied by corresponding changes i n reasoning s t r a t e g i e s . I m p l i c a t i o n s were discussed for n a t u r a l language, mathematics education, and research i n l o g i c a l reasoning (and hence c r i t i c a l t h i n k i n g ) , concept formation r u l e l e a r n i n g , and Formal Operations. P a r t i c u l a r a t t e n t i o n was given to formal l o g i c and Formal Operations as competence models. v i TABLE OF CONTENTS CHAPTER 1 Int r o d u c t i o n 1 Background 2 Syllogisms 3 T r a d i t i o n a l A n a l y s i s of Syllogisms 6 Formal Logic as a Norm 9 P h y s i c a l Referents 11 Purpose of the Study 13 The Sixteen Binary Operations 17 The Model 28 The L i n g u i s t i c Component 29 Types of L o g i c a l Expressions 29 Meanings of L o g i c a l Expressions 30 The Reasoning Component 41 The C o m p a t i b i l i t y Component 48 Statement of the Problem 52 The S y l l o g i s m Test 53 The R e f e r e n t i a l Test 54 C o m p a t i b i l i t y 56 A n a l y s i s of the Syl l o g i s m Test 56 The Norms 63 The S y l l o g i s m Test 63 The R e f e r e n t i a l Test 63 C o m p a t i b i l i t y • 63 CHAPTER 2 In t r o d u c t i o n 65 Inadequacies of Formal Logic 68 T h e o r e t i c a l Perspectives 72 C r i t i c a l Thinking 72 Formal Operational Thinking 74 The Operational Schemata * 74 The INRC Group 75 Concept Formation .. 79 " Assessment of Reasoning 84 Assessment using Syllogisms 84 Performance . 84 Ration a l e s for Performance 99 Assessment using the 4-card S e l e c t i o n Task 109 D e s c r i p t i o n 109 Performance 111 Rat i o n a l e s for Performance 120 I n t e r p r e t a t i o n s of Premises 124 Overview 133 v i i TABLE OF CONTENTS (cont.) CHAPTER 3 Design 137 Instruments and Controls 138 Design of the Tests 140 The Syllogism Test 140 The R e f e r e n t i a l Test 142 C r i t e r i a for the Dependent Measures 143 The Syllogism Test 143 The R e f e r e n t i a l Test 150 C o m p a t i b i l i t y 152 The A n a l y s i s 1 54 A n a l y t i c a l Procedures 154 The Tables 162 The Syllogism Test U s a b i l i t y Table 163 The Syllogism Test L o g i c a l Consistency Table ... 163 The Syllogism Test Binop Table 163 The R e f e r e n t i a l Test U s a b i l i t y Table 164 The R e f e r e n t i a l Class Table 164 The C o m p a t i b i l i t y Table 165 Im p l i c a t i o n s for Formal Operational Thought 166 Subjects and Controls 171 CHAPTER 4 V a l i d i t y and R e l i a b i l i t y of the Tests 174 The S y l l o g i s m Test 174 E x t e r n a l v a l i d i t y 174 I n t e r n a l V a l i d i t y 179 R e l i a b i l i t y 179 The R e f e r e n t i a l Test 186 V a l i d i t y 186 R e l i a b i l i t y 186 Results 189 The S y l l o g i s m Test Data 191 U s a b i l i t y 191 L o g i c a l Consistency 197 S y l l o g i s t i c Binops 202 The R e f e r e n t i a l Test Data 211 U s a b i l i t y 211 R e f e r e n t i a l Classes 215 Sat u r a t i o n 221 C o m p a t i b i l i t y 225 v i i i TABLE OF CONTENTS (cont.) CHAPTER 4 (cont.) Post Hoc Analyses 241 The S y l l o g i s m Test 242 P r e d i s p o s i t i o n 242 R e l i a b i l i t y of Subtests 248 The R e f e r e n t i a l Test 251 A t t r i b u t e Recognition 251 R e f e r e n t i a l Agreement . .. 252 C o m p a t i b i l i t y .. 255 C h a r a c t e r i s t i c s of the Subsample 255 Developmental Trends f o r the Subsample 258 Status of Elements 260 CHAPTER 5 Summary of Res u l t s 268 Im p l i c a t i o n s f or Natural Language 279 Im p l i c a t i o n s f o r Mathematics Education . ... 283 Problem S o l v i n g 284 Proof 294 Pedagogy 298 Im p l i c a t i o n s f or Research 304 The Sy l l o g i s m Test 304 The R e f e r e n t i a l Test 310 C o m p a t i b i l i t y .. 312 L i m i t a t i o n s of the Study 314 Recommendations for Future Research 319 REFERENCE NOTES 324 REFERENCES 325 APPENDIX A 341 APPENDIX B 3g0 GLOSSARY 3g8 i x LIST OF TABLES Table No. T i t l e page 1 'Correct' responses for four t y p i c a l s y l l o g i s m p r i n c i p l e s 5 2 Summary of two h y p o t h e t i c a l experiments 7 3 Venn diagram representation of the si x t e e n combinations of four binary elements 22 4 Piaget's s i x t e e n binary operations 25 5 O v e r a l l status of each binary element 49 6 Binops for f i r s t premises, and corresponding response sets 61 7 Comparison of binops with r u l e s i n r u l e l e a r n i n g . 81 8 Mastery of s y l l o g i s m p r i n c i p l e s 85 9 Summary of responses for the 4-card s e l e c t i o n task 111 10 Eva l u a t i o n s of binary elements f o r ' I f p then q' . 126 11 Comparison of 4-card s e l e c t i o n s with cards chosen as f a l s i f y i n g 130 12 Comparison of 4-card s e l e c t i o n s with e v a l u a t i o n s of cards 131 13 Matrix e n t r i e s for each set of answers on Page 5 of the Sy l l o g i s m Test 148 14 Procedure for d e r i v i n g r e f e r e n t i a l c l a s s e s 153 15 Examples of some nested models 158 16 Factors used i n contingency t a b l e s 162 17 Number of 'co r r e c t ' responses f o r each s y l l o g i s m . 177 18 Comparison of 'c o r r e c t ' responses with previous research 179 19 Frequency of f a i l u r e f or each S y l l o g i s m Test u s a b i l i t y c r i t e r i o n 180 20 Order by Content by S y l l o g i s t i c binop t a b l e 184 X LIST OF TABLES (cont.) Table No. T i t l e page 21 P r o p o s i t i o n by Order by Content by S y l l o g i s t i c binop table 185 22 Frequency of f a i l u r e for each R e f e r e n t i a l Test u s a b i l i t y c r i t e r i o n 187 23 Expected percentage frequencies of u s a b i l i t y f o r each f i t t e d model for S y l l o g i s m Tests 195 24 Grade by P r o p o s i t i o n by Content Order by L o g i c a l consistency t a b l e 197 25 Expected percentage frequencies of concordance for each f i t t e d model for usable Sy l l o g i s m Tests . 199 26 Grade by P r o p o s i t i o n by Content Order by S y l l o g i s t i c binop t a b l e 201 27 Grade by P r o p o s i t i o n by S y l l o g i s t i c binop t a b l e .. 202 28 C o n t r i b u t i o n of c e l l s to Pearson Chi-square for the P r o p o s i t i o n by S y l l o g i s t i c binop t a b l e C o n t r i b u t i o n of c e l l s to Pearson Chi-square for the Grade by S y l l o g i s t i c binop t a b l e 204 29 Observed and Expected values for the P r o p o s i t i o n by S y l l o g i s t i c binop t a b l e Observed and Expected values for the Grade by S y l l o g i s t i c binop t a b l e 205 30 P r o b a b i l i t i e s for comparisons of p r o p o s i t i o n s .... 206 31 Expected percentage frequencies of u s a b i l i t y for two exponential models for the R e f e r e n t i a l Test .. 214 32 Grade by P r o p o s i t i o n by R e f e r e n t i a l c l a s s t a b l e .. 216 33 Collapsed t a b l e s of Grade by R e f e r e n t i a l c l a s s for e ch p r o p o s i t i o n 218 34 P r o b a b i l i t i e s f o r e f f e c t s i n Table 33 219 35 P r o p o s i t i o n by R e f e r e n t i a l c l a s s by Sa t u r a t i o n t a b l e 222 x i LIST OF TABLES (cont.) Table No. T i t l e page 36 P r o p o s i t i o n by Sat u r a t i o n t a b l e 223 37 Grade by S a t u r a t i o n t a b l e 224 38 Grade by P r o p o s i t i o n marginal t a b l e fo r the ' c o m p a t i b i l i t y ' subsample 226 39 Grade by P r o p o s i t i o n by Case t a b l e for the ' c o m p a t i b i l i t y ' subsample 227 40 P r o p o s i t i o n by S y l l o g i s t i c binop t a b l e for the ' c o m p a t i b i l i t y ' subsample 227 41 P r o p o s i t i o n by R e f e r e n t i a l c l a s s t a b l e fo r the ' c o m p a t i b i l i t y ' subsample 228 42 Observed and expected values f o r the P r o p o s i t i o n by Case t a b l e for the ' c o m p a t i b i l i t y ' subsample .. 228 43 Grade by Case t a b l e f o r the ' c o m p a t i b i l i t y ' subsample 230 44 P r o p o s i t i o n by S y l l o g i s t i c binop by Case t a b l e fo r the ' c o m p a t i b i l i t y ' subsample 232 45 L o g i c a l consistency and c o m p a t i b i l i t y for a l l o v e r a l l usable respondents 240 46 Grade by P r o p o s i t i o n by Content by Order by P r e d i s p o s i t i o n t a b l e 242 47 Grade x P r e d i s p o s i t i o n i n t e r a c t i o n Content x P r e d i s p o s i t i o n i n t e r a c t i o n Order x P r e d i s p o s i t i o n i n t e r a c t i o n 244 48 Grade by Content by Order by predisposed Answer t a b l e 245 49 Content x Predisposed answers i n t e r a c t i o n 245 50 Observed and expected values for the Grade by Content by Order by predisposed Answer t a b l e C o n t r i b u t i o n of c e l l s to the f u l l - o r d e r i n t e r a c t i o n i n the t a b l e above 247 x i i LIST OF TABLES (cont.) Table No. T i t l e page 51 P a i r s of Syl l o g i s m subtest responses for each content order 249 52 Grade by P r o p o s i t i o n by Switching s y l l o g i s t i c binop t a b l e for the usable S y l l o g i s m Tests 250 53 Grade by A t t r i b u t e r e c o g n i t i o n table 252 54 Order by A t t r i b u t e r e c o g n i t i o n t a b l e 252 55 Grade by R e f e r e n t i a l agreement t a b l e 253 56 Expected percentage frequencies i n c o m p a t i b i l i t y subsample for t o t a l population 260 57 R e f e r e n t i a l Test summary of r e s u l t s Percentage frequency of occurrence of maximal binops of r e f e r e n t i a l c l a s s e s , f o r each p r o p o s i t i o n 280 58 S y l l o g i s m Test summary of r e s u l t s Percentage frequency of occurrence of s y l l o g i s t i c binops, for each p r o p o s i t i o n ......... 281 59 Major f a c t o r s which y i e l d responses to s y l l o g i s m s which are c o r r e c t according to formal l o g i c ; 309 x i i i LIST OF FIGURES Figure No. T i t l e page 1 The l a t t i c e s t r u c t u r e of the binary operations ... 39 2 Flowchart for t e s t a d m i n i s t r a t i o n and analyses for each respondent 53 3 Page 5 of the geometric S y l l o g i s m Test for P r o p o s i t i o n 1 146 4 Number of respondents i n S y l l o g i s m Test and R e f e r e n t i a l Test c a t e g o r i e s 190 5 Percentage u s a b i l i t y for the S y l l o g i s m Tests, for each grade 193 6 F i t t e d models for usable S y l l o g i s m Tests 196 7 F i t t e d exponential models f o r o r i g i n a l and adjusted frequencies f o r concordance i n the Syllogism Test 200 8 Cumulative percentage frequencies of s y l l o g i s t i c binops, f o r each grade 208 9 P r e f e r r e d and a l t e r n a t i v e exponential models for R e f e r e n t i a l Test u s a b i l i t y 212 10 Percentage occurrence of Case 1 c o m p a t i b i l i t y , for each p r o p o s i t i o n 231 11 Cumulative percentage occurrence of c o m p a t i b i l i t y cases, for each grade 235 12 R e f e r e n t i a l agreement f o r each grade and p r o p o s i t i o n 254 13 F i t t e d exponential model f o r developmemtal trend for membership i n the ' c o m p a t i b i l i t y ' subsample .. 259 14 Status of binary elements according to S y l l o g i s m Tests only 261 15 Status of binary elements according to R e f e r e n t i a l Tests only 263 16 O v e r a l l status of binary elements i n S y l l o g i s m Tests using r e s u l t s from both t e s t s .... 265 17 Supposed status of binary elements i n S y l l o g i s m Tests using norms of formal l o g i c 266 1 CHAPTER 1 In 1961, the Educational P o l i c i e s Commission espoused r a t i o n a l i t y as the c e n t r a l theme of American education. The Commission claimed that These ( r a t i o n a l ) powers involve the processes of r e c a l l i n g and imagining, c l a s s i f y i n g and g e n e r a l i z i n g , comparing and e v a l u a t i n g , a n a l y s i n g and s y n t h e s i z i n g , and deducing and i n f e r r i n g . These processes enable one to apply l o g i c and the a v a i l a b l e evidence to h i s ideas, a t t i t u d e s , and a c t i o n s , and to pursue b e t t e r whatever goals he may have. (p. 5) Commenting upon the nature of the implementation of programs to teach these r a t i o n a l powers, the Commission f u r t h e r suggested that Study of an ab s t r a c t subject l i k e mathematics or philosophy i n and of i t s e l f , does not n e c e s s a r i l y enhance r a t i o n a l powers, and i t i s p o s s i b l e that experiences i n areas which appear to have l i t t l e connection may i n fa c t make a s u b s t a n t i a l c o n t r i b u t i o n to r a t i o n a l development. As a case i n p o i n t , the a b i l i t i e s involved i n p e r c e i v i n g and rec o g n i z i n g patterns i n a mass of a b s t r a c t data are of considerable importance i n l e a r n i n g to analyse, deduce or i n f e r . These a b i l i t i e s may be developed i n the course of mathematical study; but they may be developed as w e l l through experiences i n a e s t h e t i c , humanistic^and p r a c t i c a l f i e l d s . Music, for example, challenges the l i s t e n e r to perceive elements of form w i t h i n the a b s t r a c t . S i m i l a r l y , v o c a t i o n a l subjects may engage the r a t i o n a l powers of p u p i l s . (p. 17) S i m i l a r sentiments have been echoed repeatedly over the years. In 1975 for example, the Conference Board of the Mathematical Sciences N a t i o n a l Advisory Committee on Mathematical Education (NACOME) recognized that 2 l o g i c a l reasoning i s one of the fundamental c h a r a c t e r i s t i c s of mathematical thought and a c r u c i a l c o n t r i b u t i o n to problem s o l v i n g . (p. 18) The Board f u r t h e r recommended that c u r r i c u l u m development was needed to produce Techniques and m a t e r i a l s which w i l l support and e f f e c t i v e l y develop a b i l i t i e s i n problem s o l v i n g , i n l o g i c a l t h i n k i n g , and c r i t i c a l t h i n k i n g . (p. 146) A review of l i t e r a t u r e reveals that i t i s apparently beyond a c t i v e dispute that the study of l o g i c a l t h i n k i n g , l i k e the study of mathematics or of reading i s a worthwhile p u r s u i t . The study of the psychology of l o g i c a l t h i n k i n g appears to have been hampered somewhat by too c l o s e an a s s o c i a t i o n with i t s a b s t r a c t e d o f f s p r i n g formal l o g i c , p a r t i c u l a r l y since Boole's (1854) comprehensive and systematic t r e a t i s e . However, r a t i o n a l i t y i s so much a part of most human communication systems, that i t i s v i r t u a l l y s e l f -c o n t r a d i c t o r y to deny i t s r o l e i n l i n g u i s t i c cohesion: "Prove to me", E p i c t e t u s was challenged, "that I should study l o g i c . " "How w i l l you know that i t i s a good proof?" (Kaplan, 1964) Background If r a t i o n a l i t y i s accepted as an important educational e n t e r p r i s e , i t s r e a l i z a t i o n requires an understanding of what i t i s , and how i t i s assessed. This s e c t i o n presents a b r i e f d e s c r i p t i o n of current research p r a c t i s e s . The d i s c u s s i o n includes a d e s c r i p t i o n of s y l l o g i s m s , current a n a l y t i c a l methods, c r i t e r i a for the assessment of c o r r e c t n e s s , and a b r i e f d e s c r i p t i o n of a r e l a t i v e l y new perspective i n which 3 attempts have been made to l i n k l o g i c a l t h i n k i n g with l i n g u i s t i c f a c t o r s . I t i s argued that current research p r a c t i c e s are both i n a p p r o p r i a t e and inadequate to describe each i n d i v i d u a l ' s reasoning p a t t e r n s . A new model i s presented i n subsequent s e c t i o n s . Syllogisms L o g i c a l t h i n k i n g has t r a d i t i o n a l l y most often been studi e d by means of t e s t s of s y l l o g i s t i c reasoning. Other means have been employed such as l o g i c a l puzzles and reasoning from t e x t , but they are not discussed here. A s y l l o g i s m i s a three-part argument. Each of the f i r s t two parts i s a statement which i s assumed to be tr u e . These are c a l l e d the f i r s t and second premises. The t h i r d part i s a conclusion which may or may not f o l l o w from only the two given premises. I f a respondent i s given a l l three parts of a s y l l o g i s m and asked to evaluate whether the conclusion n e c e s s a r i l y f o llows from only the two given premises, t h i s i s c a l l e d an eval u a t i o n item. I f a respondent i s given only the f i r s t two parts of a s y l l o g i s m and asked to supply a concl u s i o n which n e c e s s a r i l y f o l l o w s from only the two given premises, or to i n d i c a t e i f there i s no such c o n c l u s i o n , i t i s c a l l e d a completion item. In the present study, a t e s t which measures l o g i c a l t h i n k i n g by means of s y l l o g i s m s , i s c a l l e d a Sy l l o g i s m Test. Most S y l l o g i s m Tests contain only evaluation items. The c l a s s i c s y l l o g i s m which i s u s u a l l y o f f e r e d as an exemplar of a s y l l o g i s m i s as f o l l o w s : 4 A l l men are mortal [ F i r s t Premise] Socrates i s a man [Second Premise] Therefore Socrates i s mortal [Conclusion] A more t y p i c a l e v a l u a t i o n s y l l o g i s m t e s t item i s If i t i s black then i t i s a square [ F i r s t Premise] I t i s a square [Second Premise] Is i t black? The second premise i s t y p i c a l l y of one of four types. In the above example, the four types would be ' I t i s black' ' I t i s a square' ' I t i s not black', or ' I t i s not a square'. The conclusion to be evaluated u s u a l l y r e f e r s to whichever q u a l i t y i s mentioned i n the f i r s t premise, but i s not mentioned i n the second premise. In the above t e s t item, the respondent would be expected to answer e i t h e r 'yes' or 'no', or to i n d i c a t e i f the two given premises contain i n s u f f i c i e n t information to be able to answer one way or the other. For each of the four second premises above, the corresponding s y l l o g i s m i s t y p i c a l l y c a l l e d Modus Ponens, Converse, Inverse, and Modus T o l l e n s , r e s p e c t i v e l y . To t h i s author's knowledge, almost a l l analyses of syl l o g i s m s have been predicated on the b e l i e f that only one response i s c o r r e c t for each s y l l o g i s m . Each response i s d i c t a t e d by commonly accepted r u l e s of formal l o g i c . For the four t y p i c a l s y l l o g i s m s above, the 'corre c t ' responses are shown i n Table 1. The responses are determined according to the form of the argument, rather than by i t s content. Since 5 Table 1 'Correct' responses for four t y p i c a l s y l l o g i s m p r i n c i p l e s I f p then q [ F i r s t Premise] p [Second Premise] If p then q [ F i r s t Premise] q [Second Premise] q [Conclusion] Modus Ponens No conclusion p o s s i b l e Converse I f p then q [ F i r s t Premise] Not p [Second Premise] I f p then q [ F i r s t Premise] Not q [Second Premise] No conclusion p o s s i b l e Inverse Not p [Conclusion] Modus T o l l e n s the converse and inverse have no con c l u s i o n s , they are sometimes c a l l e d 'indeterminate' (Simpson & Johnson, 1966). In the above example, i f the phrase ' I t i s black' were to be replaced by another phrase (which i s symbolized by the l e t t e r p) and i f the phrase ' I t i s a square' were to be replaced by a d i f f e r e n t phrase (which i s symbolized by the l e t t e r q ) , then the form of the statement ' I f i t i s black then i t i s a square' i s ex e m p l i f i e d by the statement ' I f p then q'. Six examples of statements which have the same content but d i f f e r e n t forms are as f o l l o w s . ' I t i s black i f i t i s a square' (p i f q) ' I t i s black only i f i t i s a square' (p only i f q) ' I t i s black or square' (p or q) ' I t i s not both black and square' (Not both p and q) 6 ' I t i s black or square but not both' (p or q but not both) ' I t i s black or square or both' (p or q or both) There are many v a r i a t i o n s of these examples, concomitant with v a r i a t i o n s i n content of p and q. Other v a r i a t i o n s of s y l l o g i s m s include v a r i a t i o n s with negation, order of premises, response options, other v a r i a t i o n s of the form of the f i r s t premise (such as 'p i f and only i f q', 'Never p without q'), v a r i a t i o n s of types of both premises (such as ' A l l X are Y', 'Some X are not Y'), and others. T r a d i t i o n a l A n a l y s i s of Syllogisms For a number of d i f f e r e n t f i r s t premises, there are formal l o g i c a l r u l e s which define the c o r r e c t conclusions of s y l l o g i s m s which contain these f i r s t premises. They are not repeated here. For the s i x cases i n the previous s e c t i o n , not a l l of these r u l e s are d i f f e r e n t . For ease of reference in-the present study, the four s y l l o g i s m s i n Table 1 are i d e n t i f i e d only by t h e i r second premises. A c c o r d i n g l y , they are symbolized as <p>, <q>, <p> and <q> r e s p e c t i v e l y . The t i l d e above each l e t t e r i n d i c a t e s negation. The form of the f i r s t , premise i s l e f t open to c o n t e x t u a l f a c t o r s i n t h i s study. Hence, for example, the two s y l l o g i s m s for which t h e i r f i r s t premises are ' I f p then q' and 'p or q' r e s p e c t i v e l y , are both c a l l e d <p> syl l o g i s m s i f both of t h e i r second premises are the a s s e r t i o n of p. An important c h a r a c t e r i s t i c of research i n s y l l o g i s t i c reasoning i s the methods which have t r a d i t i o n a l l y been employed to analyse the r e s u l t s of t h i s research. The f o l l o w i n g example i l l u s t r a t e s some inadequacies of these 7 methods. Table 2 dep i c t s the r e s u l t s of two h y p o t h e t i c a l experiments i n which two d i f f e r e n t sets of f i v e respondents were asked to evaluate the four syllogisms immediately above. Table 2 Summary of two h y p o t h e t i c a l experiments Experiment 1 Experiment 2 Syllogisms Respondent <p> <q> < P > <g> <p> <q> < P > <g> 1 q ? ? P q P 7 P 2 q P g p P g P 3 q P g P q 7 g P 4 7 P g 7 q P g 7 5 q g P q ? g P Formal Logic q 7 ? P q ? 7 P Percentage c o r r e c t 80 40 20 80 80 40 20 80 As an example, respondent number 2 i n experiment 1 was given a s y l l o g i s m of the form I f i t i s black then i t i s a square I t i s not black Is i t a square? This corresponds to the <p> s y l l o g i s m . A formal l o g i c a l response i s that there i s not enough information given i n the premises to determine whether or not i t i s a square. This response i s symbolized by a '?' i n the t h i r d column of the f i r s t t a b l e i n Table 2. Respondent number 2 answered ' I t i s not a square'. This response i s represented by g i n the same column. T y p i c a l l y , formal l o g i c i s used to determine the correctness of responses. A c c o r d i n g l y , 80% of respondents i n 8 experiment 1 c o r r e c t l y evaluated the <p> s y l l o g i s m . The frequencies of c o r r e c t responses to sy l l o g i s m s i n each experiment are given i n Table 2. I t i s apparent that the frequencies of c o r r e c t responses in the second experiment are i d e n t i c a l to the corresponding frequencies i n the f i r s t experiment. However, there are vast d i f f e r e n c e s i n patterns of responses of the p a r t i c i p a n t s in each experiment. For example, i n the second experiment no respondents gave the same set of four answers as d i d the f i r s t respondent i n the f i r s t experiment. The second, t h i r d and f o u r t h respondents of the f i r s t experiment a l s o have no i d e n t i c a l counterparts i n the second experiment. The f i f t h respondent i n the f i r s t experiment has two i d e n t i c a l counterparts i n the second experiment. The two counterparts are respondents 3 and 5. With very few exceptions, present s t y l e s of a n a l y s i s of these t e s t s would be unable to detect the d i f f e r e n c e s which have been described above. U n t i l very r e c e n t l y , only column frequencies have been c a l c u l a t e d , so that i t was not p o s s i b l e to even c h a r a c t e r i z e the d i f f e r e n c e s across rows i n Table 2, not to mention to i n t e r p r e t these d i f f e r e n c e s . T a p l i n (1971), T a p l i n and Staudenmayer (1973), T a p l i n , Staudenmayer and Taddonio (1974), Staudenmayer (1975) and others have made s u b s t a n t i a l progress i n t h i s respect, but some d i f f i c u l t i e s s t i l l remain. This problem i s one of a number which are addressed i n the present study. 9 Formal Logic as a Norm As with the a d m i n i s t r a t i o n of many t e s t s , i t seems reasonable to expect that the purpose of a d m i n i s t r a t i o n of any t e s t s of l o g i c a l t h i n k i n g which use norms of formal l o g i c as c r i t e r i o n measures, would be to evaluate i n s t r u c t i o n i n formal l o g i c . However, t h i s i s very r a r e l y the case f o r research i n l o g i c a l t h i n k i n g , since respondents have been d e l i b e r a t e l y chosen f o r t h e i r formal l o g i c n a i v e t e . The only a l t e r n a t i v e purpose appears to be that formal l o g i c i s used as a measure of n a t u r a l reasoning s k i l l s . That i s to say, n a t u r a l reasoning as developed through each respondent's exposure to i n t e r p e r s o n a l communication ( p a r t i c u l a r l y language), and without e x p l i c i t i n s t r u c t i o n i n reasoning. This type of reasoning i s c a l l e d n a t u r a l l o g i c ( L akoff, 1970). This e v a l u a t i o n approach would appear to be reasonable i f formal l o g i c and n a t u r a l l o g i c were much the same. There are a number of people however, who have s t r o n g l y advocated that t h i s i s not the case. As an example, Lewis (1912), Strawson (1952), Quine (1972), Young (1972), and Ennis (1976) have presented arguments to support the p o s i t i o n that i n n a t u r a l l o g i c , statements of the form ' I f p then q' are not o r d i n a r i l y claimed to be true i f p i s f a l s e . This i s at variance w i t h one of the norms of formal l o g i c i n which the (supposedly equivalent) expression 'p=q' i s s a i d to be v a l i d i f p i s f a l s e . There are other s i m i l a r examples by these and other authors which a t t e s t to the e x i s t e n c e of many d i f f e r e n c e s between n a t u r a l l o g i c and formal l o g i c . Some of these examples are presented i n Chapter 2. D i f f e r e n c e s such 10 as the one which i s described above, have prompted Chomsky (1955) to remark that The well-known discrepancy between the m a t e r i a l c o n d i t i o n a l of l o g i c and the ' i f - t h e n ' of E n g l i s h ... should be enough to warn anyone not to make a b l i n d leap from mathematical systems to ordin a r y l i n g u i s t i c behavior. In l i g h t of the d i f f e r e n c e s between formal l o g i c and n a t u r a l l o g i c , i t does not seem appropriate to t h i s author, to continue to use formal l o g i c as a normative system for n a t u r a l l o g i c . I t seems to be as inappropriate as using a t e s t of Fortr a n programming to measure essay w r i t i n g s k i l l s . In summary, two major f a c t o r s which l i e at the b a s i s of research i n l o g i c a l t h i n k i n g have been examined. These are, the t r a d i t i o n a l a n a l y s i s of s y l l o g i s m s , and the use of formal l o g i c as a normative system. Arguments have been presented to demonstrate inadequacies of each f a c t o r . There seems to be l i t t l e point i n weakening the foundations of present research i n l o g i c a l t h i n k i n g , without p r o v i d i n g a v i a b l e a l t e r n a t i v e . The present study presents an a l t e r n a t i v e normative system other than formal l o g i c . Subsequently, i t i s shown how t h i s system was used to analyse research r e s u l t s i n such a way that the weaknesses i n a n a l y s i s which have been described above, were avoided. Some measure of freedom from the c o n s t r a i n t s of formal l o g i c has been achieved by st u d i e s which have examined d i f f e r e n t meanings of l o g i c a l terms. This has been achieved by a s s o c i a t i n g sentences i n which l o g i c a l terms have been embedded, with p h y s i c a l r e f e r e n t s . These s t u d i e s are discussed below. 11 P h y s i c a l Referents Research has shown that respondents f r e q u e n t l y do not respond to s y l l o g i s m s i n ways that are expected according to accepted norms of formal l o g i c . Some s p e c i f i c patterns of e r r o r s have been reported r e g u l a r l y i n the research l i t e r a t u r e . This suggests that e i t h e r formal l o g i c i s an inadequate norm, or that i t i s an adequate norm but that there are other reasons why these trends e x i s t in the experimental data. U n t i l 1960, a l l experimental data appear to have been analysed under the assumption that the l a t t e r case was true. F a i l u r e to f i n d adequate reasons to e x p l a i n the evidence l e d Henle (1960, 1962) to suggest that the f a u l t may not l i e with respondents' reasoning, but rather with i d i o s y n c r a t i c i n t e r p r e t a t i o n s of premises (among other f a c t o r s ) . This was s t r o n g l y supported by Gardiner's (1965) data even though he d i d not measure these i n t e r p r e t a t i o n s . This l i n e of thought has given momentum to attempts by some researchers to understand why these e r r o r s occur, by using means other than S y l l o g i s m Tests, to determine how the f i r s t premises i n s y l l o g i s m s are i n t e r p r e t e d by respondents. Two prototypes of research appear to be u s e f u l i n determining meanings of l o g i c a l sentences. In the f i r s t prototype, the experimenter attempts to f i n d how various l o g i c a l connectives, q u a l i f i e r s and q u a n t i f i e r s (such as 'or', 'and', 'not', 'some' and ' a l l ' ) are i n t e r p r e t e d . The respondent i s u s u a l l y confronted with a c o l l e c t i o n of objects of various s i z e s , shapes, c o l o u r s and other a t t r i b u t e s , and the experimenter requests that the respondent f o l l o w a command 12 such as 'give me a l l the things which are black or t r i a n g u l a r ' . The way i n which the l o g i c a l expression 'black or t r i a n g u l a r ' i s i n t e r p r e t e d by each respondent i s i n f e r r e d according to the nature of the s e l e c t e d set of o b j e c t s . Examples of these studies are those by N i t t a and Nagano (1966), Neimark (1970), Neimark and S l o t n i c k (1970), Suppes and Feldman (1971)., P a r i s (1973), Johansson and S j o l i n (1975), Roberge (1975), Neimark and Chapman (1975), Johansson (1977) and Juraschek (1978). The second prototype i s u s e f u l for determining meanings of l o g i c a l expressions which are not amenable to the f i r s t prototype (such as, ' I f i t i s black then i t i s a square'). The most common form i s c a l l e d the E v a l u a t i o n task (Johnson-L a i r d & Tagart, 1969; L e g r e n z i , 1970; Bree & Coppens,•1976, and o t h e r s ) . The respondent i s u s u a l l y confronted with cards which show p i c t u r e s of various combinations of the t r u t h and f a l s i t y of two p r o p o s i t i o n s . The respondent i s given a ( c o n d i t i o n a l ) r u l e and i s required to evaluate each card to determine whether i t i s considered to make the r u l e true or f a l s e , or whether the card i s i r r e l e v a n t to the status of the r u l e . With each of the above prototypes, an e s s e n t i a l part of the experiment i s the use of (representations of) p h y s i c a l r e f e r e n t s to determine various meanings of given l o g i c a l expressions. Accordingly, t h i s type of t e s t i s c a l l e d a R e f e r e n t i a l Test. The nature of the present study required the a d m i n i s t r a t i o n of a R e f e r e n t i a l Test, but precluded the use of the f i r s t prototype above. Hence, a type of E v a l u a t i o n 13 task was used. However, i t i s argued below that the Evaluat i o n task, as i t has been used i n the past, places undue r e s t r i c t i o n s on the range of p o s s i b l e meanings of l o g i c a l p r o p o s i t i o n s . Consequently, the R e f e r e n t i a l Test which was used i n the present study was designed so that the f u l l range of p o s s i b l e meanings of these p r o p o s i t i o n s was presented to each respondent. This t e s t was administered to measure each respondent's meaning of e i t h e r a c o n d i t i o n a l or a b i c o n d i t i o n a l p r o p o s i t i o n . These meanings were then incorporated i n t o a set of norms f o r assessing d i f f e r e n t q u a l i t i e s of reasoning in S y l l o g i s m Tests. The present author has been able to l o c a t e only two st u d i e s in which both Syllogism Tests and R e f e r e n t i a l Tests were administered to the same population sample. These are, Kuhn (1977) and Marcus and Rips (1979). In n e i t h e r of these st u d i e s were i n t r a s u b j e c t comparisons made between r e s u l t s of the two t e s t s . Hence, there are no known precedents which may act as a guide f o r these comparisons. The methods adopted i n the present study enable one to examine degrees to which each respondent's responses for S y l l o g i s m Tests and R e f e r e n t i a l Tests are compatible. Purpose of the Study In view of Henle's (1960, 1962) comments, and current research p r a c t i c e s as described i n Chapters 1 and 2, i t seems n a t u r a l to determine the extent to which reasoning patterns i n S y l l o g i s m Tests depend upon i d i o s y n c r a t i c i n t e r p r e t a t i o n s of premises. A c c o r d i n g l y , the purpose of t h i s study was to 14 examine the degrees of c o m p a t i b i l i t y between the apparent meanings of s p e c i f i c l o g i c a l expressions as used i n s y l l o g i s t i c reasoning, and the a c t u a l meanings of the same l o g i c a l expressions when a p p l i e d to p h y s i c a l r e f e r e n t s . S p e c i f i c a l l y , t h i s involved the f o l l o w i n g three tasks. The f i r s t task was to i n t e r p r e t the r e s u l t s of s y l l o g i s t i c reasoning t e s t s . A Syllogism Test was constructed for t h i s study i n which each respondent was presented with four s y l l o g i s m s , each of which had the same f i r s t premise. The form of four of these s y l l o g i s m s i s shown i n Table 1. Each respondent was assigned e x a c t l y one of four forms of l o g i c a l expressions for f i r s t premises. Two d i f f e r e n t contents were used for a l l s y l l o g i s m s . One content involved squares and c i r c l e s which were e i t h e r black or white. The other content involved numbers which were e i t h e r large or s m a l l , and which ended with e i t h e r a 1 or a 0. Before each s y l l o g i s m , each respondent was asked to i n d i c a t e whether the f i r s t and second premises were thought to be compatible. I f the premises of a s y l l o g i s m were thought to be incompatible, the respondent was not presented with that s y l l o g i s m . Otherwise, each respondent was asked two questions, each of which evaluated a conclusion for each s y l l o g i s m . Some sample S y l l o g i s m Tests are to be found i n Appendix A (p. 341). They are described more f u l l y i n Chapter 3. I t has been argued that there are s i g n i f i c a n t d e f i c i e n c i e s i n the power of present analyses of t e s t s of s y l l o g i s t i c reasoning. Hence, i t was decided that a d e s c r i p t i v e framework was needed to make i t p o s s i b l e to 15 express f u l l y each respondent's l o g i c a l t h i n k i n g p a t t e r n . The framework which was developed was of such a nature that Inhelder and Piaget's (1958) binary operations seemed to play a v i t a l r o l e . A d e s c r i p t i o n of the i n t e r p r e t a t i o n of the binary operations which was used i n t h i s study, i s deferred u n t i l the next s e c t i o n of t h i s chapter. This i n t e r p r e t a t i o n of the binary operations i s followed by a method for i n t e r p r e t i n g the r e s u l t s of s y l l o g i s t i c reasoning t e s t s . The i n t r a s u b j e c t r e s u l t s so i n t e r p r e t e d , allowed a complete c h a r a c t e r i z a t i o n to be made of each subject's t o t a l p a t t e r n of responses. The second task was to decide how to t e s t the meanings of l o g i c a l expressions as these are a p p l i e d to p h y s i c a l r e f e r e n t s . A R e f e r e n t i a l Test was constructed f o r t h i s purpose. In the present study, t h i s t e s t e n t a i l e d presenting each respondent with 32 d i f f e r e n t sets of objects ( i n c l u d i n g both c o n t e n t s ) , such as black squares, black c i r c l e s and white c i r c l e s . A l l d i f f e r e n t combinations were presented to each respondent. For each s e t , the respondent was asked to imagine one ( u n s p e c i f i e d ) object removed from the se t . Without any fu r t h e r information being given, the respondent was then asked to i n d i c a t e whether the removed object could be s a i d to s a t i s f y the c o n d i t i o n s of a given l o g i c a l expression (such as 'If i t i s black then i t i s a square'). An a f f i r m a t i v e response was used to i n f e r that t h i s respondent considered that the set of objects was of such a nature as to be s u f f i c i e n t to as s e r t the v a l i d i t y of the l o g i c a l expression for any member of the whole set. The various s t r u c t u r e s of 1 6 these sets of o b j e c t s , and the corresponding l o g i c a l expressions are described i n the next s e c t i o n of t h i s chapter. Some sample R e f e r e n t i a l Tests are to be found i n Appendix A. The t h i r d task was to construct ways to compare the r e s u l t s of Syl l o g i s m Tests with the r e s u l t s of R e f e r e n t i a l Tests. By comparing each i n d i v i d u a l ' s r e s u l t s of the two t e s t s , i t i s hoped that f u r t h e r i n s i g h t may be gained i n t o the nature of responses f o r Sy l l o g i s m Tests which have been obtained i n the past. The c o m p a t i b i l i t y c o n s t r u c t s which were used i n the present study are described i n the model which i s presented i n t h i s chapter. I t i s hoped that the co n s t r u c t s which were developed from the three tasks which have been described above, may be used as a new set of norms f o r the assessment of n a t u r a l l o g i c , i n which l i n g u i s t i c anomalies may be sanctioned, and which allows a more complete a n a l y s i s of reasoning patterns to be made. To t h i s end, a model which may act as a d e s c r i p t i v e framework for the a n a l y s i s of l o g i c a l t h i n k i n g i s proposed below. This model has three f a c e t s . F i r s t , i t contains new perspe c t i v e s and methods f o r determining meanings of l o g i c a l expressions. Second, i t suggests procedures which enable a more complete a n a l y s i s of l o g i c a l reasoning to be made. T h i r d , i t incorporates a r a t i o n a l e for making comparisons between meanings of l o g i c a l expressions, and the reasoning s t r a t e g i e s which are used i n s y l l o g i s m t e s t s i n which these l o g i c a l expressions are embedded. The model i s of such a nature that the c l a r i t y of i t s pr e s e n t a t i o n depends upon the c l a r i t y of Inhelder and Piaget's 17 term 'binary operation'. Hence, a d e s c r i p t i o n of the model i s preceded by an e x p l i c i t assignment of a meaning to t h i s term. The Sixteen Binary Operations Both of the p u b l i c a t i o n s The Growth of L o g i c a l Thinking  from Childhood to Adolescence (Inhelder & P i a g e t , 1958) and T r a i t e de Logique (Piaget, 1949) are c e n t r a l to the d i s c u s s i o n of the binary operations. For the sake of b r e v i t y , the former i s r e f e r r e d to as GLT and the l a t t e r as TL. An examination of r e l e v a n t l i t e r a t u r e which has discussed Inhelder and Piaget's work, reveals an apparently prolonged lack of consensus among dis c u s s a n t s on the nature of the s i x t e e n binary operations. This and other f a c t o r s have l e d to a number of systematic a t t a c k s upon the e n t i r e c o n c e p t u a l i z a t i o n of formal operations (Parsons, 1960; Ennis, 1975; Strauss & Kroy, 1977; B r a i n e r d , Note 1). Ennis (1975) f o r example, demonstrated that what he considered to be the most p l a u s i b l e i n t e r p r e t a t i o n s which could reasonably be made and s t r o n g l y supported, nevertheless s u f f e r e d from a number of undeniable i n c o n s i s t e n c i e s , apparent c o n t r a d i c t i o n s and unreasonable consequences. This does not n e c e s s a r i l y r e f u t e Inhelder and Piaget's work as Ennis has claimed, but may instead suggest d e f i c i e n c i e s i n Ennis' well-argued i n t e r p r e t a t i o n of t h e i r work, thereby h i g h l i g h t i n g the great d i f f i c u l t i e s i n i n t e r p r e t a t i o n . A d e t a i l e d a n a l y s i s of a number of p o s s i b l e i n t e r p r e t a t i o n s of meanings of the binary operations has y i e l d e d the f o l l o w i n g i n t e r p r e t a t i o n , which i s used i n the present study. 18 The f i r s t concept to be discussed i s the concept of a p r o p o s i t i o n a l f u n c t i o n . A p r o p o s i t i o n a l f u n c t i o n i s c l a s s i f i e d as a sentence which has no inherent t r u t h value, but which may become e i t h e r a true or a f a l s e statement by making reference to e i t h e r the a t t r i b u t e s of an object or to the occurrence of an event. As an example of a p r o p o s i t i o n a l f u n c t i o n which r e f e r s to a t t r i b u t e s , the statement 'x i s black' i s a p r o p o s i t i o n a l f u n c t i o n which may become e i t h e r true or f a l s e , depending upon the colour of the object to which 'x' r e f e r s . As an example of a p r o p o s i t i o n a l f u n c t i o n which r e f e r s to an event, the statement 'the s t r i n g i s lengthened' i s a p r o p o s i t i o n a l f u n c t i o n which may become e i t h e r true or f a l s e , depending upon the event to which the statement r e f e r s . A consequence of t h i s d e f i n i t i o n i s that a p r o p o s i t i o n a l f u n c t i o n may become e i t h e r true or f a l s e , or i t may be i r r e l e v a n t to the object or event i n qu e s t i o n . Two examples for which a p r o p o s i t i o n a l f u n c t i o n i s i r r e l e v a n t , are the reference of the p r o p o s i t i o n a l f u n c t i o n 'the s t r i n g i s lengthened', to a rock being dropped, and the reference of the p r o p o s i t i o n a l f u n c t i o n 'the number x i s odd*, to a pen. The wording of the p r o p o s i t i o n a l f u n c t i o n i s important since i t i s impossible for the p r o p o s i t i o n a l f u n c t i o n 'x i s an odd number' to be i r r e l e v a n t , for any replacement of 'x'. This i s not true of the p r o p o s i t i o n a l f u n c t i o n 'the number x i s odd'. Next, c o n s i d e r a t i o n i s given to circumstances under which p r o p o s i t i o n a l functions may r e f e r to t h e i r r e f e r e n t s . Consider any set of coloured shapes, and suppose that any one of these shapes may act as an operand for both of the 19 p r o p o s i t i o n a l f u n c t i o n s p = 'x i s black', and q = 'x i s a square'. This means that i f the v a r i a b l e 'x' i s replaced by a reference to a black c i r c l e then p would become a true statement and q would become a f a l s e statement. The negations of p and q are denoted by p and 2[ r e s p e c t i v e l y . Hence, p = 'x i s not black', and g = 'x i s not a square'. The simultaneous c o n f i r m a t i o n of p and q i s denoted by p.q. This i s not the same meaning given to 'p.q' i n formal l o g i c , i n which s i m u l t a n e i t y i s not required. Hence, •p.q = 'x i s black and square'. This i s not the same as 'x i s black and y i s a square'. The set of coloured shapes may be subdivided i n t o four d i s j o i n t subclasses i n the f o l l o w i n g way. The subset of a l l black squares i s the subclass of shapes which simultaneously s a t i s f y both p and q. That i s , each member s a t i s f i e s p.q. The subset of a l l black non-squares i s the subclass of shapes which simultaneously s a t i s f y both p and q\ That i s , each member s a t i s f i e s p.q". The subclasses of obj e c t s which s a t i s f y p.q and p.q" are defined s i m i l a r l y . Each of the expressions p.q, p.q", p.q and p.cj i s c a l l e d a binary element. This i s i n accordance with terminology which i s used i n GLT. Any set of coloured shapes may be subdivided i n t h i s way, always with the p o s s i b i l i t y that some subclasses may be empty. Consider now, the more general case where p and q are 20 any two p r o p o s i t i o n a l f u n c t i o n s , and that each term which i s described above, assumes the more general sense. A set of objects or events to which a p a r t i c u l a r (or s t i p u l a t e d ) p and q r e f e r i s c a l l e d a r e f e r e n t i a l s e t . Examples of such r e f e r e n t i a l sets are 1. The set of rods i n Inhelder and Piaget's ' f l e x i b l e rod' experiment, with p =. .' i t i s t h i n ' , and q = ' i t i s f l e x i b l e ' . 2. The set of known animals (often used as examples i n TL) with p = ' i t i s a mammal' , and q = ' i t i s v e r t e b r a t e ' . 3. The events which occur during Inhelder and Piaget's (1958) 'pendulum' experiment, with p = 'the length of the s t r i n g changes', and q = 'the period of the pendulum changes'. In examples 1 and 2, the r e f e r e n t i a l sets are o b j e c t s , the subclasses of which e i t h e r e x i s t or do not. In example 3, the r e f e r e n t i a l set i s a set of events, the subclasses of which e i t h e r occur or do not. I f a r e f e r e n t i a l set contains at l e a s t one object or event f o r which both p and q are simultaneously t r u e , then the binary element p.q i s s a i d to be v a l i d a t e d by that object or event. The other three binary elements are s a i d to be v a l i d a t e d i n a s i m i l a r way. Given two p r o p o s i t i o n a l functions p and q, and a corresponding r e f e r e n t i a l s e t , each of the four binary elements p.q, p.q", p.q and p.q" e i t h e r may or may not be 21 v a l i d a t e d by members of that set. Since there are these two p o s s i b i l i t i e s for the v a l i d a t i o n of each of the four binary elements, an a p p l i c a t i o n of some simple combinatorics reveals that there are 2" = 16 p o s s i b l e combinations of v a l i d a t i o n s of these four binary elements. Each of these s i x t e e n combinations i s symbolized i n Table 3. In each diagram, an unshaded area i n d i c a t e s that there are no objects or events which v a l i d a t e the unshaded binary element. Each shaded area i n d i c a t e s that the corresponding binary element i s v a l i d a t e d by the r e s p e c t i v e r e f e r e n t i a l s e t . Each of the s i x t e e n expressions which are shown at the bottom of each box i n Table 3 i n d i c a t e s which binary elements are v a l i d a t e d by members of the set, and which binary elements are not v a l i d a t e d . Elements which are not v a l i d a t e d are enclosed i n square brackets, as with '[p.S[]' i n diagram 11. Elements which are v a l i d a t e d are not enclosed i n brackets. I t could be argued that the meaning which has been a t t r i b u t e d to these s i x t e e n expressions i n t h i s study i s i n f a c t what i s meant by Inhelder and Piaget's term 'binary operation'. This i s by no means c e r t a i n . However, i n order to r e f l e c t the analogy ( i f not the i d e n t i t y ) between these meanings, each expression i s c a l l e d the binop of that s e t , with respect to the two given p r o p o s i t i o n a l f u n c t i o n s . Hence, a binop c o n s i s t s of four binary elements, symbolized i n the manner shown i n Table 3. I f i t i s considered that the term 'binop' as i t i s defined i n t h i s study, i s i d e n t i c a l to what Inhelder and Piaget (1958) have c a l l e d 'binary o p e r a t i o n ' , a comparison of Venn diagram T a b l e 3 r e p r e s e n t a t i o n of the s i x t e e n combinations of f o u r b i n a r y elements P q o. [ p . q W t p . q M P - q M p . q ] 4. [p .q]v p . q v [p .q ]v [p .q ] 8. p.q v [ p . q ] v [ p . q ] v [ p . q ] 1 . t p . q ] v [ p . q ] v t P - q l v P .q 5 . [ p . q l v p.q v[p.q] v p.q 9. p.q v [ p . q ] v [ p . q ] v p.q p-q 2. [ p . q ] v [ p . q ] v P q v[p.q] 6. [ p . q j v p.q v p.q v[p.q] 3. [ p . q ] v [ p . q ] v p.q v p.q 7. [ p . q j v p.q v p.q v p.q 10. p.q v [ p . q ] v p.q v[p.q] 11. p.q v [ p . q ] v p.q v p.q p q 1 2 . p.q v p.q v [ p . q ] v [ p . q ] 13. p.q v p.q v[p.q] v p.q p.q 14. p.q v p.q v p.q v t p q ] 15. p.q v p.q v p.q v p.q 23 symbolisms i s i n order. Inhelder and Piaget have used symbolism which i s i d e n t i c a l to the symbolism which i s used i n the present study, with the exception that bracketed terms are missing i n t h e i r work. Thus, the binop p.q v [ p . g ] v [ p . q ] v p . g i n the present study i s expressed as p.q v p.g by Inhelder and P i a g e t . Two important p o i n t s should be made i n reference to the symbolization which i s used for the binops. 1. The symbol 'v' does not mean 'or' i n the usual l o g i c a l sense, but i s more a p p r o p r i a t e l y i n t e r p r e t e d as 'and as w e l l ' . 2. The symbol '.' does not mean 'and' i n the usual l o g i c a l sense, but i s more a p p r o p r i a t e l y i n t e r p r e t e d as a more r e s t r i c t i v e 'and simultaneously'. To i l l u s t r a t e these i n t e r p r e t a t i o n s , consider Inhelder and Piaget's (1958) binary operation for Equivalence. The usual expression which i s given i n GLT i s p.q v p.g If the binary operations (as used by Inhelder and Piaget) are i n t e r p r e t e d i n the same way as the binops are i n t e r p r e t e d i n the present study, the binary operation above would mean: There i s at l e a s t one object (or event) i n the r e f e r e n t i a l set for which p and q are simultaneously true and as w e l l there i s at l e a s t one object (or event) i n the r e f e r e n t i a l set for which p and g are simultaneously true and as w e l l there are no obj e c t s (or events) i n the r e f e r e n t i a l set for which e i t h e r p and g are simultaneously true or p and q are simultaneously t r u e . The l a s t c o n d i t i o n of the non-existence of the missing 'elements' (as they are c a l l e d by Inhelder and Piaget) i n a binary operation i s an important one, since many paradoxes 24 become manifes t wi thout i t . S i m i l a r problems occur i f ' v ' i s i n t e r p r e t e d as ' o r ' i n e i t h e r of the usua l weak ( i n c l u s i v e ) or s trong ( e x c l u s i v e ) senses . Note that the presence of the symbol p ( for example) i n a b inary opera t ion does not i n d i c a t e the absence of r e f e r e n t s which s a t i s f y p . I t i n d i c a t e s the presence of r e f e r e n t s which s a t i s f y p. In the present s tudy , the terms ' b i n o p ' and ' b i n a r y o p e r a t i o n ' have been.used synonymously, a l though the l a t t e r term has been used only w i t h i n the context of P i a g e t ' s work. • A l t o g e t h e r , there are s i x t e e n b inary o p e r a t i o n s . Each one represents a d i f f e r e n t combinat ion of absence and presence of c o n j u n c t i o n s of a f f i r m a t i v e and negat ive forms of two p r o p o s i t i o n a l f u n c t i o n s . To c o n v e n i e n t l y i l l u s t r a t e t h i s p r o p o s i t i o n a l pas de deux, each b i n a r y opera t ion i s n u m e r i c a l l y l a b e l l e d i n Table 4 accord ing to the b i n a r y r e p r e s e n t a t i o n s of the numbers 0 through 15, w i t h 1 r epre sen t ing the presence of an element i n a r e f e r e n t i a l s e t , and 0 r e p r e s e n t i n g absence. The order which has been ass igned to the b i n a r y elements ( p . q , p.cj , P .q, p.q") i s the one which i s used by Inhelder and Piaget (1958). I t i s recognized that any other order c o u l d have been chosen, prov ided i t was s p e c i f i e d i n the present t e x t . The b i n a r y opera t ions are i l l u s t r a t e d i n Table 4 i n numerica l o r d e r , w i t h Inhelder and P i a g e t ' s name and symbols for each. A simple r e f e r e n t i a l set i s g iven to exempl i fy each one, w i t h p = ' the number i s o d d ' , and q = ' the number i s greater than 10 ' . A l p h a b e t i c a l l e t t e r s are i n c l u d e d i n each r e f e r e n t i a l s e t , to P i a g e t ' s s T a b l e 4 I x t e e n b i n a r y o p e r a t i o n s B1 n a r y e l e m e n t s C o m p l e t e d i s j u n c t i v e n o r m a l f o r m P i a g e t ' s s y m b o l s E x a m p l e o f a r e f e r e n t i a l s e t w i t h p = ' t h e n u m b e r i s o d d ' , a n d q = ' t h e n u m b e r i s > 1 0 ' N o . p . q p q p . q P i a g e t ' s n a m e s 0 0 0 0 0 ( 0 ) v ( 0 ) v ( 0 ) v ( 0 ) ( 0 ) C o m p l e t e n e g a t i o n ( o r c o n t r a d i c t i o n ) { a . b . c , k, d , h , f > 1 0 0 0 1 p . q p . q C o n j u n c t i v e n e g a t i o n < e , 2 . 8 , 4 , 6 , a , n > 2 0 0 1 0 P q P q I n v e r s e o f c o n v e r s e I m p l i c a t i o n ; N e g a t i o n o f r e c i p r o c a l i m p l i c a t i o n ( 1 4 , 1 8 , c , 1 2 , d , 2 0 ) 3 0 O 1 1 P-q v p . q Ptq] I n v e r s e o f i n d e p e n d e n c e o f p r e l a t i o n t o q ; N e g a t i o n o f p i n < 1 6 , 8 , a , 2 , 1 2 . 1 8 , 6 } 4 0 1 0 0 P - q P - q I n v e r s e o f i m p l i c a t i o n ; N o n i m p l i c a t i o n < 1 . e , 3 , 9 , w , 7 , f } 5 0 1 0 1 p . q v p . q q [p ] I n d e p e n d e n c e o f q i n r e l a t i o n N e g a t i o n o f a f f i r m a t i o n o f q t o P : < 2 , 3 . m, 7 , 8 , 1 , 6 > 6 0 1 1 0 p . q v p . q p w q R e c i p r o c a l E x c l u s i o n ( 3 , 1 2 , 9 , g , 1 , 1 6 , 18 ) 7 0 1 1 1 p . q v p . q v p . q p/q I n c o m p a t i b i l i t y ; N e g a j u n c t i o n { 1 6 . a , 3 , 8 . 4 , 1 2 , 1 > 8 1 0 0 0 p-q p .q C o n j u n c t i o n { 1 3 , 1 1 , 1 7 , d , k, 1 5 , e } 9 1 0 0 1 - p . q v p . q p = q o r p g q E q u 1 v a 1 e n c e < 1 3 . 4 , r , 8 . 1 1 . 2 . z ) 1 0 1 0 1 o p . q v p . q q[p] I n d e p e n d e n c e o f q i n r e l a t i o n A f f i r m a t i o n o f q t o P : { 1 4 . 1 6 , h , 1 3 , 1 8 , 1 7 , 11 > 1 1 1 0 1 1 p . q v p . q v p . q p =q I m p l i c a t i o n { 8 , 1 6 , 1 9 , 4 , 1 3 , h , 2 ) 1 2 1 1 0 0 p . q v p . q p[q] I n d e p e n d e n c e o f p i n r e l a t i o n A f f i r m a t i o n o f p t o q; { 1 3 , 1 7 , f , 5 , 9 , 3 . 11 > 1 3 1 1 o 1 p . q v p . q v p . q q D p C o n v e r s e i m p l i c a t i o n ; R e c i p r o c a l i m p l i c a t i o n { 3 , 8 , e , 1 7 , 4 , 1 3 , 6 ) 14 1 1 1 o p . q v p . q v p . q p v q D 1 s j u n c t 1 o n { 1 5 . 5 , 1 2 . 1 9 , g , 3 , 7 } 1 5 1 1 1 1 p . q v p . q v p . q v p . q (p*q) C o m p l e t e a f f i r m a t i o n ( o r t a u t o l o g y ) { 1 3 , 7 , 1 6 , b , 4 , 1 4 , 6 ) 26 i l l u s t r a t e that there may be objects to which n e i t h e r p nor q r e f e r , thereby i l l u m i n a t i n g the nature of complete negation (binary operation 0). An examination of GLT and TL reveals that in the former t e x t , p r o p o s i t i o n a l functions are mostly events, whereas i n the l a t t e r they are c a t e g o r i c a l (that i s , r e f e r r i n g to a t t r i b u t e s of t h i n g s ) . Further, Inhelder and Piaget do not cl a i m that t h e i r subjects e x p l i c i t l y generate the binary operations. In f a c t , they have c l e a r l y denied that t h i s i s the case (Inhelder & Piaget, 1958, p. 310). The i n t e r p r e t a t i o n which i s described above has been used i n an attempt to minimize i n t e r p r e t i v e d i f f i c u l t i e s which are apparent i n Inhelder and Piaget's work (see for example, Parsons, 1960; Ennis, 1975; Brainerd, Note 1). Nevertheless, there do appear to be some problems which remain. As an example, the expression (p a q).(q a p) = ( p = q ) has been used by Inhelder and Piaget (1958, pp. 300, 301). The symbol ' => ' i s used to represent i m p l i c a t i o n . L i n g u i s t i c a l l y , the above expression makes sense, since i t i s not regarded as c o n t r a d i c t o r y to say ( f o r example), ' I f i t i s mine then i t i s black and i f i t i s black then i t i s mine'. In other words, i t i s acceptable by l i n g u i s t i c standards. However, a c l o s e r examination of the symbolized statements reveals the f o l l o w i n g observations. Since each of 'p 3 q', 'q 3 p' and 'p = q' have been defined i n GLT i n terms of binary operations, t h i s cannot be i n t e r p r e t e d as a new d e f i n i t i o n of e i t h e r side of the equation. Note however, that 27 the a s s e r t i o n 'p a q' a f f i r m s the existence of the element p.q and denies the existence of the element p.q". Simultaneously, the a s s e r t i o n 'q z> p' denies the existence of the element p.q which 'p D q' a f f i r m s , and a f f i r m s the existence of the element p.Sf which 'p 3 q ? denies. These c o n t r a d i c t i o n s e s t a b l i s h that the expression above i s at variance with the i n t e r p r e t a t i o n of the binary operations as they have been described above. S i m i l a r c o n t r a d i c t i o n s may be found i n the statement "... p[q] and q[p] are both true at the same time ..." (GLT, p. 302). A number of c o n t r a d i c t i o n s such as the ones above, may be avoided i f a binary operation i s i n t e r p r e t e d as d e s c r i b i n g a c l a s s of r e f e r e n t i a l sets which s a t i s f y only the c o n d i t i o n of the absence of the missing elements. That i s , i f the binary operation p.q v p.q" (number 9) for example, a c t u a l l y c o n s i s t e d of the c l a s s of four r e f e r e n t i a l sets which correspond to what has been defined above, as binary operations 0, 1,8 and 9. However, t h i s i n t e r p r e t a t i o n i s i n c o n f l i c t with other things which Inhelder and Piaget have s a i d , and there i s more c o n f l i c t with t h i s i n t e r p r e t a t i o n than with the one which i s espoused i n the present study. Further, there are a number of other c o n f l i c t s i n Inhelder and Piaget's work, which the a l t e r n a t i v e i n t e r p r e t a t i o n of the term 'binary operation' above, does not r e s o l v e . R e l a t i o n s h i p s between.language and l o g i c are discussed in the model below, using the meaning of the binops which has 28 been described i n t h i s s e c t i o n . The Model The model i n the present study has three components. They have been l a b e l l e d ( r e s p e c t i v e l y ) . The L i n g u i s t i c Component, The Reasoning Component, and The C o m p a t i b i l i t y Component. The l i n g u i s t i c component serves to counterbalance the one-sided perspective which i s inherent i n a purely l o g i c a l approach. I t allows important d i s t i n c t i o n s between l i n g u i s t i c and l o g i c a l meanings which Strawson (1952), Ryle (1954), Chomsky (1955), M i t c h e l l (1962), Dik (1968), Quine (1972) and others have discussed, but which have r a r e l y reached the experimental l e v e l . In the reasoning component which f o l l o w s , a number of assumptions are presented which p e r t a i n to s t r a t e g i e s which are used i n order to respond to items i n the S y l l o g i s m Test which was used i n the present study. These assumptions allowed inferences to be made with respect to the nature of reasoning patterns of respondents, when the Sy l l o g i s m Tests were analysed. The d e s c r i p t i v e aspects of the reasoning component (namely, the binops as described above), owe much of t h e i r genesis to Inhelder and Piaget's work and to papers by Ennis (1975) and Parsons ( i 9 6 0 ) . Other aspects of the reasoning component have been deri v e d from a synthesis of research i n t h i s area. The c o m p a t i b i l i t y component describes methods which 29 allow an a n a l y s i s of the S y l l o g i s m Test to be made in concert with the R e f e r e n t i a l Test. This a n a l y s i s y i e l d s r e s u l t s which are more comprehensive and more complete than an a n a l y s i s of the S y l l o g i s m Test alone. The three components are as f o l l o w s . The L i n g u i s t i c Component Types of L o g i c a l Expressions. Inhelder and Piaget (1958) do not appear to have ever made a c l e a r d i s t i n c t i o n between the binops of r e f e r e n t i a l sets (the binary operations, i n t h i s case) and the l i n g u i s t i c counterparts which are commonly used to describe them. Instead, frequent references have been made to the term 'binary operations', the l o g i c a l - l i n g u i s t i c ambiguity of which i n h i b i t s a systematic e x p l o r a t i o n of e i t h e r i n t e r p r e t a t i o n . Subsequent d i s c u s s i o n s based on t h e i r s i x t e e n binary operations have always been encumbered by t h i s omission (among other i n t e r p r e t i v e d i f f i c u l t i e s ) . I t i s intended to c l a r i f y t h i s d i s t i n c t i o n i n t h i s s e c t i o n . There are many s y n t a c t i c a l l y d i f f e r e n t expressions which are intended to convey information about the binops of r e f e r e n t i a l s e t s . Rather than p r o v i d i n g a comprehensive l i s t , i t should s u f f i c e here to d i s t i n g u i s h between two types of the most pervasive expressions: the p r o p o s i t i o n a l and c a t e g o r i c a l expressions. In the i n t e r e s t s of b r e v i t y , t h i s d e s c r i p t i o n contains only expressions which involve two p r o p o s i t i o n a l f u n c t i o n s . With t h i s r e s t r i c t i o n , p r o p o s i t i o n a l expressions are of the form ' i f p then q', 'not both p and q', 'p i f and only i f q', and so on. C a t e g o r i c a l expressions are of the form ' a l l X are Y', 'no X are Y', 'some X are Y', and 'some X 30 are not Y'; i n other words, the c l a s s i c a l A, E, I and 0 sentence types ( r e s p e c t i v e l y ) of t r a d i t i o n a l s y l l o g i s m s . A l l of these types of sentences are r e f e r r e d to as l o g i c a l  expressions. I t should be noted that i f the p r o p o s i t i o n a l functions p and q are events, then a corresponding r e f e r e n t i a l set may be described by some appropriate p r o p o s i t i o n a l expressions ( u s u a l l y intended to r e f l e c t causation or independence) but not u s u a l l y by c a t e g o r i c a l expressions. I f however, p and q are c a t e g o r i c a l p r o p o s i t i o n a l f u n c t i o n s , then appropriate d e s c r i p t i v e expressions of r e f e r e n t i a l sets may be e i t h e r p r o p o s i t i o n a l ('If i t i s a square then i t i s black') or c a t e g o r i c a l ( ' A l l squares are black.') The next question to be addressed i s the circumstances under which a l o g i c a l expression may be used i n reference to objects or events. More p r e c i s e l y , what s a l i e n t c h a r a c t e r i s t i c s are possessed by the objects or events to which a l o g i c a l expression r e f e r s , i n order that these c h a r a c t e r i s t i c s may be compatible with a given person's meaning of the l o g i c a l expression? Present t h i n k i n g on t h i s t o p i c seems to be f a i r l y c o n s i s t e n t , but depends upon whether the l o g i c a l expression i n question i s p r o p o s i t i o n a l or c a t e g o r i c a l . The d i s c u s s i o n below i s r e s t r i c t e d to p r o p o s i t i o n a l expressions. Meanings of L o g i c a l Expressions. With respect to p r o p o s i t i o n a l expressions, an examination was undertaken of underlying assumptions behind the c o n s t r u c t i o n of l o g i c a l t r u t h t a b l e s , and of the nature of procedures used i n both 31 research i n Concept Formation r u l e l e a r n i n g , and various R e f e r e n t i a l Tests. Without exception, i t was found that i t was considered that a l o g i c a l expression can apply to a s i n g l e object (or event), but i t was not considered that a l o g i c a l expression may apply to groups of objects (or events) as a  u n i f i e d whole. As an example, standard r u l e s of formal l o g i c determine that the statement ' I f i t i s black then i t i s a square' i s a true statement, when r e f e r r i n g only to a black square. I t i s a l s o considered to be true when r e f e r r i n g only to a white square, and i t i s a l s o considered true when r e f e r r i n g only to a white c i r c l e . I t i s considered f a l s e when r e f e r r i n g only to a black c i r c l e . However, i t does not seem unreasonable to c l a i m that a statement such as ' I f i t i s black then i t i s a square' i s true when r e f e r r e d to any member of a r e f e r e n t i a l set c o n s i s t i n g of black squares and white c i r c l e s (see below). The f o l l o w i n g extension of the c o n d i t i o n s under which a l o g i c a l expression may be confirmed, could help to resolve t h i s i s s u e , while l e a v i n g the door open for a modified system of formal l o g i c to be developed, which i s more compatible with n a t u r a l l o g i c . Suppose that one i s confronted with a box of black squares. I f a d e s c r i p t i o n i s made of one of these shapes without knowing e x a c t l y to which object the statement r e f e r s , i t seems that i t i s usual that i t would be c a l l e d a 'black square' or 'black and square', but that some people might consider i t reasonable to a l s o say ' i f i t i s black then i t i s a square'. I f however, the box contained black squares and 32 white c i r c l e s , the expression 'black and square' would not seem to apply, but the expression ' I f i t i s black then i t i s a square' would probably q u i t e often be used, as may other l o g i c a l expressions, by the same people. The major point here i s that the given expression a p p l i e s to any one of the objects i n the box as a whole. The f o l l o w i n g example demonstrates what t h i s h o l i s t i c nature means. I f presented with a box of black squares and white c i r c l e s , and asked to describe any (unknown) one of the shapes upon removal from the box, i t appears that most people would acknowledge that the statement ' I f i t i s black then i t i s a square' would be t r u e . However, i f the contents of the box were not considered as a whole, but as a c o l l e c t i o n of black squares and-a c o l l e c t i o n of white c i r c l e s , i t seems u n l i k e l y that many people would use t h i s same sentence to describe any member of the c o l l e c t i o n of white c i r c l e s . In other words, while the statement does not apply to a part of the contents of the box, i t may w e l l apply to the whole contents of the box. This i s at variance with norms of formal l o g i c . The reason appears to be t h a t , w i t h i n the context of formal l o g i c , the a s s e r t i o n of a l o g i c a l expression decrees the absence of p a r t i c u l a r binary elements, but not the presence of any p a r t i c u l a r elements. For the statement ' I f p then q' for example, formal l o g i c decrees the absence of p.g, and the presence of any other (unspecified) s i n g l e binary element. S i m i l a r l y , for the statement 'p or q', formal l o g i c decrees the absence of p.g and the presence of any other (unspecified) s i n g l e element. Hence, i f the presence of a 33 s p e c i f i c element i s e s s e n t i a l f o r confirm a t i o n of a l o g i c a l expression by a respondent, i t w i l l not be i n d i c a t e d by using only the norms of formal l o g i c to evaluate responses. Experimentally speaking, there appears to be some evidence that the presence of some s p e c i f i c elements i s necessary, for many people to a s s e r t at l e a s t some l o g i c a l statements (as suggested for example, by the work of Wason, 1966; P e e l , 1967; Johnson-Laird & Tagart, 1969; and o t h e r s ) . The present study has not only attempted to e s t a b l i s h that t h i s i s so, but a l s o determined e x a c t l y which binary elements are required by each respondent e i t h e r to n e c e s s a r i l y e x i s t or to n e c e s s a r i l y not e x i s t , i n order to confirm a l o g i c a l expression. 'Necessity' means that the f a i l u r e of a binary element to s a t i s f y i t s r e s p e c t i v e existence c r i t e r i o n i s s u f f i c i e n t to cla i m that the l o g i c a l expression i s f a l s e , or otherwise inappropriate to a s s e r t . Four l o g i c a l expressions were examined i n t h i s way. The h o l i s t i c nature of the sets of r e f e r e n t s i s an e s s e n t i a l idea, for without i t , r u l e s for the confirmation of l o g i c a l expressions reduce to r u l e s which may be found i n formal l o g i c . Within the context of the h o l i s t i c nature of the r e f e r e n t i a l s e t s , i t appears that with some research i n concept formation r u l e l e a r n i n g and i n standard E v a l u a t i o n t a s k s , the s e r i a l p r e s e n t a t i o n of exemplars r e s t r i c t s the sets of candidates for con f i r m a t i o n of l o g i c a l expressions to be r e s t r i c t e d tosingle-member r e f e r e n t i a l sets which correspond only to binops 34 [p.q]v[p.g]v[p.q]v p.g, [p.q]v[p.g]v p.q v [ p . g ] r [p.q]v p.g v[p.q]v[p.g] or p.q v[p.g]v[p.q]v[p.g]. These are numbers 1, 2, 4 and 8 i n Tables 3 and 4. Subsequent r e s u l t s which do not conform to the expectations of formal l o g i c may be due at l e a s t i n p a r t , to procedures and t h e o r e t i c a l perspectives which preclude the use of any other binary r e f e r e n t i a l s e t s . The f i n a l question to be addressed i n t h i s s e c t i o n i s the means by which the n e c e s s i t y of e i t h e r the presence or absence of s p e c i f i c elements i n a r e f e r e n t i a l set may be determined for each respondent, when the respondent i s given a l o g i c a l expression. This i s tantamount to determining the meaning of the l o g i c a l expression for each respondent. An underlying assumption i n the present study i s that each respondent has a meaning which i s as s o c i a t e d with each given l o g i c a l expression. In t h i s study, i t i s assumed that t h i s meaning c o n s i s t s of a c o n d i t i o n which i s attached to each of the binary elements p.q, p.g, p.q and p.g as they occur i n r e f e r e n t i a l s e t s . For each element, i t i s assumed that each respondent considers i t to be e i t h e r n e c e s s a r i l y present, n e c e s s a r i l y absent, or p o s s i b l y present and p o s s i b l y absent. Given two p r o p o s i t i o n a l f u n c t i o n s p and q, and a binary r e f e r e n t i a l s e t , there i s e x a c t l y one binop which corresponds to t h i s s e t , which involves only p and q. Other questions such as the representativeness of t h i s r e f e r e n t i a l set as a sample from a l a r g e r r e f e r e n t i a l s e t , or the means by which the c o r r e c t binop i s confirmed, are not discussed here. Suppose that a respondent i s confronted with a 35 r e f e r e n t i a l set which corresponds to one of the s i x t e e n binops which have been summarized i n Table 3. In t h i s s e t , some of the binary elements p.q, p.?, p.q and p.2f may be v a l i d a t e d and some may not. Suppose a l s o that the respondent's a t t e n t i o n i s drawn to only one of the objects (or events) i n the r e f e r e n t i a l s e t , without being t o l d which one. In the present study, t h i s was accomplished by i n d i c a t i n g that one of the objects was to be removed from a ( h y p o t h e t i c a l ) box. I f the respondent i s asked to decide whether i t i s c o r r e c t to a s s e r t a given l o g i c a l expression i n reference to one of the o b j e c t s , the respondent may respond e i t h e r a f f i r m a t i v e l y or n e g a t i v e l y . I f the respondent responds n e g a t i v e l y , i t i s considered to i n d i c a t e that the given r e f e r e n t i a l set was r e j e c t e d e i t h e r because of the presence of at l e a s t one binary element which i s n e c e s s a r i l y absent, or because of the absence of at l e a s t one binary element which i s n e c e s s a r i l y present. In t h i s instance, no attempt was made i n t h i s study to d i s t i n g u i s h between these two a l t e r n a t i v e s . Consequently, a negative response was ignored. I f the respondent responds a f f i r m a t i v e l y , i t i s considered to i n d i c a t e that a l l binary elements which are considered to be n e c e s s a r i l y absent are i n fa c t absent, and that a l l elements which are considered to be n e c e s s a r i l y present are i n f a c t present. In t h i s case, the binop of the r e f e r e n t i a l set i s noted. I f t h i s procedure i s repeated for a l l p o s s i b l e r e f e r e n t i a l sets (see Table 3 ) , the binops of those r e f e r e n t i a l sets for which a f f i r m a t i v e responses are given by 36 the respondent may be examined to determine which binary elements are always present, which elements are always absent, and which elements are sometimes present, sometimes absent. The two types of i n v a r i a n t elements were c a l l e d necessary elements. The elements which are not i n v a r i a n t were c a l l e d p o s s i b l e elements. In the present study, i f the element p.q was deemed to be n e c e s s a r i l y present, i t was denoted by the symbol p.q. I f i t was deemed to be n e c e s s a r i l y absent, i t was denoted by [p.q]. I f i t was a p o s s i b l e element, i t was denoted by (p.q). The other three elements p.g, p.q and p.g were denoted s i m i l a r l y . For each respondent, the c l a s s of a l l r e f e r e n t i a l sets for which a f f i r m a t i v e answers were given was denoted i n a way which i s c o n s i s t e n t with the notation which has been used for the binops. This n o t a t i o n a l s o r e f l e c t s the nature of the n e c e s s i t y and p o s s i b i l i t y of each element. As an example, consider the expression p.q v [p.g] v (p.q) v (p.g). This expression denotes a c l a s s of r e f e r e n t i a l sets for a respondent who considered that the element p.q i s n e c e s s a r i l y present, the element p.g i s n e c e s s a r i l y absent, and e i t h e r or both of the elements p.q and p.g may or may not be present. Table 14 (Chapter 3, page 153) shows how t h i s expression was derived f o r a h y p o t h e t i c a l respondent, w i t h i n the design c o n s t r a i n t s of the R e f e r e n t i a l Test which was used i n the present study. The f o l l o w i n g example demonstrates how t h i s expression might be derived from a simpler R e f e r e n t i a l Test. 37 Let p = ' i t i s black', and q = ' i t i s a square'. Suppose that a respondent has been assigned the l o g i c a l expression ' I f i t i s black then i t i s a square'. In t h i s study, the respondent would be confronted with a l l s i x t e e n combinations of the presence and absence of black squares, black c i r c l e s , white squares and white c i r c l e s . Each combination ( r e f e r e n t i a l set) would be purported to be i n a box. For each set, the respondent would be i n s t r u c t e d that one of the shapes shown i s to be removed from the box. The respondent would be asked, "Can we say, 'If i t i s black then i t i s a square'?", i n reference to the shape which i s to be removed. Suppose that the respondent r e p l i e s 'yes' only to r e f e r e n t i a l sets which correspond to the four binops p.q v[p.g]v[p.q]v[p.g], p.q v[p.g]v[p.q]v p.g, p.q v[p.g]v p.q v[p.g] and p.q v[p.g]v p.q v p.g. A v i s u a l examination of these four binops reveals the f o l l o w i n g information. • The element p.q i s always present. That i s , i t i s i n v a r i a n t . • The element p.g i s always absent. That i s , i t i s a l s o i n v a r i a n t . • The elements p.q and p.g are sometimes present and sometimes absent. That i s , n e i t h e r element i s i n v a r i a n t . The expression which would be assigned to t h i s respondent i s p.q v [p.g] v (p.q) v (p.g). I t should be noted that d i f f e r e n t c l a s s e s of r e f e r e n t i a l sets may generate the same expression. For 38 example, the two r e f e r e n t i a l sets which correspond to the binops p.q v[p.q]v[p.q]v p.g and p.q v[p.g]v p.q v[p.g] both y i e l d the expression p.q v [p.g] v (p.q) v (p.g). So a l s o do the three r e f e r e n t i a l sets which correspond to the binops p.q v[p.g]v[p.q]v[p.g], p.q v[p.g]v[p.q]v p.g and p.q v[p.g]v p.q v[p.g]. In order to at l e a s t p a r t i a l l y overcome t h i s problem of non-uniqueness, the operator { } i s used to denote a c l a s s of r e f e r e n t i a l sets which correspond to a l l p o s s i b l e r e f e r e n t i a l sets which generate the operand. Thus, an expression such as {p.q v [p.g] v (p.q) v (p.g)} i s c a l l e d a r e f e r e n t i a l c l a s s . A l l r e f e r e n t i a l c l a s s e s are of the form {p.q v p.g v p.q v p.g}, with e i t h e r square brackets, round brackets or no brackets around each of the four binary elements. Hence there are 3* = 81 r e f e r e n t i a l c l a s s e s . I f a r e f e r e n t i a l c l a s s contains k elements which are p o s s i b l e (denoted by round b r a c k e t s ) , then i t i s composed of 2^ binops. The l a t t i c e s t r u c t u r e of the binops, based on c l a s s i n c l u s i o n , i s shown i n Figure 1. Each r e f e r e n t i a l c l a s s c o n s i s t s of a subset of the binops. The l a t t i c e s t r u c t u r e i s of such a nature that each r e f e r e n t i a l c l a s s contains a maximal element. The r e f e r e n t i a l c l a s s {p.q v [p.g] v (p.q) v (p.g)} for example, c o n s i s t s of r e f e r e n t i a l sets which correspond to 15. p.q v p.q v p.q v p.q 0. [ p . q ] v [ p . q ] v [ p . q ] v [ p . q ] F i g u r e 1. T h e l a t t i c e s t r u c t u r e o f t h e b i n a r y o p e r a t i o n s T h e p a r t i a l o r d e r i s i n d u c e d b y s e t i n c l u s i o n . co 40 the binops p.q v[p.g]v[p.q]v[p.g], p.q v[p.g]v[p.q]v p.g, p.q v[p.g]v p.q v[p.g] and p.q v[p.g]v p.q v p.g (numbers 8, 9, 10 and 11). The maximal element of the r e f e r e n t i a l c l a s s {p.q v [p.g] v (p.q) v (p.g)} i s the r e f e r e n t i a l set corresponding to the binop p.q v[p.g]v p.q v p.g. Thus, the maximal element of a r e f e r e n t i a l c l a s s corresponds to the binop which i s obtained by a f f i r m i n g the existence of a l l p o s s i b l e binary elements (round parentheses) i n the operand. V i s u a l l y , t h i s i s the same as removing a l l round parentheses. This property was used subsequently i n the a n a l y s i s of s y l l o g i s t i c reasoning. F i n a l l y , the construct which i s c a l l e d s a t u r a t i o n below, was used i n an attempt to keep account of the non-uniqueness of c l a s s e s of r e f e r e n t i a l sets which generate the same r e f e r e n t i a l c l a s s . Suppose that a r e f e r e n t i a l c l a s s has k elements which are p o s s i b l e . The f o l l o w i n g three circumstances may occur. 1. I f the respondent chose a l l 2 r e f e r e n t i a l sets of the r e f e r e n t i a l c l a s s as confirming the l o g i c a l expression, then the respondent was s a i d to have a saturated r e f e r e n t i a l c l a s s . 2. I f a respondent d i d not have a saturated r e f e r e n t i a l c l a s s , but the i n c l u s i o n of binop 0 would make i t so, then the respondent was s a i d to have a c o n d i t i o n a l l y  saturated r e f e r e n t i a l c l a s s . 3. I f a respondent d i d not have e i t h e r a saturated or a c o n d i t i o n a l l y saturated r e f e r e n t i a l c l a s s , the respondent was s a i d to have an unsaturated r e f e r e n t i a l c l a s s . As an example, the three binops 41 [p.q]v[p.g]v[p.q]v p.g, p.q v[p.g]v[p.q]v[p.g] and p.q v[p.g]v[p.q]v p.g (numbers 1, 8 and 9) comprise the c o n d i t i o n a l l y saturated r e f e r e n t i a l c l a s s {(p.q) v [p.g] v [p.q] v (p.g)}. In t h i s s e c t i o n , a method by which various meanings may be ass o c i a t e d with l o g i c a l expressions has been presented. These meanings have been c a l l e d ' r e f e r e n t i a l c l a s s e s ' . I t now remains to describe how these meanings may be incorporated i n t o a method by which responses for S y l l o g i s m Tests are i n t e r p r e t e d . This i s the focus of the f o l l o w i n g s e c t i o n . The Reasoning Component In the i n t e r e s t s of parsimony, t h i s d i s c u s s i o n i s r e s t r i c t e d to syll o g i s m s which involve e i t h e r two or three p r o p o s i t i o n a l f u n c t i o n s . Table 1 i l l u s t r a t e s syllogisms which inv o l v e two p r o p o s i t i o n a l f u n c t i o n s . Examples which use three p r o p o s i t i o n a l functions are A l l A are B Some B are C (Evaluation or Completion response) and If p then q If q then r (Evaluation or Completion response) Syllogisms u s u a l l y i n v o l v e only p r o p o s i t i o n s or only c a t e g o r i c a l statements. An exception i s the s o - c a l l e d ' c l a s s i c s y l l o g i s m ' on page 3. In t h i s study, a s y l l o g i s m i s defined as an argument which has three p a r t s . They are, a f i r s t premise, a second 42 premise and a co n c l u s i o n . The f i r s t premise may be any l o g i c a l expression ( e i t h e r p r o p o s i t i o n a l or c a t e g o r i c a l ) which inv o l v e s e x a c t l y two p r o p o s i t i o n a l f u n c t i o n s . This includes any of the l o g i c a l expressions which have been described above, whether they are p r o p o s i t i o n a l or c a t e g o r i c a l . The second premise may be e i t h e r the a s s e r t i o n or d e n i a l of one of the p r o p o s i t i o n a l f u n c t i o n s i n the f i r s t premise, or i t may be any l o g i c a l expression which i n v o l v e s e x a c t l y two p r o p o s i t i o n a l f u n c t i o n s , e x a c t l y one of which i s in common with the f i r s t premise. I f the second premise i s an a s s e r t i o n or d e n i a l of one of the p r o p o s i t i o n a l f u n c t i o n s i n the f i r s t premise, the conclusion ( i f one e x i s t s ) i s e i t h e r the a s s e r t i o n or d e n i a l of the other p r o p o s i t i o n a l f u n c t i o n in the f i r s t premise. I f the second premise i s a l o g i c a l expression, the conclusion ( i f one e x i s t s ) i s a l s o a l o g i c a l expression which involves the two p r o p o s i t i o n a l f u n c t i o n s which are not i n common between the f i r s t and second premises. In Chapter 5, an attempt has been made to answer the question, 'How does one determine the l o g i c a l n e c e s s i t y of a concl u s i o n ? ' . U n t i l then, the determination of l o g i c a l n e c e s s i t y of a conclusion w i l l remain moot. Formal l o g i c has not been used as a normative system to answer t h i s question. As a replacement for formal l o g i c , responses to sy l l o g i s m s have been c l a s s i f i e d according to a number of co n s t r u c t s which are described below. Appropriate r e s t r i c t i o n s to these co n s t r u c t s do enable a formal l o g i c a l c l a s s i f i c a t i o n of responses to be made. As such, these c o n s t r u c t s subsume t h e o r e t i c a l i n t e r p r e t a t i o n s which use formal l o g i c as a 43 d e s c r i p t i v e and t h e o r e t i c a l base. However, they were not formulated i n such a way that t h e i r primary o b j e c t i v e was to f a c i l i t a t e the use of formal l o g i c as a normative system. Consider a respondent who i s confronted with a s y l l o g i s m , the f i r s t premise of which involves the two p r o p o s i t i o n a l f u n c t i o n s p and q. The meaning which i s a s s o c i a t e d with the f i r s t premise may be determined by the R e f e r e n t i a l Test which has been described above. I t c o n s i s t s of a c o l l e c t i o n of c o n d i t i o n s which are associated with each of the binary elements p.q, p.g, p.q and p.g. However, i n a S y l l o g i s m Test, no respondents were supplied with the f u l l range of these four elements. A c e n t r a l assumption of the present study i s that each respondent attempted to generate (a f a c s i m i l e of) a l l four elements, and sometimes f a i l e d to do so. I t i s assumed that the respondent reasoned only with those elements which were self-generated. As an example, suppose that the meaning that a respondent a s s o c i a t e s with the f i r s t premise ' I f p then q' i s the r e f e r e n t i a l c l a s s {p.q v [p.g] v (p.q) v (p.g)}. Further, suppose that the respondent f a i l s to generate (a f a c s i m i l e of) the element p.q. In t h i s case, only the elements p.q, p.g and p.g would be a v a i l a b l e for use i n the s y l l o g i s m . For these three elements, i f d e c i s i o n s f o r elements i n the S y l l o g i s m Test are compatible with the st a t e d meaning above, then the element p.q would be considered to be n e c e s s a r i l y present, the element p.g would be considered to be n e c e s s a r i l y absent, and the element p.g would be considered to 44 be p o s s i b l y present and p o s s i b l y absent. Next, suppose the respondent accepts the a s s e r t i o n of the second premise. The second premise i n the s y l l o g i s m would be e i t h e r another l o g i c a l expression, or the a f f i r m a t i o n or d e n i a l of e i t h e r p. or q. Syllogisms for which the second premise i s another l o g i c a l expression are beyond the scope of t h i s study. Hence, the second premise would be e i t h e r the a s s e r t i o n or the d e n i a l of e i t h e r p or q. Each of these four a l t e r n a t i v e s are considered i n turn below. • I f the second premise i s p, the only element which i s thought to be present, which contains p, i s the element p.q. Hence, the conclusion would be q. • I f the second premise i s q, the only element which i s thought to be present, which contains q, i s the element p.q. Hence, the conclusion would be p. • I f the second premise i s p, the only element which i s thought to be present, which contains p, i s the element p.g. Hence, the conclusion would be g. • I f the second premise i s g, the only element which i s thought to be present, which contains g, i s the element p.g. Hence, the conclusion would be p. Hence, the c o n d i t i o n a l l o g i c a l expression ' I f p then q' would have been t r e a t e d as i f the maximal element of the respondent's meaning of t h i s expression was the binop p.q v[p.g]v[p.q]v p.g (number 9 ) . Note that t h i s r e s u l t d i d not occur because the maximal element of the respondent's meaning was t h i s binop. I t occurred because the maximal element of the meaning of the f i r s t premise was the binop p.q v[p.g]v p.q v p.g (number 1 1 ) , and because the respondent f a i l e d to generate the element p.q i n the Syllogism Test. In t h i s case, the respondent would 45 appear to have reasoned with the p r o p o s i t i o n ' I f p then q' as i f i t were a b i c o n d i t i o n a l p r o p o s i t i o n . The observation that the respondent i n the above example, t r e a t e d the l o g i c a l expression 'If p then q' as i f the maximal element of the meaning of t h i s expression was the binop p.q v[p.q]v[p.q]v p.g i s of c e n t r a l importance i n the a n a l y s i s of the S y l l o g i s m Test i n t h i s study. The binop which was i n f e r r e d from each respondent's pattern of responses, i s c a l l e d the s y l l o g i s t i c  binop of the f i r s t premise of each respondent's s y l l o g i s m . In the example above, the respondent would be c l a s s i f i e d as using binop p.q v[p.g]v[p.q]v p.g (number 9) i n the S y l l o g i s m Test. The means by which each respondent's s y l l o g i s t i c binop was i n f e r r e d , i s o p e r a t i o n a l l y defined i n Chapter 3. I t i s a l s o discussed i n a simpler form i n the s e c t i o n l a b e l l e d 'Analysis of the S y l l o g i s m Test' near the end of t h i s chapter. A consequence of the above r a t i o n a l e for the a n a l y s i s of s y l l o g i s m s i s that the second premise may be incompatible with the f i r s t premise. Two cases are considered, to i l l u s t r a t e t h i s p o i n t . In the f i r s t case, suppose that a respondent a s s o c i a t e s the r e f e r e n t i a l c l a s s {p.q v [p.g] v [p.q] v [p.g]} with a given l o g i c a l expression. Further, suppose that t h i s respondent generates a l l four elements i n the S y l l o g i s m Test. If d e c i s i o n s for elements i n the S y l l o g i s m Test are compatible 46 with the above meaning, the respondent should decide that the element p.q i s necessarily present, and that a l l three other elements are necessarily absent. Thus, the only element with which to reason in the Syllogism Test would be the element p.q. In the second case, suppose that a respondent associates the r e f e r e n t i a l class {(p.q) v [p.g] v [p.q] v (p.g)} with the same l o g i c a l expression. Further, suppose that t h i s respondent generates only the elements p.q, p.g and p.q. If decisions for elements in the Syllogism Test are compatible with t h i s meaning, the respondent should decide that the element p.q may or may not be present, and that neither p.g nor p.q are present. The element p.g would not be considered by the respondent, since i t was not generated. Thus, in t h i s case also, the only element with which to reason in the Syllogism Test would be the element p.q. It i s apparent that two d i f f e r e n t meanings have been used in the two examples above, and that in each case, the only element which may be considered to be present i s the element p.q. If the second premise of the syllogism i s p, then the conclusion would be q, and i f the second premise of the syllogism i s q, then the conclusion would be p. However, the assertion of either p or g in the second premise would be incompatible with the respondent's bel i e f that only the element p.q may be present. Hence in the present study, each syllogism was presented to each respondent only i f the respondent agreed that the two premises were compatible. 47 A f i n a l example serves to complete the range of response options i n the S y l l o g i s m Test. Suppose that a respondent a s s o c i a t e s the r e f e r e n t i a l c l a s s {p.q v [p.g] v (p.q) v (p.g)} with the l o g i c a l expression which i s the f i r s t premise i n the Syllogism Test. Further, suppose that the respondent generates a l l four elements i n the S y l l o g i s m Test. I f d e c i s i o n s f o r elements i n the S y l l o g i s m Test are compatible with the above meaning, the respondent should decide that the element p.q i s n e c e s s a r i l y present, that the element p.g i s n e c e s s a r i l y absent, and that each of the elements p.q and p.g may or may not be present. Thus, the elements p.q, p.q and p.g w i l l not be considered to be absent. Each of the four a l t e r n a t i v e s for the second premise are considered i n turn below. • I f the second premise i s p, the only element which i s thought to be present, which contains p, i s the element p.q. Hence, the conclusion would be q. • I f the second premise i s q, there are two elements which may contain q. They are the elements p.q and p.q. Hence, i t would be impossible to determine whether the conc l u s i o n i s p or p. • I f the second premise i s p, there are two elements which may contain p. They are the elements p.q and p.g. Hence, i t would be impossible to determine whether the conclusion i s q or g. • I f the second premise i s g, the only element which i s thought to be present, which contains g, i s the element p.g. Hence, the conclusion would be p. A primary o b j e c t i v e of the S y l l o g i s m Test i n t h i s study was to determine each respondent's s y l l o g i s t i c binop. The a b i l i t y to do so, required that each respondent reason i n a l o g i c a l l y c o n s i s t e n t manner. I t was recognized i n t h i s study 48 that not a l l respondents may reason i n t h i s way. The nature of l o g i c a l consistency and of l o g i c a l i n c o nsistency are discussed towards the end of t h i s chapter. For present purposes, i t s u f f i c e s to say only that a l l respondents were c l a s s i f i e d according to t h e i r l o g i c a l c o nsistency: respondents for whom i t was p o s s i b l e to i n f e r a s y l l o g i s t i c binop were be c l a s s i f i e d as concordant. Otherwise, a respondent was c l a s s i f i e d as discordant. The C o m p a t i b i l i t y Component The s y l l o g i s t i c binop which i s a s s o c i a t e d with each concordant respondent i n the Syllogism Test i n d i c a t e s which elements that respondent has acknowledged may be present, and which elements are not acknowledged to be present. I f an element was acknowledged to be present i n the S y l l o g i s m Test, the model which i s used i n the present study assumes that • the element was generated by the respondent, and • the respondent considered that e i t h e r i t was n e c e s s a r i l y present, or that i t was p o s s i b l y present, p o s s i b l y absent. If an element was not acknowledged to e x i s t i n the S y l l o g i s m Test, the model assumes that e i t h e r • the element was generated by the respondent and i t s presence was denied (that i s , i t s absence was necessary), or • the element was not generated by the respondent. The R e f e r e n t i a l Test i s one i n which a l l combinations of elements were presented to each respondent. No respondents were required to generate any of the elements i n the R e f e r e n t i a l Test (as opposed to the S y l l o g i s m T e s t ) . The r e f e r e n t i a l c l a s s which i s a s s o c i a t e d with each respondent i n 49 the R e f e r e n t i a l Test, i n d i c a t e d which elements the respondent considered to be e i t h e r n e c e s s a r i l y present, p o s s i b l y present and p o s s i b l y absent, and n e c e s s a r i l y absent. Thus, there are three a l t e r n a t i v e s for each element i n the R e f e r e n t i a l Test. In the Sy l l o g i s m Test, there are two a l t e r n a t i v e s for each element. These a l t e r n a t i v e s are that e i t h e r an element i s acknowledged to be present, or that i t s presence i s not acknowledged. Hence, there are 2 x 3 a l t e r n a t i v e s for each element when the elements are compared i n the Syl l o g i s m Test and the R e f e r e n t i a l Test. Table .5 contains the d e c i s i o n which was accorded each element, when each of these s i x contingencies occurred. This d e c i s i o n a p p l i e d to each element Table 5 O v e r a l l s t a t u s of each binary element Status i n R e f e r e n t i a l Test Status i n N e c e s s a r i l y P o s s i b l y present, N e c e s s a r i l y S y l l o g i s m Test present p o s s i b l y absent absent Acknowledged N e c e s s a r i l y P o s s i b l y present Incompatible present p o s s i b l y absent Not Not Not N e c e s s a r i l y acknowledged generated generated absent as i t was used by each respondent i n the Sy l l o g i s m Test. Thus, i f a respondent used s y l l o g i s t i c binop p.q v[p.g]v[p.q]v p.g (number 9) in the Sy l l o g i s m Test, and r e f e r e n t i a l c l a s s {p.q v [p.g] v (p.q) v (p.g)} in the R e f e r e n t i a l Test, Table 5 shows that the d e c i s i o n for o v e r a l l s t a t u s of each element i n the Sy l l o g i s m Test would be 50 as f o l l o w s . 1. p.q was generated and i s n e c e s s a r i l y present. 2. p.g was generated and i s n e c e s s a r i l y absent. 3. p.q was not generated i n the Sy l l o g i s m Test. 4. p.g was generated and i s p o s s i b l y present, p o s s i b l y absent. As another example, i f a respondent used s y l l o g i s t i c binop p.q v p.g v p.q v p.g (number 15) in the Syl l o g i s m Test, and r e f e r e n t i a l c l a s s {p.q v [p.g] v [p.q] v (p.g)} i n the R e f e r e n t i a l Test, the d e c i s i o n for each element i n the Sy l l o g i s m Test would be as f o l l o w s . 1. p.q was generated and i s n e c e s s a r i l y present. 2. p.g was acknowledged i n the Syl l o g i s m Test but denied i n the R e f e r e n t i a l Test. These r e s u l t s are incompatible. 3. p.q was acknowledged i n the Syl l o g i s m Test but denied i n the R e f e r e n t i a l Test. These r e s u l t s are incompatible. 4. p.g was generated and i s p o s s i b l y present, p o s s i b l y absent. For each respondent, a d e c i s i o n was made for a l l four elements, using the procedure which i s described above. I f for a given respondent, any element showed i n c o m p a t i b i l i t y between Syllogism Test and R e f e r e n t i a l Test, that respondent was c l a s s i f i e d as having Case 4 c o m p a t i b i l i t y . A more appropriate l a b e l may be ' i n c o m p a t i b i l i t y ' . Degrees of c o m p a t i b i l i t y are described i n d e t a i l i n Chapter 3. If Case 4 d i d not occur, the s y l l o g i s t i c binop was a member of the respondent's r e f e r e n t i a l c l a s s . A respondent who i n the Syllogism Test, generated every element of the s e l e c t e d r e f e r e n t i a l c l a s s i n the R e f e r e n t i a l Test (regardless 51 of whether or not the elements were subsequently thought to be present), was deemed to have Case 1 c o m p a t i b i l i t y . In t h i s case, the s y l l o g i s t i c binop corresponded to the maximal element of the respondent's r e f e r e n t i a l c l a s s . Note that t h i s does not depend upon a r e f e r e n t i a l c l a s s being saturated. I f a respondent had neither Case 1 nor Case 4 c o m p a t i b i l i t y , the respondent was c l a s s i f i e d as having e i t h e r Case 2 or Case 3 c o m p a t i b i l i t y . Case 2 c o m p a t i b i l i t y occurred i f the respondent's s y l l o g i s t i c binop was a member of the group of r e f e r e n t i a l . s e t s which generated the r e f e r e n t i a l c l a s s . Otherwise, Case 3 c o m p a t i b i l i t y occurred. The d i f f e r e n c e between Case 2 c o m p a t i b i l i t y and Case 3 c o m p a t i b i l i t y may be i l l u s t r a t e d by the f o l l o w i n g example. Suppose that a respondent used s y l l o g i s t i c binop p.q v[p.g]v[p.q]v p.g (number 9) i n the S y l l o g i s m Test, and suppose that the r e f e r e n t i a l c l a s s {p.q v [p.g] v (p.q) v (p.g)} was used i n the R e f e r e n t i a l Test. I f the group of r e f e r e n t i a l sets which were accepted by the respondent, was composed of r e f e r e n t i a l sets which corresponded to binops p.q v[p.g]v[p.q]v[p.g], p.q v[p.g]v[p.q]v p.g and p.q v[p.g]v p.q v p.g (numbers 8, 9 and 11), Case 2 c o m p a t i b i l i t y occurred. This i s because the s y l l o g i s t i c binop p.q v[p.g]v[p.q]v p.g (number 9) i s included i n the group of r e f e r e n t i a l sets which were accepted by the respondent. On the other hand, i f the same r e f e r e n t i a l c l a s s was composed of r e f e r e n t i a l sets which 52 corresponded to binops p.q v[p.q]v[p.q]v[p.g], p.q v[p.g]v p.q v[p.g] and p.q v[p.g]v p.q v p.g (numbers 8, 10 and 11), Case 3 c o m p a t i b i l i t y occurred. This i s because the r e f e r e n t i a l set which corresponded to s y l l o g i s t i c binop p.q v[p.g]v[p.q]v p.g (number 9) was not included i n the group of r e f e r e n t i a l sets which were accepted by the respondent. Statement of the Problem This s e c t i o n replaces the usual statement of research hypotheses. Since the model which i s used i n the present study i s a d e s c r i p t i v e model rather than a p r e d i c t i v e one, i t i s i n a ppropriate generate hypotheses to t e s t p r e d i c t i o n s which do not e x i s t . The population sample c o n s i s t e d of i n t a c t c l a s s e s i n each of Grades 6, 8, 10 and 12. Each respondent was given a s i n g l e l o g i c a l expression embedded i n a S y l l o g i s m Test and a R e f e r e n t i a l Test. Four l o g i c a l expressions were used i n t h i s study. They are the p r o p o s i t i o n s of the form P r o p o s i t i o n 1: I f p then q P r o p o s i t i o n 2: p i f and only i f q P r o p o s i t i o n 3: I f p then q and i f q then p P r o p o s i t i o n 4: I f p then q and i f not p then not q. The problems to be resolved i n the present study are stated as the f o l l o w i n g research questions. 53 The S y l l o g i s m Test The S y l l o g i s m Tests were c o n s t r u c t e d i n such a way that there were three d i s t i n c t phases i n the ana lyses of r e s u l t s . These phases are shown i n F i g u r e 2. In the f i r s t phase, each Ass ign p r o p o s i t i o n Admini s t e r S y l l o g i s m Test Admin i s t e r R e f e r e n t i a l Test usable? y_L Unusable Concordant? ^Yes Discordant S y l l o g i s t i c Binop Yes^ Usable? .No Referent i a l C l a s s Unusable C o m p a t i b i l i t y F i g u r e 2. F lowchart for t e s t a d m i n i s t r a t i o n and analyses for each respondent. re spondent ' s r e s u l t s were c l a s s i f i e d as e i t h e r being usable or unusab le . In the second phase, a l l usable r e s u l t s were c l a s s i f i e d as be ing e i t h e r concordant or d i s c o r d a n t . In the t h i r d phase, a l l concordant r e s u l t s were c l a s s i f i e d accord ing 54 to t h e i r s y l l o g i s t i c binops. The questions to be resolved i n the S y l l o g i s m Test were as f o l l o w s : Question S1 . Are there any unusable responses? I f so, are they of such a nature that they need a t t e n t i o n i n any future t e s t s of s y l l o g i s t i c reasoning? Question S2. Does u s a b i l i t y depend, upon the p r o p o s i t i o n used? Question S3. Are there developmental trends i n the u s a b i l i t y of responses? I f so, what i s t h e i r f u n c t i o n a l nature? Question S4. Is the 'concordance' construct a v i a b l e one? Question S5. Do the frequencies of occurrence of concordance depend upon the p r o p o s i t i o n used? Question S6. I f the 'concordance' construct i s v i a b l e , are there developmental trends i n the frequencies of occurrence of concordance? I f so, what i s t h e i r f u n c t i o n a l nature? Question S7. Do the s y l l o g i s t i c binops conform to the expectations of formal l o g i c ? •Question S8. Do the frequencies of occurrence of the s y l l o g i s t i c binops depend upon the p r o p o s i t i o n which i s used? Question S9. Are there developmental trends i n the frequencies of occurrence of each s y l l o g i s t i c binop? Question S10. Are there developmental trends i n the frequencies of occurrence of the s y l l o g i s t i c binops for each p r o p o s i t i o n ? The R e f e r e n t i a l Test The R e f e r e n t i a l Tests were a l s o constructed i n such a way that respondents were required to pass s t r i n g e n t c r i t e r i a for u s a b i l i t y . Figure 2 d i s p l a y s the two a n a l y t i c a l phases f o r the R e f e r e n t i a l Test. As with the S y l l o g i s m Tests, the 55 u s a b i l i t y c r i t e r i a ensured reasonable v a l i d i t y and r e l i a b i l i t y of t e s t r e s u l t s . The questions to be resolved i n .the R e f e r e n t i a l Test were as f o l l o w s : Question R1. Are there any unusable responses? I f so, are they of such a nature that they need a t t e n t i o n i n any future R e f e r e n t i a l Tests? Question R2. Does u s a b i l i t y depend upon the p r o p o s i t i o n used? Question R3. Are there developmental trends i n the u s a b i l i t y of responses? I f so, what i s t h e i r f u n c t i o n a l nature? Question R4. Is the ' h o l i s t i c ' construct a v i a b l e one? That i s , do the r e f e r e n t i a l c l a s s e s chosen f a i l to conform to the expectations of formal l o g i c ? Question R5. Are there developmental trends i n the frequencies of occurrence of each r e f e r e n t i a l c l a s s ? Question R6. Do the frequencies of occurrence of the r e f e r e n t i a l c l a s s e s depend upon the p r o p o s i t i o n which i s used? Question R7. Are there developmental trends i n the nature of the r e f e r e n t i a l c l a s s e s for each p r o p o s i t i o n ? Question R8. Is the ' s a t u r a t i o n ' construct a v i a b l e one? Question R9. Do the frequencies of occurrence of s a t u r a t i o n depend upon the p r o p o s i t i o n which i s used? Question R10. Are there developmental trends i n the frequencies of occurrence of s a t u r a t i o n ? Question R11. Are there developmental trends i n the dependence of s a t u r a t i o n on the p r o p o s i t i o n used? 56 C o m p a t i b i l i t y These questions a p p l i e d to only those respondents whose S y l l o g i s m Test response sets were concordant, and who had usable R e f e r e n t i a l Tests (see Figure 2). Question C1 . Is the ' c o m p a t i b i l i t y ' c o nstruct a v i a b l e one? That i s , are the responses for the S y l l o g i s m Tests compatible with the responses for the R e f e r e n t i a l Tests i n ways which are u n l i k e l y to occur by chance? Question C2. Does the degree of attainment of c o m p a t i b i l i t y depend upon the p r o p o s i t i o n which i s used? Question C3. Does the degree of attainment of c o m p a t i b i l i t y manifest developmental trends? Question C4. Are there developmental trends i n the dependence of c o m p a t i b i l i t y upon the p r o p o s i t i o n which i s used? As a consequence of the r e s o l u t i o n s of Questions S5, S6 and C4, a r e s o l u t i o n was sought for the f o l l o w i n g question, i n the area of Formal Operations. Question F. To what extent do Grade 12 students e i t h e r e x h i b i t or f a i l to e x h i b i t Formal Operational behaviour? The nature of developmental trends was a l s o discussed i n t h i s context. A n a l y s i s of the S y l l o g i s m Test In the model f o r t h i s study, a d e s c r i p t i o n has been given of the means by which i t was assumed that each respondent responds to s y l l o g i s m s . I t remains to reverse t h i s procedure, namely, given a set of responses f o r s y l l o g i s m s , what may be i n f e r r e d by the experimenter? This s e c t i o n d e s c r i b e s inferences which may be made by an experimenter who 57 administers a S y l l o g i s m Test i n which each s y l l o g i s m i s presented no more than once. The design which was used i n the present study was more complex, but the i n f e r e n t i a l process was s i m i l a r . The purpose here i s to i l l u s t r a t e the i n f e r e n t i a l process. In a d d i t i o n , t h i s s e c t i o n presents a more d e t a i l e d account of the concept ' l o g i c a l consistency' and i t s two values, 'concordance' and 'discordance'. This concept was introduced above, i n the d i s c u s s i o n of the reasoning component of the model. A f u l l account of these f a c t o r s i s given i n Chapter 3. In the S y l l o g i s m Test which was used i n t h i s study, one of four p r o p o s i t i o n s was randomly assigned to each respondent. The p r o p o s i t i o n was used as the f i r s t premise i n <p>, <q>, <p> and <q> s y l l o g i s m s . For each s y l l o g i s m , allowance was made for the p o s s i b i l i t y that the f i r s t and second premises may be incompatible. Thus, the questions which followed the a s s e r t i o n of the f i r s t premise of each s y l l o g i s m were of two types. The f i r s t question was used to determine whether or not the second premise was compatible with the f i r s t . Thus, for the <q> s y l l o g i s m for example, each respondent was given the f i r s t premise and then asked a question of the form 'Can q be true?' . The <q> s y l l o g i s m was subsequently administered only i f an a f f i r m a t i v e response was given. The responses would have been analysed i n the f o l l o w i n g way i f each s y l l o g i s m was presented no more than once. If the second premise was deemed to be incompatible with the f i r s t premise, i t was i n f e r r e d that both binary elements which contained the r e s p e c t i v e p r o p o s i t i o n a l f u n c t i o n 58 were considered by the respondent to be absent. Thus, i f the answer for "Can q be true?' was 'no', i t was i n f e r r e d that the respondent considered that both of the elements p.q and p.q were absent. I f on the other hand, the second premise of a s y l l o g i s m was considered by the respondent to be compatible with the f i r s t premise, i t was i n f e r r e d that e i t h e r or both of the elements which contain the p r o p o s i t i o n a l f u n c t i o n in the second premise may be present. Thus, i f q was acknowledged to be p o s s i b l e , i t was i n f e r r e d that the respondent considered e i t h e r the element p.q or the element p.q or both elements, to at l e a s t p o s s i b l y ( i f not n e c e s s a r i l y ) be present. In order to determine which of these three options was t r u e , the s y l l o g i s m was then presented to the respondent. I f the response f o r the <q> s y l l o g i s m was that p i s t r u e , i t was i n f e r r e d that the respondent considered that the element p.q was present, while the element p.q was not. I f the response for the <q> s y l l o g i s m was that p was not true (so that p was t r u e ) , i t was i n f e r r e d that the element p.q was present, while the element p.q was not. I f the response for the <q> s y l l o g i s m was that p may or may not be t r u e , i t was i n f e r r e d that both of the elements p.q and p.q could be present, and hence that i t was impossible for the respondent to determine whether p or p was t r u e . I f t h i s process i s repeated for a l l four of the <p>, <q>, <p> and <q> s y l l o g i s m s , i t i s p o s s i b l e that c o n t r a d i c t i o n s may occur. For example, suppose that a respondent answers 'no' i n the <g> s y l l o g i s m to the question 'Is p t r u e ? * . This means that the element p.g i s acknowledged 59 to be present, while the element p.g i s not acknowledged to be present. Suppose that the same respondent i n d i c a t e s that p i s incompatible with the f i r s t premise of the <p> s y l l o g i s m . This means that neither of the elements p.q and p.g are acknowledged to be present. Hence, on the one hand, p.g has been acknowledged to be present, while on the other hand, the presence of p.g has been denied. The p o t e n t i a l for c o n t r a d i c t i o n s of the above type could be considered to be a s i g n i f i c a n t problem, since i t may appear that no account has been taken of t h i s contingency i n the foregoing d i s c u s s i o n of the assumptions i n t h i s study. However, a s o l u t i o n for t h i s problem has not been precluded i n the d i s c u s s i o n of the means by which respondents use t h e i r meanings of given l o g i c a l statements to answer s y l l o g i s m s . In t h i s d i s c u s s i o n , i t was assumed that each respondent has a meaning which i s associated with each l o g i c a l expression. Subsequently, some or a l l of the binary elements were presumed to have been generated i n the Syl l o g i s m Test. The responses which were given as a consequence, depended upon the assumption of consistency between the d e c i s i o n s which were made for each element i n the Syl l o g i s m Test, and the corresponding meanings i n the R e f e r e n t i a l Test. I t i s recognized i n t h i s study that t h i s consistency may not be forthcoming. I t was assumed that f a i l u r e to be c o n s i s t e n t r e s u l t e d from e i t h e r (or both) of two f a c t o r s . One f a c t o r was that the set of binary elements which were generated by the respondent may not have remained s t a b l e throughout the S y l l o g i s m Test. 60 This may have occurred i f ( f o r example) a respondent generated a l l four elements for the <p> s y l l o g i s m , but f a i l e d to r e t a i n the element p.q for the <q> s y l l o g i s m . The other f a c t o r was that a l l binary elements which were generated in the S y l l o g i s m Test remained s t a b l e throughout the t e s t , but the respondent simply reasoned i n a c o n t r a d i c t o r y manner. In e i t h e r case, the respondent was deemed to have f a i l e d to coordinate the binary elements which were generated. Hence, i f a subject responded i n the Syllogism Test i n such a way that c o n t r a d i c t i o n s occurred with respect to the existence of elements, that subject was s a i d to be d i s c o r d a n t . I f there were no c o n t r a d i c t i o n s of t h i s type, the subject was s a i d to be concordant. I t i s a simple matter at t h i s stage to determine the occurrence of c o n t r a d i c t i o n s of the above type. An ordered quadruple of four responses f o r a given l o g i c a l expression (one each f o r the <p>, <q>, <p> and <q> sy l l o g i s m s r e s p e c t i v e l y ) i s c a l l e d a response s e t . Table 6 contains a l l and only those response sets which lack such l o g i c a l c o n t r a d i c t i o n s and a c c o r d i n g l y , they are c a l l e d concordant response s e t s . For a given concordant response set of a l o g i c a l expression w i t h i n a s y l l o g i s m , the corresponding binop (that i s , the binop to the l e f t of Table 6 i n the same row) i s c a l l e d the s y l l o g i s t i c binop of the statement. This term has been used above, i n the d e s c r i p t i o n of the reasoning component of the model f o r t h i s study. Since there are four a l t e r n a t i v e s ('yes', 'no', 'maybe' and a n u l l response) for each of four s y l l o g i s m s , there are 61 Table 6 Binops for f i r s t premises, and corresponding response sets Binops 1 for Response s e t s 2 f i r s t premise f o r s y l l o g i s m s P o s s i b l e v e r b a l No. p.q p.g p.q p.g d e s c r i p t i o n <p> <q> <p> <g> 0 0 0 0 0 Nothing i s true or f a l s e • • • - • 1 0 0 0 1 Not p and not q - • • g P 2 0 0 1 0 q but not p • P q m 3 0 0 1 1 Not p, independent of q • P 7 P 4 0 1 0 0 p but not q g • • P 5 0 1 0 1 Not q, independent of p g • g 7 6 0 1 1 0 p or q but not both g P q P 7 0 1 1 1 Not both p and q g P 7 7 8 1 0 0 0 p and q q p • • 9 1 0 0 1 p i f and only i f q q p g P 10 1 0 1 0 q, independent of p q ? q u 1 1 1 0 1 1 If p then q q ? 7 P 12 1 1 0 0 p, independent of q ? P • P 1 3 1 1 0 1 If q then p ? P g 7 1 4 1 1 1 0 p or q or both ? 7 q P 15 1 1 1 1 p and q are independent ? 7 7 7 10 = absence of binary element 1 = presence of binary element 2 ? = indeterminate response • = impossible to a s s e r t second premise, given f i r s t premise 4* = 256 p o s s i b l e response s e t s . Only 16 of these are concordant. The c o n d i t i o n of concordance allows the i d e n t i f i c a t i o n of a unique binop which serves as a model for each concordant respondent, for any given l o g i c a l expression 62 as f i r s t premise i n a s y l l o g i s m . T a p l i n , Staudenmayer and Taddonio (1974), Staudenmayer (1975) and others, have attempted to derive s i m i l a r s y l l o g i s t i c binops from t h e i r s y l l o g i s t i c reasoning t e s t s , but some weaknesses i n t h e i r conception of t h i s construct have precluded a thorough treatment of t h i s idea (see Chapter 2). However, they concluded that From these f i n d i n g s therefore i t i s not necessary to conclude that c o n d i t i o n a l reasoning has become more l o g i c a l with i n c r e a s i n g age. An equally proper conclusion to draw would be that the meaning of the c o n d i t i o n a l connective has changed with age, and that reasoning at a l l ages i s l o g i c a l . The present experiment does not enable us to make a c l e a r d e c i s i o n between these a l t e r n a t i v e s , (pp. 370, 371) These authors d i d not describe what they meant by the word ' l o g i c a l ' i n t h e i r experiment, but i t seems l i k e l y that they had a c r i t e r i o n i n mind which i s somewhat akin to 'concordance' as i t i s defined i n the present study. I f t h i s i s the case, i t may be argued that the present study does in f a c t enable a c l e a r d e c i s i o n to be made between these authors' a l t e r n a t i v e s , by r e s o l v i n g Question S6 above. A re-examination of the response sets i n the h y p o t h e t i c a l experiments which are summarized i n Table 2 (page 7) reveals the f o l l o w i n g i n f ormation. In Experiment 1, four out of f i v e respondents were concordant. The s y l l o g i s t i c binops which may be derived from t h e i r response sets are p.q v[p.g]v[p.q]v p.g (number 9, t w i c e ) , p.q v[p.g]v p.q v p.g (number 11), and p.q v p.g v[p.q]v p.g (number 13). In Experiment 2, no respondents were concordant. These r e s u l t s may be obtained from Table 6. 63 The Norms In former t e s t s of reasoning, formal l o g i c has almost always been used as a normative system. These norms have been used to c l a s s i f y responses i n t e s t s of reasoning as e i t h e r c o r r e c t or i n c o r r e c t . I t has been argued i n t h i s chapter that the norms of formal l o g i c do not provide a c l a s s i f i c a t i o n system which i s appropriate to evaluate n a t u r a l l o g i c . The norms which are presented below are o f f e r e d as an a l t e r n a t i v e c l a s s i f i c a t i o n system with which reasoning with n a t u r a l l o g i c may be evaluated. The Sy l l o g i s m Test This t e s t measures patterns of reasoning with l o g i c a l expressions. Norm S: The respondent i s concordant. The R e f e r e n t i a l Test This t e s t measures meanings of l o g i c a l expressions. Norm R1: The respondent has a saturated r e f e r e n t i a l c l a s s . Norm R2: The respondent has a c o n d i t i o n a l l y saturated r e f e r e n t i a l c l a s s . C o m p a t i b i l i t y The r e s u l t s of R e f e r e n t i a l Tests and Syl l o g i s m Tests may be used to determine c o m p a t i b i l i t y between meanings and reasoning p a t t e r n s . Norm C1: The respondent s a t i s f i e s Case 1 c o m p a t i b i l i t y . Norm C2: The respondent s a t i s f i e s Case 2 c o m p a t i b i l i t y . Norm C3: The respondent s a t i s f i e s Case 3 c o m p a t i b i l i t y . 64 I t should be noted that the norms which are described above are independent of formal l o g i c , and as such, l i b e r a t e the assessment of reasoning from the formalism which gives r i s e to i t s greatest c r i t i c i s m . Norms of formal l o g i c may be imposed by r e q u i r i n g ( f o r example) that a respondent a s s o c i a t e s the meaning (that i s , the r e f e r e n t i a l c l a s s ) {(p.q) v [p.g] v (p.q) v (p.g)} with the l o g i c a l expression ' I f p then q', and that the respondent s a t i s f i e s Norms R, S and C1 above. However, no attempts have been made i n t h i s study to c l a s s i f y responses as c o r r e c t or i n c o r r e c t . The purpose of the s i x norms above i s to provide a framework with which responses may be c l a s s i f i e d , but which a l s o permit e x t e r n a l c r i t e r i a to be e a s i l y . imposed i f they are deemed to be appropriate. 65 CHAPTER 2 Introduct ion An examination only of p u b l i c a t i o n s which are devoted to mathematics teaching and l e a r n i n g has revealed the f o l l o w i n g change of focus on l o g i c over the past 50 years. E a r l i e r p u b l i c a t i o n s (Fawcett, 1935; Ulmer, 1937; Lazar, 1938; Rosskopf & Exner, 1955) focussed on the use of mathematics as a v e h i c l e for teaching l o g i c a l t h i n k i n g . Later p u b l i c a t i o n s focussed on the teaching of l o g i c per se (Suppes & B i n f o r d , 1965; Eisenberg & McGinty, 1974, 1975), or on the assessment of l o g i c a l t h i n k i n g ( P e e l , 1967; Roberge, 1969, 1972, 1975; O'Brien, Shapiro & R e a l i , 1971; O'Brien, 1972, 1973, 1975; Shapiro & O'Brien, 1973; Jansson, 1974, 1978; Gregory & Osborne, 1975; Damarin, 1977a, 1977b; Easterday & Henry, 1978; Juraschek, 1978; A d i , Karplus & Lawson, 1981). Although these l a t e r s t u d i e s often j u s t i f i e d t h e i r work i n terms of i m p l i c a t i o n s f o r mathematics, nevertheless the types of research questions which were asked d i d not s i g n i f i c a n t l y d i f f e r from those i n the l a r g e r body of research found mostly in psychology j o u r n a l s . Consequently, the review of l i t e r a t u r e f o r the present study was not r e s t r i c t e d to jo u r n a l s of a mathematical genre. To do so would unduly c o n s t r a i n attempts to determine the current st a t u s of the assessment of l o g i c a l t h i n k i n g . The assessment of l o g i c a l t h i n k i n g i s s i g n i f i c a n t only i n s o f a r as l o g i c a l t h i n k i n g i t s e l f i s be l i e v e d to play a meaningful r o l e i n at l e a s t some valued contexts. To t h i s 66 end, the r o l e of l o g i c a l t h i n k i n g i n the r e l a t i v e l y r e c e n t l y -acquired paradigm of s o - c a l l e d ' s c i e n t i f i c i n q u i r y ' has been discussed by Cohen and Nagel (1934), Kuhn (1962), Campbell and Stanley (1963), Kaplan (1964), C a t t e l l (1966), Popper (1973), Hempel (1973), and many others. A p p l i c a t i o n s of l o g i c a l t h i n k i n g to s p e c i f i c areas other than s c i e n t i f i c i n q u i r y have been described by Anderson, Marcham, and Dunn (1944), Lewis (1950), M i l l e r (1955), Suppes (1957), Hyram (1957), Ennis and Paulus (1965), Wilder (1967), and Hrabi (1967). P r o b a b i l i s t i c aspects of l o g i c a l t h i n k i n g have been presented by Strawson (1952), Pblya (1954), M i l l (1973) and Popper (1973). F i n a l l y , some r e s t r i c t i o n s which apply to the power of l o g i c i n mathematics have been described by Hadamard (1945) and K l i n e (1973, 1976). If i t i s accepted that l o g i c a l t h i n k i n g i s an important c o n s t r u c t , as i s suggested by the evidence presented i n the paragraph above, the assessment of l o g i c a l t h i n k i n g becomes an important instrument of change. As such, the underlying assumptions of assessment c r i t e r i a should be beyond a c t i v e d i s p u t e . The underlying assumption of most assessment c r i t e r i a i s that formal l o g i c i s the most appropriate means by which to judge acceptable performance i n l o g i c a l t h i n k i n g ^ A d i r e c t consequence i s that each l o g i c a l expression i s to be i n t e r p r e t e d i n only one way. The present chapter argues that the adequacy of formal l o g i c as a normative system may be challenged on a number of f r o n t s , and there are grounds for b e l i e v i n g the model presented i n Chapter 1 of t h i s study to be more adequate. 67 Since t h i s new model gives a prominent r o l e to Piaget's 16 binary operations as they have been defined i n the present study, i t s v i a b i l i t y i s supported by t r a c i n g the s i g n i f i c a n t r o l e which the binary operations play i n three current schools of research in human reasoning. Moreover, i t i s argued that current research provides s u b s t a n t i a l e m p i r i c a l support for the model. An examination of two current assessment paradigms reveals considerable evidence that formal l o g i c i s an inadequate competence model of behaviour i n common reasoning t a s k s . The subsequent emergence of the E v a l u a t i o n task i s noted i n both assessment paradigms. This task i s a r e s t r i c t e d form of the R e f e r e n t i a l Test as described i n the proposed model, and p e r t a i n s d i r e c t l y to the l i n g u i s t i c component of i t . The f i r s t s e c t i o n of t h i s chapter contains a summary of the inadequacies of the current formal l o g i c normative system for the assessment of l o g i c a l t h i n k i n g . The second s e c t i o n discusses three current t h e o r e t i c a l frameworks, i n which Piaget's 16 binary operations appear to play a c e n t r a l r o l e . To assess e m p i r i c a l support for the model, the t h i r d s e c t i o n undertakes a systematic review of two experimental paradigms: (a) the use of s y l l o g i s m s , and (b) the use of the 4-card s e l e c t i o n task. For each paradigm, performance i s f i r s t d e scribed, focussing not upon underlying processes, but upon f a c t o r s which inf l u e n c e responses, using formal l o g i c as the norm for assessment. For each paradigm, a d e s c r i p t i o n of performance i s followed by a presentation of models which purport to e x p l a i n behaviour. I t should be seen that w i t h i n 68 each paradigm, r e l i a n c e on norms of formal l o g i c has given way to greater awareness of the r o l e of d i f f e r e n t i n t e r p r e t a t i o n s of premises i n the tasks themselves. These i n t e r p r e t a t i o n s may d i f f e r from those accorded by formal l o g i c . In the fo u r t h s e c t i o n , the assessment of d i f f e r e n t i n t e r p r e t a t i o n s of premises i s t r e a t e d f u l l y . I t i s argued that the Evaluat i o n task which i s used for t h i s assessment, provides e m p i r i c a l j u s t i f i c a t i o n f or the use of the R e f e r e n t i a l Test i n the present study. A case i s made for fu r t h e r refinements of use of the R e f e r e n t i a l Test i n conjunction with the S y l l o g i s m Test, y i e l d i n g the i n t r a s u b j e c t construct which has been c a l l e d C o m p a t i b i l i t y i n the present study. The overview presented i n the f i n a l s e c t i o n of t h i s chapter places the r o l e of i n t e r p r e t a t i o n s of premises w i t h i n the context of the competence / performance d i s t i n c t i o n espoused by Chomsky (1965), F l a v e l l and Wohlwill (1969), Neimark (1975, 1979) and others. Inadequacies of Formal Logic In Chapter 1, i t was argued that formal l o g i c i s an inadequate normative system for n a t u r a l l o g i c . Many authors have described d i f f e r e n c e s between these two systems. In order to i l l u s t r a t e some of t h e i r arguments below, c o n d i t i o n a l sentences i n n a t u r a l l o g i c have been-written i n the form ' I f p then q', and t h e i r counterparts i n formal l o g i c have been w r i t t e n i n the form 'paq'. These i l l u s t r a t i v e arguments are r e s t r i c t e d to c o n d i t i o n a l sentences, although there are 69 s i m i l a r such arguments for c o n j u n c t i o n , d i s j u n c t i o n , negation and l o g i c a l q u a n t i f i e r s . The question a r i s e s of the c o n d i t i o n s under which statements may be v a l i d a t e d w i t h i n each l o g i c a l system. That i s , are there r u l e s i n one l o g i c a l system which do not apply to the other? For each of the numbered r u l e s l i s t e d below, the authors who are l i s t e d f o l l o w i n g the r u l e , have claimed that the given r u l e holds in one l o g i c a l system but i t s counterpart does not hold in the other. They have supported t h e i r claims with examples i n which p and q are replaced by meaningful phrases. Some r u l e s ( i n symbolic form) which are purported to hold for formal l o g i c but t h e i r counterparts do not hold for na t u r a l l o g i c are as f o l l o w s . 1. 'p=>q' i s v a l i d i f p i s f a l s e . Supported by Lewis (1912), Strawson (1952), Chipman (1971), Quine (1972), Young (1972), Ennis (1976), Kapadia (1976) . 2. 'p=>q' i s v a l i d i f q i s t r u e . Supported by Lewis (1912), Strawson (1952), Chipman (1971), Quine (1972), Young (1972). 3. ' p s q ' i s c o n s i s t e n t with 'paEf' . This i s a consequent of 1 above, for i f p i s f a l s e , both p s q and p => q" are v a l i d . Supported by Lewis (1912), Strawson (1952), M i t c h e l l (1962), Young (1972). 70 4. 'p=>q' i s v a l i d i f both p and q are tr u e . Lewis (1912), Strawson (1952), Chomsky (1955), M i t c h e l l (1962), Peel (1967), Quine (1972), Ennis (1976), and Braine (1978) have argued that i n n a t u r a l l o g i c , there should e x i s t some kind of reasonable explanation why the f u l f i l m e n t of p i n some way f u l f i l s q. 5. ' p s q ' i s equivalent to •'gap'. Supported by Strawson (1952), Watling (1953), Braine . (1978). 6. • ' p 3 ( p 3 q ) ' and ' p a l q a p ) ' are always v a l i d . These observations f o l l o w from 1, 2 and 4 above. The fa c t that these expressions are v a l i d may be seen from an a n a l y s i s of t r u t h t a b l e s . Supported by Lewis (1912) . Further, i t could be argued that 7. The expression 'p only i f q' i s considered to be equivalent to ' i f not q then not p' by both l o g i c i a n s and l i n g u i s t s , and equivalent to ' i f p then q' by l o g i c i a n s but not by l i n g u i s t s . Consider f o r example, p = 'the egg breaks', and q = 'the egg i s dropped'. The expression ' i f the egg breaks then the egg i s dropped' reverses temporal order and the sense of causation which i s i m p l i c i t i n the expression 'the egg breaks only i f the egg i s dropped'. Consequences of t h i s d i f f e r e n c e in meaning have been presented by Braine (1978). Some r u l e s which are purported to hold for n a t u r a l l o g i c but 71 t h e i r counterparts do not hold f o r formal l o g i c are as f o l l o w s . 8. ' ( i f p then q) and ( i f p then £[)' i s always f a l s e . Supported by Strawson (1952). 9. ' I f ( i f p then q) then ( i f p then 2f) ' i n many contexts. Supported by Geis and Zwicky (1971), Staudenmayer (1975), Evans and Newstead (1977). 10. ' I f p then p' i s always i n v a l i d to a s s e r t . Supported.by M i t c h e l l (1962), Young (1972). Ennis (1975) has a l s o supported the d i s t i n c t i o n s l i s t e d as 1 and 4 above (and some others which are not l i s t e d above). However, he made these d i s t i n c t i o n s i n order to d i s t i n g u i s h between h i s v e r s i o n of Piaget's p r o p o s i t i o n a l l o g i c and n a t u r a l l o g i c , rather than between formal l o g i c and n a t u r a l l o g i c . The terms 'paradoxes of co n f i r m a t i o n ' and 'paradoxes of m a t e r i a l i m p l i c a t i o n ' have been used to describe some of the above apparent d i s c r e p a n c i e s between formal l o g i c and n a t u r a l l o g i c . Strawson (1952) has pointed out that they are not paradoxes at a l l , since the two l o g i c a l systems are d i s t i n c t . This point of view i s not without i t s c r i t i c s ( f o r example, R u s s e l l , 1960; F a r i s , 1962; C l a r k , 1971), but i t i s the perspecti v e which has been adopted for the present study. The above arguments which weaken the use of formal l o g i c as a normative system are supplemented below by a d e s c r i p t i o n of three broad research r a t i o n a l e s whose per s p e c t i v e s provide t h e o r e t i c a l support for the model described i n Chapter 1. 72 T h e o r e t i c a l Perspectives A number of models u n d e r l i e research i n l o g i c a l t h i n k i n g . No attempt i s made here to c a t e g o r i z e these models. Instead, a t t e n t i o n i s d i r e c t e d towards three of the t h e o r e t i c a l frameworks i n which P i a g e t 1 s 16 binary combinations appear to play a c e n t r a l r o l e . They are, C r i t i c a l Thinking, Formal Operations and Concept Formation. C r i t i c a l Thinking There are of course, many models or d e s c r i p t i o n s of pa r t s of what i s commonly c a l l e d C r i t i c a l Thinking (see Ennis, 1962, for a re p r e s e n t a t i v e l i s t ) . The model which was proposed by Ennis (1962) has been chosen as an example of these models, since i t was deemed t o be concise yet comprehensive. He has l i s t e d twelve aspects of C r i t i c a l Thinking together with b r i e f but i l l u m i n a t i n g d e s c r i p t i o n s of each. His l o g i c a l dimension, c r i t e r i a l dimension and pragmatic dimension, together e x h i b i t a d i f f e r e n t p r o f i l e for each aspect. The twelve aspects are as f o l l o w s . 1. Grasping the meaning of a statement. 2. Judging whether there i s ambiguity i n a l i n e of reasoning. 3. Judging whether c e r t a i n statements c o n t r a d i c t each other. 4. Judging whether a c o n c l u s i o n f o l l o w s n e c e s s a r i l y . 5. Judging whether a statement i s s p e c i f i c enough. 6. Judging whether a statement i s a c t u a l l y the a p p l i c a t i o n of a c e r t a i n p r i n c i p l e . 7. Judging whether an observation statement i s r e l i a b l e . 8. Judging whether an i n d u c t i v e c onclusion i s warranted. 73 9. Judging whether the problem has been i d e n t i f i e d . 10. Judging whether something i s an assumption. 11. Judging whether a d e f i n i t i o n i s adequate. 12. Judging whether a statement made by an a l l e g e d a u t h o r i t y i s acceptable. I t i s apparent that l o g i c a l t h i n k i n g plays a dominant r o l e i n numbers 3, 4 and 8, and at l e a s t an e s s e n t i a l r o l e i n numbers 2, 6 and 10. N a t u r a l l y enough, a number of authors have j u s t i f i e d t h e i r work with l o g i c a l t h i n k i n g by v i r t u e of i t s e s s e n t i a l r o l e i n C r i t i c a l Thinking (Eisenberg & McGinty, 1975; Dienes & Golding, 1966; Paulus, 1967; H i l l , 1961; Grandstaff 1964; Aylesworth & Reagan, 1969; Ennis & Paulus, 1965). In f a c t , the above model has generated s u b s t a n t i a l research which has been d i r e c t e d to l o g i c a l aspects of the model. A l l of the above authors who assessed l o g i c a l t h i n k i n g , have done so with s y l l o g i s m s . The r o l e of the 16 binary operations i n s y l l o g i s t i c reasoning has been described i n Chapter 1. The Ennis model i s c h a r a c t e r i z e d by i t s dominantly b e h a v i o u r i s t i c p r e d i s p o s i t i o n f o r research, and i t s d i l u t i o n and de-emphasis of s o - c a l l e d ' s c i e n t i f i c i n q u i r y ' (as opposed for example, to models due to Dewey, 1933, and Black, 1952). I t i s a l s o c h a r a c t e r i z e d by i t s non-adherence to stage-bound developmental growth and i t s general i n c o m p a t i b i l i t y with Piaget's Formal Operational stage (see Ennis, 1975, 1976). 74 Formal Operational Thinking Formal Operational t h i n k i n g includes a l l aspects of s c i e n t i f i c i n q u i r y which focus upon the formation and t e s t i n g of hypotheses by means of measurement, observation, a b s t r a c t i o n and c o n t r o l of v a r i a b l e s . S c i e n t i f i c i n q u i r y i s i n e x t r i c a b l y bound to Piaget's work, as a glance at v i r t u a l l y any j o u r n a l which i s devoted to Science Education w i l l a t t e s t . The key references which describe t h i s part of Piaget's work are T r a i t e de Logique (Piaget, 1949), and The Growth of  L o g i c a l Thinking from Childhood to Adolescence (Inhelder and P i a g e t , 1958). These p u b l i c a t i o n s are r e f e r r e d to as TL and GLT r e s p e c t i v e l y , as mentioned i n Chapter 1. An examination of GLT reveals that Formal Operational thought c o n s i s t s of the i n t e g r a t i o n of three c o n s t r u c t s . They are, the s i x t e e n binary operations, the o p e r a t i o n a l schemata, and the INRC group. The s i x t e e n binary operations have been discussed at length i n Chapter 1, as an e s s e n t i a l part of the model fo r t h i s study. The other two c o n s t r u c t s are b r i e f l y discussed below. The Operational Schemata. Neimark (1975) has l i s t e d eight schemata i n Formal Operations. They are, c o m b i n a t o r i a l operations, p r o p o r t i o n s , c o o r d i n a t i o n of two systems of reference, mechanical e q u i l i b r i u m , p r o b a b i l i t y , c o r r e l a t i o n s , m u l t i p l i c a t i v e compensations and forms of conservation. Inhelder and Piaget's d e s c r i p t i o n of the nature of the o p e r a t i o n a l schemata (GLT p. 106) i s e i t h e r too ambiguous or too l o o s e l y t r a n s l a t e d to a f f o r d a c l e a r understanding of t h e i r meaning. However, the most frequent sense i n which the term i s used suggests that they are the means by which 75 e m p i r i c a l problems which require t h e i r use (such as those which have been used i n GLT) are solved by manipulating v a r i a b l e s i n such a way that the power of the binary operations may be invoked as hypotheses. This i n t e r p r e t a t i o n i s supported by the authors' c l a i m that From the s t a r t , the combinatorial system becomes an instrument of c o n c l u s i v e deduction. (GLT p. 121). The problems i n Chapters 9 to 14 of GLT focus upon systems i n e q u i l i b r i u m , and as such re q u i r e an understanding of compensation i n a number of contexts. However, compensation i s not l i s t e d as a schematum but rather i s used as a r a t i o n a l e to introduce the INRC group. The INRC Group. The nature of the INRC group may be i n t e r p r e t e d i n two d i f f e r e n t ways. I n f o r m a l l y , i t may be considered to be the means by which an experimenter has complete f l e x i b i l i t y i n the generation of a l t e r n a t e hypotheses (that i s , f l e x i b i l i t y i n switching from one binary operation to another as hypotheses for the nature of the l o g i c a l r e l a t i o n s h i p between relevant experimental v a r i a b l e s ) and consequently i n the e l i m i n a t i o n of r i v a l hypotheses. Hypotheses are e l i m i n a t e d by the manipulation of extraneous v a r i a b l e s , e i t h e r by e s t a b l i s h i n g independence (binary operation 15), or by the more common reduction of d i m e n s i o n a l i t y of higher-order operations by maintaining f i x e d values f o r these v a r i a b l e s (commonly c a l l e d 'experimental c o n t r o l ' ). Formally however, I , N, R, and C are considered as transformations of the binary operations as f o l l o w s : 76 (a) I ( d e n t i t y ) transforms a binary operation i n t o i t s e l f . (b) N(egation) transforms a binary operation i n t o i t s complement with respect to binary operation 15 (complete a f f i r m a t i o n ) (c) R ( e c i p r o c i t y ) transforms a binary operation i n t o the one which corresponds to the negation of a l l p r o p o s i t i o n a l f u n c t i o n s . (d) C ( o r r e l a t i v i t y ) transforms a binary operation i n t o the one found by applying f i r s t N, then R to the r e s u l t . For example, I(p v q) = p v q That i s , p.q v p.g v p.q v[p.g] —-> p.q v p.g v p.q v[p.g] N(p v q) = p.g That i s , p.q v p.g v p.q v[p.g] > [p.q]v[p.g]v[p.q]v p.g R ( p v q ) = p v g That i s , p.q v p.g v p.q v[p.g] — H> [p.q]v p.g v p.q v p.g C(p v q) = p.q That i s , p.q v p.g v p.q v[p.g] > p.q v[p.g]v[p.q]v[p.g]. With a l i t t l e work, i t may be shown that these four transformations form an Abelian group under the commutative operation 'and then' (so that N.C i s the transformation which i s equivalent to f i r s t performing C and then N, or N and then C) which i s isomorphic to the K l e i n 4-group. 1 There seems to be s u b s t a n t i a l d i s p a r i t y however, between the formal meanings of I, N, R and C, and the senses in which they are introduced i n GLT from Chapter 8 onwards. 1The present author has found no evidence of any p r a c t i c a l or t h e o r e t i c a l s i g n i f i c a n c e of t h i s isomorphism. 77 For example, on page 130 of GLT, the change of the c o n d i t i o n a l " i f the b a l l i s slowing down then f r i c t i o n a l forces are a c t i n g upon i t " ( w r i t t e n as "p = (q v r v s v t v ...)") to the c o n t r a p o s i t i v e c o n d i t i o n a l " i f there are no f r i c t i o n a l forces then the b a l l w i l l not slow down" ( w r i t t e n as q'.r'.S. t r . . . s p ) i s c a l l e d an "inverse transformation" (p. 131) or at l e a s t an "i n v e r s e " (p. 130), and the word " i n v e r s e " i s i d e n t i f i e d with "negation" (pp. 105, 131). C l e a r l y , since the c o n t r a p o s i t i v e c]3p (= q.p v q.p v q.p) i s e x a c t l y the same binary operation as p s q (= p.q v p.q v p.g), t h i s i n v e r s i o n corresponds to the transformation I i n the INRC group, and not N, whereas the v e r b a l d e s c r i p t i o n of t h i s operation ("...what should be the r e s u l t of the negation of a l l of these f a c t o r s " ) suggests that the operation i s R and neither I nor N. As another example of i n t e r p r e t i v e d i f f i c u l t i e s , the word " r e c i p r o c i t y " (or "symmetry") i s described as c h a r a c t e r i s t i c of l a t t i c e s (GLT p. 105). The most s i g n i f i c a n t symmetrical c h a r a c t e r i s t i c of l a t t i c e s i s the dualism between 'meet' and ' j o i n ' 1 . The l a t t i c e s t r u c t u r e shown i n Figure 1 (page 39), suggests that t h i s r e c i p r o c i t y r e f e r s to the operation C ( f o r which C(p v q) = p.q, and C(p.q) = p v q) rather than R. C l e a r l y , such l a b i l e semantics preclude easy i n t e r p r e t a t i o n s of the INRC group. I t should be noted that the above d e s c r i p t i o n a p p l i e s to only one of four stages of e p i s t e m o l o g i c a l development p o s i t e d by Piaget and h i s c o l l a b o r a t o r s , and a d e s c r i p t i o n of 1 T h i s means that v a l i d formulae which hold for the l a t t i c e , are a l s o v a l i d i f 'meet' (.) and ' j o i n * (v) are interchanged. 78 the remaining three stages i s beyond the scope of t h i s work. Some d e s c r i p t i o n s which place the stage of Formal Operations i n t h i s context may be found i n p u b l i c a t i o n s by F l a v e l l (1963), Neimark (1975) and Neimark and Chapman (1975). C r e d i b i l i t y of the Formal Operations model appears to be hampered by a lack of symbolic and semantic c l a r i t y (which may be due i n part to the d i f f i c u l t i e s of t r a n s l a t i o n ) , over-i n t e r p r e t a t i o n of r e s u l t s (that i s , making claims which are d i f f i c u l t to j u s t i f y on the basi s of experimental r e s u l t s alone - see Bynum, Thomas & Weitz, 1972; Weitz, Bynum, Thomas & Steger, 1973), a lack of o p e r a t i o n a l d e f i n i t i o n s and poor a t t e n t i o n to l i n g u i s t i c a m b i g u i t i e s . The Formal Operations model i s c h a r a c t e r i z e d by i t s c l a i m to be d e s c r i p t i v e rather than p r e s c r i p t i v e , and by i t s adherence to a stage-bound developmental growth. The question of stage development has been under dispute f o r many years. Piaget has modified h i s e a r l i e r claims somewhat, r e q u i r i n g stage development to be o r d e r - i n v a r i a n t but not a g e - i n v a r i a n t . Further, the overwhelming evidence that the stage which i s manifest by many i n d i v i d u a l s i s t a s k - s p e c i f i c , has been acknowledged by Piaget (1972) who recognized that h o r i z o n t a l decalage (as i t has been named) weakens h i s claims of t r a n s f e r a b i l i t y across tasks. This i s a major concession by Piaget to the d i s t i n c t i o n between competence and performance, espoused by F l a v e l l and Wo h l w i l l (1969), Neimark and Chapman (1975), Moshman (1977), Neimark (1975, 1979) and others (for example Chomsky, 1965, i n l i n g u i s t i c s ) . I t i s c h a r a c t e r i s t i c of t h i s d i s t i n c t i o n that Piaget's work i s recognized as 79 conceptually convenient i d e a l i z a t i o n s of the growth of i n t e l l i g e n c e rather than as r e p r e s e n t a t i v e accounts of a c t u a l e p i s t e m o l o g i c a l phenomena. This i s not beyond precedent, since most mathematical models of r e a l phenomena are not exact images of r e a l i t y , but are nevertheless h i g h l y i n s t r u c t i v e , mnemonic and p r e d i c t i v e . Concept Formation Consider Inhelder and Piaget's s i x t e e n binary operations with the r e s t r i c t i o n that the p r o p o s i t i o n a l functions are c a t e g o r i c a l , and not events. They may be used as a d e s c r i p t i v e system whereby a l l combinations of the presence or absence of any two a t t r i b u t e s of any c l a s s of objects may be symbolized. In the i n t e r e s t s of non-t r i v i a l i t y , binary operations 0 and 15 (complete negation and complete a f f i r m a t i o n ) may be e l i m i n a t e d . Further, i t i s apparent that some binary operations are redundant since i t i s p o s s i b l e to choose the two p r o p o s i t i o n a l functions i n e i t h e r order - that i s , as a d e s c r i p t i v e system, i t i s unnecessary to include ' I f q then p' i f we already have ' I f p then q'. Hence, i n the i n t e r e s t s of parsimony the remaining fourteen binary operations may be d i v i d e d i n t o equivalence c l a s s e s , so that two binary operations are equivalent i f one may be transformed i n t o the other by commuting the c o n s t i t u e n t p r o p o s i t i o n a l f u n c t i o n s . E x a c t l y one member may be s e l e c t e d to represent each c l a s s , so that binary operations 2, 5, 10 and 13 may be e l i m i n a t e d , l e a v i n g ten binary operations to act as a n o n - t r i v i a l parsimonious d e s c r i p t i v e system of a l l combinations of two a t t r i b u t e s . 80 I n t e r e s t i n g l y , , t h i s set of ten binary operations i s p r e c i s e l y the set which has been used as the pool from which samples have been chosen by researchers i n Concept Formation r u l e l e a r n i n g . S u r p r i s i n g l y , there appears to be no evidence in the research l i t e r a t u r e that any authors have made t h i s connection with Inhelder and Piaget's binary operations. Neisser and Weene (1962) for example, have attempted to h i e r a r c h i c a l l y c l a s s i f y these ten "types of a t t r i b u t e s " i n some order of d i f f i c u l t y (see Table 7) but no mention has been made of Piaget or any of h i s work. In Concept Formation research, a concept appears to be perceived as an e n t i t y which has two components - i t s d e f i n i n g a t t r i b u t e s (the p r o p o s i t i o n a l functions) and the ru l e which governs the r e l a t i o n s h i p between them (the form of the binary o p e r a t i o n s ) . G e n e r a l l y , research i n Concept Formation appears to be of two types. (a) E a r l y research ( f o r example, Bruner, Goodnow & A u s t i n , 1956) focussed upon the determination of f a c t o r s which govern e i t h e r the i d e n t i f i c a t i o n of relevant a t t r i b u t e s ( a t t r i b u t e i d e n t i f i c a t i o n ) , given the r u l e which governs t h e i r r e l a t i o n s h i p ( u s u a l l y conjunction - binary operation 8) or the s u c c e s s f u l attainment of concepts i n which ne i t h e r a t t r i b u t e s nor r u l e s were given. (b) In the mid-1960' s, the second type of research emerged i n i t s own r i g h t ( f o r example, Neisser & Weene, 1962; Haygood & Bourne, 1965; Bourne, 1966) which examined f a c t o r s which govern the l e a r n i n g of the forms of the r e l a t i o n s h i p s between two given a t t r i b u t e s ( r u l e l e a r n i n g ) . In a l l cases the experimenter t y p i c a l l y s e l e c t e d one or more r u l e s from the ten binary operations i n 81 Table 7 Comparison of binops with r u l e s i n r u l e l e a r n i n g Name and Symbolic D e s c r i p t i o n of p o s i t i v e Designation instance Binary elements p.q p.g p.q p.g Level I Presence (A) A must be present 1 1 0 0 Absence (-A) A must not be present 0 0 1 1 (complement of presence) Level II Conjunction (A.B) Both A and B must be 1 0 0 0 present D i s j u n c t i o n (AvB) E i t h e r A or B must be 1 1 1 0 present E x c l u s i o n (A.-B) A must be present and 0 1 0 0 B not present D i s j u n c t i v e E i t h e r A or B or both 0 1 1 1 absence (-Av-B) must be absent Conjunctive A and B must both be 0 0 0 1 absence (-A.-B) absent I m p l i c a t i o n A may be absent, but 1 0 1 1 (-AvB) i f A i s present then B must be a l s o Level I I I E i t h e r / or E i t h e r A or B must be 0 1 1 0 (A.-B)v(-A.B) present, but not both Both / n e i t h e r together Both A and B must 1 0 0 1 (A.B)v(-A.-B) be present, unless n e i t h e r i s Note. The r u l e s given i n the f i r s t two columns are from Neisser & Weene (1962), Table 1. Their symbols have been repeated here. The t h i r d column i s for comparative purposes, using symbolization from the present study. Table 7, and chose which values of which a t t r i b u t e s to use from a population of stimulus o b j e c t s . This stimulus population t y p i c a l l y c o n s i s t e d of e x a c t l y one instance of every combination of about three to f i v e values of about 4 or 82 5 a t t r i b u t e s 1 . Thus, given a r u l e and a choice of a t t r i b u t e values, the stimulus population may be c l a s s i f i e d i n t o objects which s a t i s f y the c o n d i t i o n s of the binary operation ( c a l l e d 'exemplars') and those which do not ('non-exemplars'). Within the context of the model for the present study, the set of exemplars corresponds p r e c i s e l y to the r e f e r e n t i a l set which was mentioned i n Chapter 1. Hence, the binary operations are seen to play a c e n t r a l r o l e i n Concept Formation research. In the d i s c u s s i o n s above, the 16 binary operations may be seen to act as a u n i f y i n g thread i n C r i t i c a l Thinking ( v i a l o g i c a l reasoning), Formal Operations, and Concept Formation research. However, i n s p i t e of t h i s observation, no connections appear to have been made i n the l i t e r a t u r e between the ten r u l e s i n Table 7 and Inhelder and Piaget's binary operations. This may be due i n part to the lack of communication between these schools of research, and i n part to the d i f f e r e n t i a l demand c h a r a c t e r i s t i c s of experimental paradigms. In experiments which t e s t f o r the existence of Formal Operations for example, the subject i s i n complete c o n t r o l of i n i t i a t i n g , designing and executing t e s t s f or the existence of each binary element i n a binary o p e r a t i o n , with f u l l f a c i l i t i e s f or record-keeping and redress, and a r e l a t i v e l y small r e f e r e n t i a l s e t . None of these contingencies u s u a l l y operate i n st u d i e s of r u l e l e a r n i n g , wherein the experimenter i s u s u a l l y i n c o n t r o l of order of p r e s e n t a t i o n , 1For example, 3 values of c o l o u r , 3 of s i z e , 4 of shape and 2 of thickness would y i e l d 3 x 3 x 4 x 2 = 72 d i f f e r e n t stimulus o b j e c t s . 83 frequency, feedback and so on, and a r e l a t i v e l y large number of s t i m u l i are presented s e r i a l l y . The only p u b l i c a t i o n s which were evident, which discussed even two of these three schools of research were those by M i l l e r (1955), Youniss and Furth (1964), Larson (1964), McAloon (1969) and Ennis (1975, 1976). In most cases, one of the three schools of research was advocated at the expense of.another. Notwithstanding these authors' lack of support, the t h e o r e t i c a l p resentations above are considered s u f f i c i e n t to e s t a b l i s h some v i a b i l i t y of the model presented i n Chapter 1. Since the present study i s devoted to the assessment of l o g i c a l reasoning, i t would be incomplete to omit a summary of the present s t a t e of the a r t . To t h i s end, two common assessment paradigms have been s e l e c t e d for p r e s e n t a t i o n . They are, the use of syllogisms (see Table 1), and the use of the 4-card s e l e c t i o n task ( i n i t i a t e d by Wason, 1966, 1968). In both of these paradigms, assessment i s p r e d i c a t e d on the b e l i e f that behaviour which i s not i n accordance with norms of formal l o g i c i s i n c o r r e c t . I t i s shown w i t h i n each paradigm, that d i s s a t i s f a c t i o n with the adequacy of formal l o g i c has l e d to attempts to f i n d explanations of behaviour i n which adherence to norms of formal l o g i c has been suspended. 84 Assessment of Reasoning In the i n t e r e s t s of b r e v i t y , the only p r o p o s i t i o n to be discussed i n d e t a i l i s the c o n d i t i o n a l p r o p o s i t i o n of the form ' I f p then q'. This precludes d e t a i l e d d i s c u s s i o n of c o n j u n c t i o n , d i s j u n c t i o n , negajunction and a l l c a t e g o r i c a l s y l l o g i s m s (those using ' a l l ' , 'some', 'not'). Further, r e s t r i c t i o n to the two assessment paradigms noted on page 83, precludes d e t a i l e d d i s c u s s i o n of research i n concept r u l e -l e a r n i n g , research of reasoning i n text (such as d e t e c t i v e s t o r i e s ) , 3-term s e r i e s problems ( o r d i n a l l o g i c ) and P i a g e t i a n research. Assessment using Syllogisms The next 14 pages are devoted to the f a c t o r s which a f f e c t performance with s y l l o g i s m s . I t should be emphasized that performance as reported i n t h i s s e c t i o n , p e r t a i n s to c o m p a t i b i l i t y with norms of formal l o g i c , which have been shown to be of doubtful v a l i d i t y . Hence, terms such as ' c o r r e c t ' , ' e a s i e r ' , 'improvement', and ' e r r o r s ' , p e r t a i n to these norms. Performance. The f i r s t group of f a c t o r s to be discussed are w i t h i n - s u b j e c t s f a c t o r s . Ennis' (1976) d i s t i n c t i o n between l o g i c a l p r i n c i p l e s , content and complexity has been used i n t h i s respect. Probably the most concise report on the development of competence with d i f f e r e n t p r i n c i p l e s of sentence reasoning ( s y l l o g i s m s ) has been given by Roberge (1972). He summarized the developmental patterns from Grades 4 to 12 which have been observed by s i x of the more s i g n i f i c a n t s t u d i e s i n the p e r i o d 85 1965-1970, over eleven p r i n c i p l e s . This i s only about a t h i r d of the number of p r i n c i p l e s l i s t e d by Ennis (1976) but i t i s s t i l l s u f f i c i e n t l y r e p r e s e n t a t i v e to give some i n d i c a t i o n of s i g n i f i c a n t trends. A l l but one of these s t u d i e s used some consistency c r i t e r i o n f o r mastery of each p r i n c i p l e , which requires that each respondent should c o r r e c t l y answer t e s t questions for each p r i n c i p l e s u f f i c i e n t l y often that the p r o b a b i l i t y of ob t a i n i n g the c o r r e c t answers by chance alone are l e s s than .05 for each p r i n c i p l e . The data are summarized i n Table 8. They show gross d i f f e r e n c e s i n mastery between the four most commonly i n v e s t i g a t e d p r i n c i p l e s . Note that the frequencies Table 8 Mastery of s y l l o g i s m p r i n c i p l e s Pr inc i p l e 1 Grade l e v e l s <p> <q> <p> <g> Middle Elementary 50 - 55 2 - 3 2 - 3 25 - 30 Upper Secondary 95 - 100 8 - 1 2 8 - 1 2 55 - 60 'Figures shown are percentage mastery of each p r i n c i p l e Note. The s t u d i e s reported by Roberge (1972) were those by Ennis and Paulus (1965), Gardiner (1965), Howell (1965), Martens (1967), M i l l e r (1968, 1969) and Roberge (1970a). for the <q> and <p> s y l l o g i s m s are approximately what one would expect by chance alone, with a 95% c r i t e r i o n l e v e l for mastery. The trends shown i n Table 8 are a l s o supported by Paulus (1967), C a r r o l l (1970), T a p l i n (1971), Ennis (1971), O'Brien (1972, 1973), Eisenberg and McGinty (1974) and others, 86 w i t h some minor v a r i a t i o n s . The e n t r i e s are very approximate s ince there i s much var i ance between f requencies reported by researchers due to f a c t o r s such as content , n e g a t i o n , c r i t e r i o n l e v e l and other t e s t i n g f a c t o r s , and i t i s d i f f i c u l t to ob ta in a p r e c i s e f i g u r e i n each case . The i n t e n t i s only to i n d i c a t e gross d i f f e r e n c e s between p r i n c i p l e s over the age l e v e l s i n v e s t i g a t e d . The content dimension i s s u f f i c i e n t l y complex that i t may be argued that no model e x i s t s to adequately descr ibe content e f f e c t s . W i l k i n s (1928) proposed the o r i g i n a l content model ( F a m i l i a r , Sugges t ive , Symbol ic , U n f a m i l i a r ) for c a t e g o r i c a l s y l l o g i s m s , and t h i s has been r e f i n e d and improved by Gardiner (1965) and Ennis (1976). The present author has i d e n t i f i e d 16 s tud ie s i n which content was compared i n c o n d i t i o n a l s y l l o g i s m s . They a r e , T h i s t l e t h w a i t e (1950), Ennis and Paulus (1965), Paulus (1967), Martens (1967) ,. M i l l e r (1968), McAloon (1969), C a r r o l l (1970), Roberge and Paulus (1971), Ennis (1971), O ' B r i e n (1972, 1973), Shapiro and O ' B r i e n (1973), Kodroff and Roberge (1975), Staudenmayer (1975), Jansson (1977), and Roberge (1978). A s i m i l a r number of s tud ie s which used c a t e g o r i c a l s y l l o g i s m s was a l s o found. The nature of the c o n c l u s i o n s below was the same i n e i t h e r ca se . The 27 c l a s s i f i c a t i o n s of content which were found i n these s tud ie s for c o n d i t i o n a l s y l l o g i s m s were grouped i n the f o l l o w i n g ways. Twelve c a t e g o r i e s appeared to r e f l e c t content which was f a m i l i a r and for which respondents would not o r d i n a r i l y be expected to have predisposed answers. F i v e 87 c a t e g o r i e s appeared to r e f l e c t . u n f a m i l i a r content f o r which p r e d i s p o s i t i o n would a l s o not be expected. Seven c a t e g o r i e s appeared to r e f l e c t content f o r which i t would o r d i n a r i l y be expected that respondents would be predisposed to answer i n ways which were compatible with a p r i o r i b e l i e f s or p r e j u d i c e s . The remaining three content c a t e g o r i e s c o u l d not be c l a s s i f i e d i n the above ways, s i n c e l i k e l y p r e d i s p o s i t i o n was u n p r e d i c t a b l e . The present author found i t d i f f i c u l t to determine s p e c i f i c reasons why r e s e a r c h e r s have i n v e s t i g a t e d e f f e c t s of d i f f e r e n t c o ntents i n t h e i r measures of s y l l o g i s t i c r e asoning. The p o i n t of these e f f o r t s appeared to be to determine i f responses to c o n c l u s i o n s of s y l l o g i s m s were i n f l u e n c e d by t h e i r p e r c e i v e d f a c t u a l s t a t u s (True, F a l s e , or N e i t h e r ) , or by the form of the argument. Fur t h e r refinements of content then become i r r e l e v a n t . If t h i s i s the case, i t seems n a t u r a l to compare a response to a c o n c l u s i o n of a s y l l o g i s m e i t h e r with the p e r c e i v e d f a c t u a l s t a t u s ( p r e d i s p o s i t i o n ) of the c o n c l u s i o n , or with the response to a c o n c l u s i o n of a c o n t r o l s y l l o g i s m of the same form, f o r which there i s no predisposed response f o r i t s c o n c l u s i o n . Both of these procedures would n e c e s s i t a t e the r e p o r t i n g of i n t r a s u b j e c t comparisons. The present author has been unable to l o c a t e any s t u d i e s of any c o n d i t i o n a l s y l l o g i s m s which have e i t h e r r e p o r t e d or c a r r i e d out any of these comparisons. Within c a t e g o r i c a l s y l l o g i s m s , i t appears that only J a n i s and F r i c k (1944), Thouless (1959), Henle and Michael (1956), Kaufman and G o l d s t e i n (1967), and to some extent R e v l i s and L e i r e r 88 (Note 2) have attempted t o e m p i r i c a l l y e s t a b l i s h p r e d i s p o s i t i o n s towards c o n c l u s i o n s f o r each respondent. In J a n i s and F r i c k ' s study f o r example, i t was shown t h a t respondents tended t o a c c e p t c o n c l u s i o n s w i t h which they agreed, and r e j e c t c o n c l u s i o n s w i t h which they d i s a g r e e d . However, norms of f o r m a l l o g i c were used ( r a t h e r than c o n t r o l s y l l o g i s m s ) , t o c l a i m t h a t they would not have o t h e r w i s e done so. V a l i d i t y of f i n d i n g s i s t h e r e f o r e q u e s t i o n a b l e . R e s u l t s were mixed, w i t h the g e n e r a l c o n c l u s i o n b e i n g t h a t p r e d i s p o s i t i o n sometimes a f f e c t s performance i n ways c o m p a t i b l e w i t h p r i o r b e l i e f s . The l a c k of i n t r a s u b j e c t comparisons d i d not e n a b l e t h i s c o n c l u s i o n t o be a p p l i e d t o i n d i v i d u a l s r a t h e r than t o s y l l o g i s m q u e s t i o n s . A l a c k of both of the above two t y p e s of comparisons (response v s . b e l i e f , or response v s . n e u t r a l c o n t e n t response) made i t d i f f i c u l t f o r the p r e s e n t a u t h o r t o determine what was b e i n g measured when d i f f e r e n t c o n t e n t s were compared. The best t h a t may be s a i d i s t h a t t h e r e appear t o be s t r o n g i n t e r a c t i o n s of c o n t e n t w i t h p r i n c i p l e s , i f c o n t e n t c o n t a i n s c o n c l u s i o n s f o r which p r e d i s p o s i t i o n can be supposed ( C a r r o l l , 1970; J a n s s o n , 1977). T h i s may be e x p e c t e d s i n c e the p r e s e n c e or absence of n e g a t i o n i n d i f f e r e n t p r i n c i p l e s produces d i f f e r e n t l o g i c a l c o m p a t i b i l i t i e s w i t h b e l i e f s . S y s t e m a t i c e f f e c t s of c o n t e n t a r e not apparent i n p r e s e n t r e s e a r c h . E x i s t i n g c a t e g o r i e s of c o n t e n t appear t o be too cumbersome, and e x i s t i n g a n a l y t i c a l p r o c e d u r e s seem t o l a c k i n c i s i v e power. Perhaps a more a p p r o p r i a t e d i r e c t i o n f o r r e s e a r c h i n t h i s a r e a i s the approach which has been emerging 89 in the analyses of the 4-card s e l e c t i o n task (see below). As a consequence of t h i s new approach with c o n d i t i o n a l s y l l o g i s m s , Staudenmayer (1975) has concluded that ... subjects reasoning with meaningful p r o p o s i t i o n s c a u s a l l y r e l a t e d are more i n c o n s i s t e n t than those reasoning with a b s t r a c t m a t e r i a l . The complexity f a c t o r below, i s considered to include negation, mode, and order of premises. Other f a c t o r s have been suggested (Ennis, 1976), but they are not addressed here. Negation i s recognized as a general i n h i b i t o r of performance over a wide v a r i e t y of tasks (Hovland & Weiss, 1953; Wason & Jones, 1963; Wason, 1965; Suppes & Feldman, 1971; Van Duyne, 1974; Helsabeck, 1975). As a r e s u l t , respondents tend to avoid using negative information where p o s s i b l e (Donaldson, 1959) and to take a longer time processing negative information than a f f i r m a t i v e information (Wason, 1959, 1965; Wason & Jones, 1963). H i l l (1961), Roberge (1969) and Jansson (1974) found that the presence of negation i n the major premise i-nhibited c o r r e c t responding for c o n d i t i o n a l s y l l o g i s m s . More s i g n i f i c a n t l y , systematic analyses of negation e f f e c t s have c o n s i s t e n t l y demonstrated that a negated antecedent i s the p r i n c i p a l cause of these d i f f i c u l t i e s (O'Brien, 1972; Evans, 1972a, 1977a; Roberge, 1974, 1978; Jansson, 1977; Wildman & F l e t c h e r , 1977). Consistent s i g n i f i c a n t i n t e r a c t i o n s with p r i n c i p l e s have a l s o been found. Modus t o l l e n s arguments have been found to be p a r t i c u l a r l y d i f f i c u l t with a negated antecedent (Evans, 1972a, 1977a; Roberge, 1974, 1978), with a s i m i l a r but weaker e f f e c t for inverse arguments (Roberge, 90 1974). On the other hand, converse arguments have shown apparent f a c i l i t a t i v e e f f e c t s of negated antecedent. (Evans, 1972a, 1977a). Negation i n the consequent has g e n e r a l l y had no e f f e c t (Evans, 1972a, 1977a; Roberge, 1974, 1978; Wildman & F l e t c h e r , 1977). These r e s u l t s should be placed i n the context that negation does not appear to be used with any r e g u l a r i t y i n mathematics classrooms. This was suggested i n Gregory and Osborne's (1975) study of grade 7 classrooms, i n which heavy teacher usage of modus ponens was found. The term 'mode' means the p r i n c i p a l medium of communication. I t reduces e s s e n t i a l l y to varying degrees of concreteness, which i n rough descending order would read something l i k e , concrete o b j e c t s , p i c t u r e s of o b j e c t s , Venn diagrams, w r i t t e n , v e r b a l . Most i n v e s t i g a t o r s use the w r i t t e n mode to i n v e s t i g a t e reasoning, but a few (f o r example, Ennis, F i n k e l s t e i n , Smith & Wilson, 1969; T a p l i n , 1971) used other modes without d i s c u s s i n g the degree to which the use of these modes change reasoning processes compared with the w r i t t e n mode. Gardiner (1965) i n f a c t , pointed out that Piaget's t e s t s for Formal Operations are i n concrete form, not r e q u i r i n g (he claimed) an understanding of p r o p o s i t i o n a l s t r u c t u r e s . By way of d i r e c t comparison of modes, H i l l (1961) found that there were no s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s between w r i t t e n and ver b a l modes of Grades 1 to 3 students. Kodroff and Roberge (1975) found that students i n grades 1 to 3 could perform b e t t e r with the concrete mode than with the 91 v e r b a l mode, although no i n t e r a c t i o n with concreteness was found. This could be because they mixed modes f o r each student to the extent that the concrete mode presented a response precedent for the v e r b a l mode. F i n a l l y , the order i n which the premises were presented has been found to have no e f f e c t with e i t h e r c a t e g o r i c a l or p r o p o s i t i o n a l syllogisms (Begg & Denny, 1969; Roberge, 1970b). The second group of f a c t o r s to be discussed are between-subjects f a c t o r s . There i s an abundance of l i t e r a t u r e which suggests that the sex of the respondent does not s i g n i f i c a n t l y c o n t r i b u t e to performance d i f f e r e n c e s over a wide range of reasoning tasks and ages. No sex d i f f e r e n c e s have been found on c a t e g o r i c a l reasoning tasks for Primary school c h i l d r e n (Ennis et a l . , 1969; Ennis & Paulus, 1965; O'Brien & Shapiro, 1968; H i l l , 1961), for undergraduates (Begg & Denny, 1969; Dawes, 1964) and the i n t e r v e n i n g years (Ennis & Paulus, 1965; McAloon, 1969; Roberge & Paulus, 1971). S i m i l a r r e s u l t s have been found for c o n d i t i o n a l s y l l o g i s m s ( H i l l , 1961; O'Brien & Shapiro, 1968; Martens, 1967; Ennis, 1971; Roberge & Paulus, 1971; Sanner, 1974; Evans, 1977a; Wildman & F l e t c h e r , 1977, 1979), even when an attempt was made to teach l o g i c (Ennis et a l . , 1969; Ennis & Paulus, 1965; McAloon, 1969). The same holds for d i s j u n c t i v e s y l l o g i s m s ( H i l l , 1961; Martens, 1967), i n t e r p r e t a t i o n s of connectives not w i t h i n a s y l l o g i s t i c reasoning task ( M a t u l i s , 1969; Suppes & Feldman, 1971; P a r i s , 1973), c r i t i c a l t h i n k i n g (Hrabi, 1967) and s t r a t e g i e s of i n q u i r y (Larson, 1964). Only four studies have been found which c o n f l i c t i n any 92 way with these r e s u l t s . Gardiner (1965), Paulus (1967) and M i l l e r (1968) each found o c c a s i o n a l sex d i f f e r e n c e s although no trends were apparent. These may reasonably be i n t e r p r e t e d as Type I e r r o r . Flener (1973) found no sex d i f f e r e n c e s , but an i n t e r e s t i n g i n t e r a c t i o n between sex and the determinate / indeterminate dimension, whereby g i r l s were b e t t e r than boys with determinate items (<p> and <q>) and boys were bet t e r with indeterminate items (<p> and <q>). This suggests that boys may be more cautious than g i r l s (see below). An examination of Paulus' (1967) data however, does not show t h i s i n t e r a c t i o n . The evidence i s so overwhelming that sex d i f f e r e n c e s do not e x i s t , that Kodroff and Roberge (1975), and Neimark and Chapman (1975) have e x p l i c i t l y omitted sex as a v i a b l e f a c t o r i n t h e i r s t u d i e s . Acquiescence has been suggested as a f a c t o r of p o s s i b l e s i g n i f i c a n c e i n l o g i c a l reasoning, by S e l l s (1936) who c a l l e d i t " g u l l i b i l i t y " . A tendency to acquiesce to conclusions would i n f l a t e c o r r e c t performance for <p> s y l l o g i s m s and i n f l a t e e r r o r s of a l l others. In s p i t e of t h i s r e c o g n i t i o n of i t s r o l e as a confounding f a c t o r , only one study appears to have taken account of i t s i n f l u e n c e . T a p l i n (1971) included a s p e c i f i c question among h i s t e s t items to t e s t f o r acquiescence among undergraduates, and found about 6% of responses to t h i s item corresponded to an acquiescent response. This i n d i c a t e d that acquiescence d i d not appear to be p a r t i c u l a r l y i n f l u e n t i a l for h i s question at l e a s t . N e verthless, T a p l i n and Staudenmayer (1973), T a p l i n et 93 a l . (1974) and Staudenmayer (1975) have c o m p l e t e l y c o n t r o l l e d a c q u i e s c e n c e i n the d e s i g n of t h e i r s y l l o g i s m c o n c l u s i o n s . A c q u i e s c e n c e i s a l s o c o n t r o l l e d i n a s i m i l a r way i n the p r e s e n t study (see Chapter 3 ) . ' C a u t i o n ' i s o p e r a t i o n a l i z e d i n the p r e s e n t study as a tendency t o choose an i n d e t e r m i n a t e response. T h i s would i n f l a t e <p> and <q> s y l l o g i s m s c o r e s , and d e f l a t e o t h e r s . I t i s p r o b a b l y not a p a r t i c u l a r l y i n f l u e n t i a l v a r i a b l e , a l t h o u g h Woodworth and S e l l s (1935), and Roberge and P a u l u s (1971) suggested t h a t i t may be. P a u l u s ' (1967) d a t a support t h e i r s u g g e s t i o n . In the p r e s e n t s t u d y , c a u t i o n i s measured r a t h e r than c o n t r o l l e d . I t would be e v i d e n t i n dominant 'maybe' r e s p o n s e s , y i e l d i n g s y l l o g i s t i c b i n o p p.q v p.g v p.q v p.g (number 15) i n the S y l l o g i s m T e s t s . A u t h o r i t a r i a n i s m or f a s c i s m a p p l i e s o n l y t o c o n t e n t which i s not n e u t r a l . S h e n f e l d (1958) found t h a t i n t o l e r a n t u n d e r g r a d u a t e s tended t o respond i n c o r r e c t l y i n f a v o u r of t h e i r b i a s e s more f r e q u e n t l y than t o l e r a n t u n d e r g r a d u a t e s , but t h e r e were i n t e r a c t i o n s w i t h c o n t e n t . S h e l l e y and D a v i s (1957) found t h a t u n d e r g r a d u a t e s who s c o r e d low on a F a s c i s m s c a l e were not i n f l u e n c e d by t h e i r a t t i t u d e , whereas a h i g h F-s c o r e i n d i c a t e d t h a t a t t i t u d e towards c o n t e n t would be a s i g n i f i c a n t f a c t o r . The r e s u l t s of these s t u d i e s a re consonant w i t h r e a s o n a b l e e x p e c t a t i o n s , and c l a r i f y c o n d i t i o n s under which c o n t e n t e f f e c t s a r e l i k e l y t o o c c u r . P r a c t i c a l l y no r e s e a r c h has been l o c a t e d which has examined how a s p e c t s of c o g n i t i v e s t y l e a f f e c t s y l l o g i s t i c 94 reasoning. C a r r o l l (1970) concluded that low-achieving grade 9 students showed no general d i f f e r e n t i a l performance on s y l l o g i s m tasks when i m p u l s i v i t y / r e f l e c t i v i t y was measured. Kagan, Pearson and Welch (1966) found that r e f l e c t i v e grade 1 students performed be t t e r on t h e i r i n d u c t i o n t e s t s , than impulsive students. There were no s y l l o g i s t i c reasoning t e s t s i n t h e i r study. F i n a l l y , no studies of s y l l o g i s t i c reasoning were l o c a t e d which studied e f f e c t s of e i t h e r Witkin's (1967) f i e l d dependent / independent dimension, or of K l e i n ' s (1951) c o g n i t i v e c o n t r o l c o n s t r u c t s . Age has been considered to be a s i g n i f i c a n t f a c t o r i n the assessment of l o g i c a l reasoning over grades 1 to 12 ( H i l l , 1961; Paulus, 1967; M i l l e r , 1968; Roberge, 1969, 1970a; Ennis et a l . , 1969; Shapiro & O'Brien, 1970; Roberge & Paulus, 1971; Kodroff & Roberge, 1975), although some i n t e r a c t i o n s have been found with l o g i c a l p r i n c i p l e s (O'Brien & Shapiro, 1968; McAloon, 1969; Roberge, 1970a; Eisenberg & McGinty, 1974; T a p l i n et a l . , 1974). For example, i t i s ge n e r a l l y agreed that improvement for the indeterminate <p> and <q> syll o g i s m s does not begin u n t i l about grade 9. The most s i g n i f i c a n t f i n d i n g however, i s that a number of s t u d i e s have reported apparent behavioural r e g r e s s i o n at grade 12 and beyond, for the c o n t r a p o s i t i v e (modus t o l l e n s ) p r i n c i p l e (Jansson, 1977; Easterday & Henry, 1978; Wildman & F l e t c h e r , 1977, 1979). Wildman & F l e t c h e r (1977) have suggested that t h i s may be due to m i s i n t e r p r e t a t i o n of premises. A c l o s e i n s p e c t i o n of t h e i r data however, reveals that i t i s more l i k e l y due to a decrease in 'matching' responses (described on p. 114) with i n c r e a s i n g 95 age. This i s because a matching response y i e l d s c o r r e c t answers f o r both c o n t r a p o s i t i v e s y l l o g i s m s i n which the antecedent i s i n negated form. There are a number of c o g n i t i v e v a r i a b l e s which have been considered to a f f e c t reasoning performance. For example, some i n v e s t i g a t o r s have considered IQ to be a p o t e n t i a l l y s i g n i f i c a n t v a r i a b l e , with mixed r e s u l t s . M a t u l i s (1969), M i l l e r (1968), and Paulus (1967) each found IQ to be a s i g n i f i c a n t i n f l u e n t i a l v a r i a b l e over grades 4 to 12, although not f o r a l l ages, p r i n c i p l e s , content, and use of negation for the l a t t e r study. Ennis (1971) found high c o r r e l a t i o n s between v e r b a l IQ and t o t a l c o n d i t i o n a l l o g i c scores for grades 1 to 3, and Roy (1970) claimed that IQ in f l u e n c e d the a b i l i t y of grade 12 students to determine the v a l i d i t y of an argument, but not to prove theorems by i n d u c t i o n . On the other hand, Sanner (1974) found l i t t l e to no r e l a t i o n s h i p between performance on an a l t e r e d v e r s i o n of the Smith-Sturgeon C o n d i t i o n a l Reasoning Test, and IQ of grade 5 students. M i l l e r (1955) found no r e l a t i o n s h i p between IQ and e i t h e r the rank order of d i f f i c u l t y of 29 f a l l a c i e s i n reasoning, or the a b i l i t y to recognize these 29 f a l l a c i e s . Hyram (1957) found that IQ was not r e l a t e d to h i s own version of reasoning a b i l i t y . This question remains moot. Reading l e v e l has a l s o been thought to a f f e c t performance. Hyram (1957) f a i l e d to f i n d any s i g n i f i c a n t c o r r e l a t i o n s between h i s ve r s i o n of C r i t i c a l Thinking, and e i t h e r language l e v e l or reading l e v e l , but Gregory and Osborne (1975) i n t h e i r post fa c t o a n a l y s i s found that reading 96 l e v e l was s i g n i f i c a n t l y and p o s i t i v e l y c o r r e l a t e d with t h e i r t o t a l c o n d i t i o n a l reasoning t e s t and most subtests. The l a t t e r r e s u l t i s probably the more r e l i a b l e one, since c r i t e r i o n measures were w e l l recognized as acceptable. No other s t u d i e s appear to have t r e a t e d these f a c t o r s as independent v a r i a b l e s , although Lewis (1950) used reading l e v e l as a f a c t o r to equate treatment and c o n t r o l groups. Mathematical a b i l i t y has been studied i n t h i s context by a number of i n v e s t i g a t o r s . Gregory and Osborne (1975) found a s i g n i f i c a n t p o s i t i v e r e l a t i o n s h i p between mathematical a b i l i t y and c o n d i t i o n a l reasoning, although most of t h i s was found only with indeterminate forms. No studies appear to have been designed to examine r e l a t i o n s h i p s between mathematical a b i l i t y and c a t e g o r i c a l reasoning t a s k s . However, Eisenberg (1975) examined d i s j u n c t i v e s y l l o g i s m s , and Keating and Caramazza (1975) examined o r d i n a l l o g i c . Although Eisenberg found no s t a t i s t i c a l l y s i g n i f i c a n t e f f e c t s of mathematical a b i l i t y , he d i d f i n d an i n t e r e s t i n g i n t e r a c t i o n between mathematical a b i l i t y and determinate / indeterminate forms. S t a t i s t i c a l s i g n i f i c a n c e was not i n d i c a t e d . Contrary to what might be expected, low a b i l i t y grade 8 and 9 students were found to be b e t t e r than high a b i l i t y students with determinate d i s j u n c t i v e s y l l o g i s m s , while the reverse was true of indeterminate forms. There are a number of p o s s i b l e explanations for t h i s , however, such as high a b i l i t y students being more cau t i o u s , or low a b i l i t y students being more s u s c e p t i b l e to e f f e c t s of negation (since a negative 'atmosphere' was often confounded with c o r r e c t determinate 97 responses). In f a c t , h i s data s t r o n g l y suggest that both of these hypotheses may be tr u e . F i n a l l y , there i s strong agreement that socioeconomic status i s s t r o n g l y a s s o c i a t e d with the development of reasoning a b i l i t y (Ennis et a l . , 1969; Ennis, 1971; M a t u l i s , 1969, and to some extent Ennis & Paulus, 1965). Further, Suppes and Feldman (1971) found strong evidence to suggest d i f f e r e n t i a l i n t e r p r e t a t i o n s of l o g i c a l connectives i n 4- and 6-year-old c h i l d r e n of d i f f e r e n t socioeconomic s t a t u s , thereby lending strong evidence to suggest that l o g i c a l competence i s dependent upon environmental f a c t o r s as w e l l as maturational ones. Ennis (1971) a l s o suggested that d w e l l i n g area (Urban / Rural / Suburban) may be a s i g n i f i c a n t f a c t o r . In deference to t h i s f a c t o r , care was taken i n the present study to ensure • representativeness of socioeconomic l e v e l s i n the population sample. The t h i r d and f i n a l group of f a c t o r s are ones which do not so much a f f e c t performance as a f f e c t v a l i d i t y of t e s t s which purport to measure performance. S p e c i f i c a l l y , these are task ambiguity and response options. Each i s discussed i n turn below. Task ambiguity i s one major f a c t o r which appears to inf l u e n c e the measurement of reasoning a b i l i t y . I t i s the l i k e l i h o o d that a s i g n i f i c a n t number of respondents do not understand the nature of the task before them. Ri c h t e r (1957) and Henle (1962) for example, suggested that many respondents f a i l to d i s t i n g u i s h between l o g i c a l v a l i d i t y and f a c t u a l s t a t u s , thereby g i v i n g assent to conclusions which they agree 98 are t r u e , rather than to those they agree to be v a l i d . I t seems n a t u r a l then, to t e s t t h i s suggestion by g i v i n g the f u l l e s t i n s t r u c t i o n s p o s s i b l e . Henle and Michael (1956) found in t h i s case, a remarkable increase i n performance from an average of 53% c o r r e c t to 83% c o r r e c t for t h e i r c a t e g o r i c a l s y l l o g i s m s , but t h e i r ' f u l l e s t i n s t r u c t i o n s ' a l s o included i n s t r u c t i o n s on how to solve these s y l l o g i s m s with Venn diagrams. In the present study, students who d i d not understand the task were c l a s s i f i e d as 'unusable' (see Chapter 3) . There i s some evidence to suggest that the response option 'maybe' or i t s e q u i v a l e n t , i s not recognized by many respondents as a l e g i t i m a t e response for indeterminate s y l l o g i s m s , and t h i s could e x p l a i n a p o r t i o n of the i n c o r r e c t scores on these items (Knifong, 1974). In the present study, the nature of the 'maybe' response was c a r e f u l l y described to students, before c l a s s i f y i n g those who d i d not use i t as intended, as unusable. In summary, there appear to be f i v e major f a c t o r s which i n f l u e n c e performance with s y l l o g i s m s . They are, p r i n c i p l e s , content, negation, age and socioeconomic s t a t u s . The l o g i c a l p r i n c i p l e s f a c t o r seems to have drawn most a t t e n t i o n -c o n s i s t e n t low performance with the indeterminate items h i g h l i g h t s the inadequacies of formal l o g i c as an appropriate normative system. This concludes the d i s c u s s i o n of f a c t o r s which a f f e c t performance with s y l l o g i s t i c reasoning. E f f e c t s of i n t e r v e n t i o n are not considered here, since the present study 99 d i d not include t h i s f a c t o r . Consideration of the above f a c t o r s has l e d to the f o l l o w i n g r a t i o n a l i z a t i o n s of behaviour in s y l l o g i s t i c reasoning. Rationales for Performance. There appear to be two major (and incompatible) perspectives i n r a t i o n a l i z i n g performance. They are, (a) the use of formal l o g i c as a norm, and (b) the sanctioning of d i f f e r e n t i n t e r p r e t a t i o n s of premises i n which norms of formal l o g i c have been suspended. The major purpose of the present s e c t i o n i s to t r a c e the decreasing r o l e of the former p e r s p e c t i v e , and the i n c r e a s i n g r o l e of the l a t t e r . In the course of i t s p r e s e n t a t i o n , the emerging r o l e of s y l l o g i s t i c binops i s described. Many researchers have continued to a t t r i b u t e students' i n a b i l i t i e s to respond to s y l l o g i s m s according to norms of formal l o g i c , to f a u l t s i n t h e i r reasoning processes. For many researchers, the underlying r a t i o n a l e i s that formal l o g i c provides the best (and f o r some, the only) competence model for l o g i c a l t h i n k i n g , and experiments have been designed to determine which performance f a c t o r s (see above) a f f e c t t h e i r formal l o g i c competence. The most p e r s i s t e n t s i n g l e f a c t o r has been the continued d i s c r e p a n c i e s between the norms of formal l o g i c and performance with the s o - c a l l e d ' i n v a l i d ' or 'indeterminate' s y l l o g i s m s (the converse <q> and the inverse <p>) as may be seen i n Table 8, page 85. Some researchers have consequently l a b e l l e d these two s y l l o g i s m s 'hard' and both modus ponens and modus t o l l e n s 'easy'. In t h i s case however, the d i s t i n c t i o n between 'easy' tasks and 'hard' tasks may not be so much a question of how 100 frequently students meet c e r t a i n c r i t e r i a , but how adequately the model from which the c r i t e r i a are d e r i v e d , p r e d i c t s and explains t h e i r behaviour on a number of r e l a t e d t a s k s . The degree of a c t u a l d i f f i c u l t y may be measured more a c c u r a t e l y by some combination of students' confidence l e v e l s i n t h e i r responses, and response l a t e n c i e s . T a p l i n (1971) for example, found s i g n i f i c a n t d i f f e r e n c e s in confidence l e v e l s i n the order (from highest to lowest) <p>,<p>,<q> and <q>. S i m i l a r l y , Evans (1977a) found mean response l a t e n c i e s of 7.5, 7.9, 9.9 and 10.4 seconds for <p>,<q>, <p> and <q> syllo g i s m s r e s p e c t i v e l y . Marcus and Rips (1979) found s i m i l a r (but not i d e n t i c a l ) patterns of response l a t e n c i e s . In other words, the 'easy' <q> p r i n c i p l e e l i c i t e d lowest confidence l e v e l s and al s o took the longest time to process. Hence the formal l o g i c l a b e l s 'easy' and 'hard' appear to b e l i e the d i f f i c u l t y l e v e l s of a l l but the <p> syl l o g i s m s (modus ponens), f u r t h e r reducing c r e d i b i l i t y of the formal l o g i c model. I t might be added that T a p l i n (1971) a l s o found a s t a t i s t i c a l l y s i g n i f i c a n t negative c o r r e l a t i o n between performance and confidence for these <p> sy l l o g i s m s . The search for a more adequate r a t i o n a l e f o r ' e r r o r s ' i n reasoning was given a new meaning when Henle (1962) r a i s e d the point that Where e r r o r occurs, i t need not inv o l v e f a u l t y reasoning, but may be a fun c t i o n of the i n d i v i d u a l ' s understanding of the task or the m a t e r i a l s presented to him. Although t h i s may seem l i k e a restatement of the se c t i o n on task ambiguity, she placed t h i s observation i n a more general context o u t l i n e d by Spinoza (1930) who suggested the 101 importance of considering reasoning i n i t i a l l y as a two-stage process. F i r s t , the respondent needs to interpret the information given, and second, t h i s interpretation is used to arrive at a response of some kind. Hence, i f the respondent interprets a conditional as a biconditional, or a weak disjunction as a strong disjunction, or any proposition as any other, and responds c o r r e c t l y according to that interpretation, then i t i s not the reasoning which i s incorrect at a l l , but rather the interpretation which may be c l a s s i f i e d either as 'incorrect' or at least 'not what was expected by the experimenter'. This raises the important question of what i s being measured by errors in tests of s y l l o g i s t i c reasoning. Is i t the interpretation of the respondent which is incorrect, while the reasoning i s correct, or i s the reasoning incorrect, with correct interpretations, or are both factors operative? This new perspective requires that t r a d i t i o n a l norms of formal logic be suspended, and that methods be developed which measure each individual's meanings of l o g i c a l statements. There is abundant circumstantial evidence that a number of l o g i c a l expressions are not interpreted as expected. For example, for conditional sentences (If p then q), the following results suggest that many students treat t h i s sentence as either a biconditional (binop 9) or a conjunction (binop 8 ) . 1. Relatively good performance has been found on determinate items (<p> and <q>) and simultaneous poor performance on indeterminate items (<p> and <q> - see Table 8 ) . 1 02 2. S t a t i s t i c a l l y c o n s i s t e n t responses were obtained for indeterminate items which correspond to a b i c o n d i t i o n a l i n t e r p r e t a t i o n (Gardiner, 1965). These "misunderstandings" peaked at about the Grade 9 l e v e l . 3. A ' c o n d i t i o n a l ' marking scheme was compared with a ' b i c o n d i t i o n a l ' marking scheme, and large proportions of students s a t i s f i e d the l a t t e r and not the former scheme (Shapiro & O'Brien, 1970; B e r e i t e r & H i d i , Note 3). 4. Large negative c o r r e l a t i o n s were found between indeterminate and determinate items ( T a p l i n , 1971), between indeterminate items and the t o t a l t e s t ( M i l l e r , 1968), between indeterminate items and Mental Age i n lower grades (Paulus, 1967) and between <£5> and <q> syl l o g i s m s (Howell, 1965). In f a c t , M i l l e r ' s c o r r e l a t i o n c o e f f i c i e n t s d r a m a t i c a l l y changed sign between grades 8 and 10, and t h i s i s c o n s i s t e n t with Gardiner's (1965) f i n d i n g that c o n s i s t e n t misunderstandings of p r o p o s i t i o n s q u i c k l y attenuated a f t e r grades 8 - 9 . 5. In young (grades 1-3) c h i l d r e n , the frequency of c o r r e c t responses for <£5> syl l o g i s m s i s s i g n i f i c a n t l y higher than for <q> sy l l o g i s m s (Ennis, 1971). This i s co n s i s t e n t with a conjunctive i n t e r p r e t a t i o n . 6. Some evidence was found of a con j u n c t i v e response pattern for i n d i v i d u a l young (8-9 years old) students (Taplin et a l . , 1974). 7. Kodroff and Roberge (1975) found very l i t t l e i n t e r r o g a t i v e evidence of the use of c o n d i t i o n a l reasoning i n grades 1 to 3, contrary to what may be 103 i n f e r r e d from t h e i r q u a n t i t a t i v e data. 8. Peel (1967) found responses to h i s games which suggested that the c o n d i t i o n a l i s i n t e r p r e t e d even i n t h i s context, as i f i t were a b i c o n d i t i o n a l . S i m i l a r c i r c u m s t a n t i a l evidence for i d i o s y n c r a t i c i n t e r p r e t a t i o n s of d i s j u n c t i v e and negajunctive l o g i c a l expressions may be found i n Gardiner (1965), Howell (1965), Peel (1967), Eisenberg (1975), Roberge (1975, 1976) and Juraschek (1978). Of course, i t should be recognized that although the hypothesis that many students have e i t h e r c o n j u n c t i v e or b i c o n d i t i o n a l i n t e r p r e t a t i o n s of c o n d i t i o n a l sentences, p r e d i c t s the above r e s u l t s , nevertheless the a f f i r m a t i o n of the p r e d i c t i o n s does not prove the hypothesis. Rather than attempting to weaken a l t e r n a t e hypotheses which p r e d i c t s i m i l a r r e s u l t s , the tendency has been to e s t a b l i s h the ' i n t e r p r e t a t i o n ' hypothesis by d i r e c t measurement. There appear to have been three d i s t i n c t phases to these attempts. F i r s t , a number of researchers have advocated the existence of e i t h e r b i c o n d i t i o n a l or c o n j u n c t i v e i n t e r p r e t a t i o n s of c o n d i t i o n a l sentences, supporting t h e i r claims with measurements using c r i t e r i a which are e i t h e r l o o s e l y or incompletely defined (P e e l , 1967; O'Brien, Shapiro & R e a l i , 1971; Shapiro & O'Brien, 1973; O'Brien, 1973; Roberge, 1974; Knifong, 1974; Kuhn, 1977; Wildman & F l e t c h e r , 1977). Knifong has a l l u d e d to what Piaget c a l l s ' t r ansductive' reasoning, which Knifong and to some extent Kuhn, appear to i d e n t i f y with b i c o n d i t i o n a l reasoning. 104 Cherkes (1979) on the other hand, suggested that Knifong discussed a con j u n c t i v e i n t e r p r e t a t i o n of t r a n s d u c t i o n . O'Brien et a l . (1971) have i d e n t i f i e d what they c a l l "Math l o g i c " (the c o n d i t i o n a l i n t e r p r e t a t i o n ) , " C h i l d ' s l o g i c " (the b i c o n d i t i o n a l i n t e r p r e t a t i o n ) and f i v e " Q u a s i - c h i l d l o g i c s " (the f i r s t one of which resembles a conjunctive i n t e r p r e t a t i o n ) . The major weakness of these s t u d i e s was that i t was not c l e a r whether i n t r a s u b j e c t response patterns were determined for each respondent, or whether responses of the experimental groups were analysed, for each type of question (compare with the two h y p o t h e t i c a l experiments i n Table 2, page 7). In most cases i t appears that claims of b i c o n d i t i o n a l i n t e r p r e t a t i o n s were made only on the bas i s of responses for each indeterminate s y l l o g i s m s e p a r a t e l y . The second phase represents a s i g n i f i c a n t advance over the f i r s t phase, wherein i n t r a s u b j e c t response patterns were determined and c l a s s i f i e d , thereby overcoming the major weakness of the s t u d i e s above. A n a l y t i c a l procedures used by T a p l i n (1971) and T a p l i n and Staudenmayer (1973) have been improved f u r t h e r by T a p l i n et a l . (1974) and Staudenmayer (1975) with the use of more appropriate response options. The two main c h a r a c t e r i s t i c s of these s t u d i e s are that s t a t i s t i c a l consistency c r i t e r i a were used for each of the <p>,<q>, <p> and <q> p r i n c i p l e s , and that the dominant responses of a l l four (though i n some cases only three) p r i n c i p l e s were used i n a way which would y i e l d the same outcome as the use of Table 6 (page 61) i n the present study. These authors' procedures are more complex than the t r a n s l a t i o n of the response sets i n 1 05 Table 6 to binary operations, but y i e l d the same r e s u l t s . The most improved v e r s i o n of t h e i r procedures may be found i n Staudenmayer (1975, p. 66). Their r e s u l t s are t y p i f i e d by T a p l i n , Staudenmayer and Taddonio's (1974) study. In t h e i r study, the 50% to 60% of students who were s t a t i s t i c a l l y c o n s i s t e n t on a l l arguments showed a developmental trend from a conjunctive i n t e r p r e t a t i o n (or " t r u t h f u n c t i o n " as the authors c a l l e d i t ) to a b i c o n d i t i o n a l i n t e r p r e t a t i o n to a c o n d i t i o n a l i n t e r p r e t a t i o n over grades 3 to 11. The trend was not complete, since at the highest grade l e v e l , there were s t i l l about twice as many b i c o n d i t i o n a l i n t e r p r e t a t i o n s as c o n d i t i o n a l i n t e r p r e t a t i o n s . These i n t e r p r e t a t i o n s would be c a l l e d s y l l o g i s t i c binops 8, 9 and 11 i n the present study, although the a n a l y t i c a l procedures are somewhat d i f f e r e n t . I t was a l s o found by these authors that about 13% of the students were s t a t i s t i c a l l y c o n s i s t e n t on a l l four arguments, but c o n t r a d i c t o r y (meaning i n e f f e c t that no corresponding response set could be found i n Table 6). In the present study, these students would be c l a s s i f i e d as d i s c o r d a n t . S i m i l a r patterns of c l a s s i f i c a t i o n s were found f o r students who were s t a t i s t i c a l l y i n c o n s i s t e n t on at l e a s t one argument. In the present study, most (but probably not a l l ) of these students would a l s o be c l a s s i f i e d as d i s c o r d a n t . No developmental patterns were found by T a p l i n et a l . (1974), f o r c o n t r a d i c t i o n s i n responses. I t appears that at l e a s t two studies have been conducted since the above study, which have used s i m i l a r o p e r a t i o n a l procedures. Staudenmayer (1975) i n v e s t i g a t e d 106 s y l l o g i s t i c reasoning i n which the p r o p o s i t i o n a l f u n c t i o n s were events, rather than c a t e g o r i c a l (statements about a t t r i b u t e s of o b j e c t s ) . R e s u l t s were of l i t t l e immediate consequence to the present study. In the second study, Marcus and Rips (1979) appear to have used a weaker form of the above techniques. Their r e s u l t s p e r t a i n d i r e c t l y to the present study and they are discussed i n that context i n the s e c t i o n ' I n t e r p r e t a t i o n s of Premises' below. Although the o p e r a t i o n a l procedures i n these s t u d i e s represent a s i g n i f i c a n t breakthrough i n the a n a l y s i s of s y l l o g i s t i c reasoning, there appears to be an important conceptual weakness in t h e i r procedures. A conjunctive i n t e r p r e t a t i o n of ' I f p then q' i s one i n which only the element p.q i s acknowledged to e x i s t (binop 8). Hence, for respondents who have a co n j u n c t i v e i n t e r p r e t a t i o n , i t i s c o n t r a d i c t o r y to assert <p> and <q> s y l l o g i s m s , since no element e x i s t s f o r which e i t h e r p or g holds. I t i s d i f f i c u l t to decide how one should expect such a respondent to respond to these two s y l l o g i s m s . A reasonable expectation i s "I don't know" i n the sense "I don't know what to say", as opposed to "I don't know and i t i s impossible to know because there may be cases of p.q and p.g" (the d i s t i n c t i o n made by Knifong, 1974). T a p l i n et a l . (1974) on the other hand, chose to o p e r a t i o n a l i z e a conjunctive i n t e r p r e t a t i o n as one which was manifest by c o n t r a d i c t o r y responses. In the example below, the respondent was expected to reply "no" to each of the questions "Is q true?" and "Is not-q t r u e ? " 107 If p then q I f p then q Not p Not p Is q true? Is not-q true? Conceptually, t h i s makes some sense, since the respondent would then i n turn be denying the existence of cases of both p.q and p.c], but t h i s i s a l s o where the d i f f i c u l t y l i e s . Since p i s a premise, i t must be assumed to be t r u e , so d e n i a l of one binary element i m p l i e s the existence of the other, contrary to a conjunctive i n t e r p r e t a t i o n . An examination of Table 6 (page 61), reveals that the a s s e r t i o n of a l l four s y l l o g i s m s precludes a proper a n a l y s i s for i n t e r p r e t a t i o n s which correspond to binops 0 to 5, 8, 10 and 12. In the present study, these contingencies have been sanctioned by a s s e r t i n g each s y l l o g i s m only a f t e r e s t a b l i s h i n g the c o m p a t i b i l i t y of f i r s t and second premises. The procedures for e s t a b l i s h i n g the s y l l o g i s t i c binop of each respondent (see Chapter 3) d i f f e r somewhat from T a p l i n et a l . (1974), since they were conceived independently. The procedures i n the present study make more economical and e f f i c i e n t use of information from responses, but they do not enable a d i s t i n c t i o n to be made between discordant respondents who are s t a t i s t i c a l l y c o n s i s t e n t over p r i n c i p l e s , and those who are not. The t h i r d phase i n the attempts to d i r e c t l y determine i n d i v i d u a l s ' i n t e r p r e t a t i o n s of premises i s more d i r e c t than the procedures used by the above authors. In e f f e c t , T a p l i n et a l . (1974) used c o n s i s t e n t reasoning from s y l l o g i s m s to i n f e r s y l l o g i s t i c binops. From them, the absence or presence of each of the four binary elements may be d e r i v e d . They 108 thereby assumed that the r e s u l t i n g contingencies represented each respondent's i n t e r p r e t a t i o n of the premises. However, a new type of task has been developed, c a l l e d the E v a l u a t i o n  task, i n which the syllogisms are bypassed, and the relevance of each of the binary elements to a given ( c o n d i t i o n a l ) l o g i c a l expression i s determined d i r e c t l y . This has been accomplished by presenting respondents with representations of each of the elements p.q, p.q\ P«q and p.g ( u s u a l l y on cards) and a l o g i c a l expression ( u s u a l l y a c o n d i t i o n a l ) , and i n q u i r i n g whether each element makes the expression e i t h e r true or f a l s e , or whether i t i s i r r e l e v a n t to the status of the expression. The E v a l u a t i o n task i s c r u c i a l to the present study, because i t i s a type of R e f e r e n t i a l Test. I t i s discussed below i n the context w i t h i n which i t was developed, namely, the 4-card s e l e c t i o n task. I t i s then placed w i t h i n the context of the present study. The foregoing d i s c u s s i o n describes the dominant d i r e c t i o n of past and present attempts t o f i n d an a l t e r n a t i v e to the formal l o g i c competence model for s y l l o g i s t i c reasoning. In t h i s a l t e r n a t i v e model, competence has been assumed to r e s t upon i d i o s y n c r a t i c i n t e r p r e t a t i o n s of premises, with subsequent performance r e s t i n g upon accepted performance f a c t o r s such as content, negation, or age. During the future development of t h i s approach, i t may be necessary to p o s t u l a t e f u r t h e r competence f a c t o r s before a competence model which includes i n t e r p r e t a t i o n a l f a c t o r s gains widespread acceptance. These comments have been made because the 109 i n t e r p r e t a t i o n - performance model i s not the only a l t e r n a t i v e to the formal l o g i c model. Johnson-Laird (1975) and Braine (1978) have postula t e d a l t e r n a t i v e normative systems based upon inference schemes i n n a t u r a l language. Neither system appears to have reached the experimental l e v e l , but they do o f f e r what appears to be a v i a b l e a l t e r n a t i v e approach to f i n d i n g a replacement f o r formal l o g i c . I t i s n o t i c a b l e that n e i t h e r of these two sets of n a t u r a l inference schemes appear to have come to terms with inferences which contravene norms of formal l o g i c (such as i n v e r s i o n and conversion). However, parsimony has been s a c r i f i c e d i n the i n t e r e s t s of representativeness, l e a v i n g the door open to the c o n s i d e r a t i o n of contexts w i t h i n which v a r i o u s inference schemes may be sanctioned. The a n a l y s i s of s y l l o g i s m s i s not the only v e h i c l e by which l o g i c a l reasoning has been assessed. The s o - c a l l e d 4-card s e l e c t i o n task has gained widespread a t t e n t i o n i n the past ten years, and the r o l e of the i n t e r p r e t a t i o n -performance model i s tr a c e d below, w i t h i n i t s context. Assessment using the 4~card S e l e c t i o n Task D e s c r i p t i o n . A t y p i c a l v e r s i o n of t h i s problem i s as f o l l o w s . The subject i s presented with four cards and i s t o l d that a l l cards have l e t t e r s on one side and numbers on the other s i d e . A c o n d i t i o n a l sentence of the from ' I f p then q' i s then presented as a r u l e which may or may not be true when a p p l i e d to the cards. The components p and q are t y p i f i e d by 1 10 p = 'there i s an A on one s i d e ' , and q = 'there i s a 3 on the other s i d e ' . Only one side of the four given cards i s a c c e s s i b l e to the respondent. Two l e t t e r s and two numbers are v i s i b l e . The l e t t e r s correspond to one case each of p and p and the numbers respresent one case each of q and q". In the s e l e c t i o n task, the subject i s required to s e l e c t a l l those cards and only those cards i t i s necessary to turn over i n order to determine whether the r u l e i s true or f a l s e . V a r i a t i o n s of t h i s task include a r e s t r i c t e d number of cards (Johnson-Laird & Wason, 1970b; Lunzer, Harrison & Davey, 1972; Lunzer, 1975; Roth, 1979), v a r i a t i o n s in content - that i s , using content other than symbols and p i c t u r e s of shapes (Wason & Shapiro, 1971; Johnson-Laird, Legrenzi & L e g r e n z i , 1972; G i l h o o l y & Falconer, 1974; Bracewell & H i d i , 1974; Van Duyne, 1974), v a r i a t i o n s i n the syntax and/or the type of sentence (Legrenzi, 1970; Johnson-Laird et a l . , 1972; Bracewell & H i d i , 1974; Van Duyne, 1974) and the wording of the task i t s e l f (see below). The c r i t e r i a f or c o r r e c t performance on t h i s task are two-fold. F i r s t , the c o n d i t i o n a l r u l e (and any others, i f used) i s expected to be i n t e r p r e t e d according to standard r u l e s of formal l o g i c - that i s , only the element p.g can f a l s i f y the r u l e . Experimenters d i f f e r on whether the remaining three elements should a l l v a l i d a t e the r u l e , or whether some may be i r r e l e v a n t to the v a l i d a t i o n of the r u l e . These d i f f e r e n c e s do not a f f e c t the c r i t e r i a f or correctness. The second c r i t e r i o n i s that only cards with which i t i s  p o s s i b l e to f a l s i f y the r u l e should be s e l e c t e d . Hence the 111 ' c o r r e c t ' s o l u t i o n i s to s e l e c t the cards for which p and c] are v i s i b l e . This 'cor r e c t ' s o l u t i o n has been s t i p u l a t e d by almost a l l authors who have used t h i s task, and r a t i o n a l i z e d by Johnson-Laird and Wason (1970a), Wason and Shapiro (1971), Evans (1972a), G i l h o o l y and Falconer (1974), Wason and Evans (1975), Evans and Wason (1976) and others. The r a t i o n a l e s given r e l y on the b e l i e f that the r u l e i s true unless i t can be proven f a l s e . The appropriateness of these c r i t e r i a and the nature of the task are discussed below, a f t e r performance with t h i s task i s described. Performance. Performance on t h i s task under the standard c o n d i t i o n s described above i s t y p i f i e d by the aggregation of the r e s u l t s of four experiments which have been summarized by Johnson-Laird and Wason (1970a). The r e s u l t s are shown in Table 9. I t may be seen that expectations and performance are Table 9 Summary of responses for the 4-card s e l e c t i o n task Cards s e l e c t e d p,q p p,q,e[ P r S 1 Others T o t a l Frequency 59 42 9 5 1 3 1 28 Percentage 46.1 32.8 7.0 3.9 10.2 100.0 1 'Correct' response h i g h l y d i s c r e p a n t , as with s y l l o g i s m tasks. Considerable e f f o r t s have been devoted to both r a t i o n a l i z i n g and r e c t i f y i n g t h i s i n c o n g r u i t y . The same format i s used i n the forthcoming d i s c u s s i o n of these e f f o r t s as was used in the d i s c u s s i o n of performance with s y l l o g i s m s . 1 12 The f i r s t group of f a c t o r s to be discussed are content and complexity. The ' l o g i c a l p r i n c i p l e s ' f a c t o r does not apply to the s e l e c t i o n task. The content f a c t o r i s much simpler i n the 4-card s e l e c t i o n task than i n s y l l o g i s m s , since i t includes only two dimensions. One dimension i s the a c t u a l content used: symbolic ( l e t t e r s , numbers, c o l o u r s , shapes, etc.) vs. r e a l i s t i c ( a c t u a l o b j e c t s or p i c t u r e s of o b j e c t s ) . The second dimension i s the type of r e l a t i o n s h i p between contingencies in the r u l e used: a r b i t r a r y (back vs. front of a card, one h a l f vs. the other h a l f of a card) vs. r e a l i s t i c ( r e a l - l i f e s i t u a t i o n s ) . Wason and Shapiro (1971), Johnson-L a i r d et a l . (1972), and Van Duyne (1974) conducted experiments i n which the standard 4-card s e l e c t i o n task was compared with r e a l i s t i c content. Results i n a l l three experiments showed that the use of r e a l i s t i c content improved r e s u l t s remarkably (using the above formal l o g i c c r i t e r i a ) . Subsequently, G i l h o o l y and Falconer (1974) and Bracewell and H i d i (1974) have noted that the two e a r l i e r experiments compared a r b i t r a r y r u l e s and symbolic content, with r e a l i s t i c r u l e s and r e a l i s t i c content. When the appropriate 2 x 2 experiments were c a r r i e d out, G i l h o o l y and Falconer concluded that the d i f f e r e n t i a l e f f e c t s of content were due to the a c t u a l content used (symbolic vs. r e a l i s t i c ) and not to the type of r e l a t i o n s h i p ( a r b i t r a r y vs. r e a l i s t i c ) . On the other hand, Bracewell and H i d i concluded that the e f f e c t s of content were due to the type of r e l a t i o n s h i p and not to the a c t u a l content used. These d i r e c t l y opposed f i n d i n g s were q u i t e unexpected, since content and r e l a t i o n s h i p s used i n these 1 1 3 studies were i d e n t i c a l . However, there were some minor d i f f e r e n c e s between the two s t u d i e s , the most s i g n i f i c a n t of which i s probably the f a c t that respondents were t o l d i n Bracewell and H i d i ' s study that the c o n d i t i o n a l sentence was not r e v e r s i b l e . This was not done i n the G i l h o o l y and Falconer study. V a l i d i t y of Bracewell and H i d i ' s study i s questionable not only because of t h i s f a c t , but a l s o because of the very low frequencies of p,q card s e l e c t i o n s reported i n t h e i r study. In f a c t , these two observations may be c a u s a l l y r e l a t e d . No more work has been found by the present author on t h i s content f a c t o r , but i t appears that the two dimensions of content incorporate u s e f u l d i s t i n c t i o n s . Nevertheless, the question remains unresolved. D i s c u s s i o n of the complexity f a c t o r i s r e s t r i c t e d mainly to e f f e c t s of negation. Syntax has been examined by Legrenzi (1970), Johnson-Laird and Wason (1970b), Johnson-L a i r d et a l . (1972), Bracewell and H i d i (1974) and Van Duyne (1974). Some researchers have used the r u l e ' I f p then q' i n t h e i r d i r e c t i o n s while others have used r u l e s of the form 'Every card which p has q', i m p l i c i t l y assuming that the d i f f e r e n t syntax d i d not a f f e c t r e s u l t s . F o r t u n a t e l y , d i r e c t comparisons have shown t h i s to be the case (Van Duyne, 1974). Roberge (1969, 1970a), Roberge and Paulus (1971) and Ryoti (1972) reported data from s y l l o g i s m s which supported t h i s view. Nevertheless, other l o g i c a l l y equivalent but s y n t a c t i c a l l y d i f f e r e n t expressions have y i e l d e d d i f f e r e n t r e s u l t s . Evans and Lynch (1973) conducted an exhaustive study of 1 1 4 e f f e c t s of negation i n the r u l e by examining performance with four r u l e s of the form ' I f p then q', ' I f p then g', ' I f p then q', and ' I f p then g'. The only other 4-card s e l e c t i o n task which the present author has found i n which negation was used, was a study by Wason and Evans (1975). They compared only the f i r s t two r u l e s above and the r e s u l t s of t h e i r comparisons are subsumed under the l a r g e r study by Evans and Lynch. I t was not the i n t e n t of Evans and Lynch to study e f f e c t s of negation, so much as to use negation as an o p e r a t i o n a l v a r i a b l e to t e s t the v i a b i l i t y of two r i v a l hypotheses. These hypotheses are c a l l e d the ' v e r i f i c a t i o n ' hypothesis and the 'matching' hypothesis. The v e r i f i c a t i o n hypothesis p r e d i c t s that respondents w i l l choose only cards which v e r i f y the r u l e (that i s , p and/or q cards for ' I f p then q', p and/or g cards for ' I f p then g', and so on). The matching hypothesis p r e d i c t s that respondents w i l l match the a t t r i b u t e s mentioned i n the r u l e (that i s , p and/or q for a l l r u l e s , independent of the p o s i t i o n of negation). Thus, the matching hypothesis i s completely a l o g i c a l . An examination of the 176 cards which were s e l e c t e d in t h e i r study revealed that 39% of s e l e c t i o n s were p r e d i c t e d by both hypotheses, 28% were p r e d i c t e d by matching but not by v e r i f i c a t i o n , 21% were p r e d i c t e d by v e r i f i c a t i o n but not by matching, and 12% (almost a l l l o g i c a l l y ' correct') were p r e d i c t e d by n e i t h e r hypothesis. The e f f e c t s of negation may now be described. Since ne i t h e r of the above hypotheses were re f u t e d and both were supported, the question becomes one concerning the r e l a t i v e frequencies of v e r i f i c a t i o n vs. matching (vs. l o g i c a l ) 1 15 s t r a t e g i e s for s o l u t i o n . In t h i s case, matching appears to be the more frequent s t r a t e g y , and for these respondents, negation has no e f f e c t whatsoever. For respondents who use a v e r i f i c a t i o n s t r a t e g y , negation (and i t s absence) determines p r e c i s e l y which cards are s e l e c t e d . No other s t r a t e g i e s appear to be known, to determine f u r t h e r e f f e c t s of negation. The second group of performance f a c t o r s to be discussed, are between-subjects f a c t o r s . There appears to be p r a c t i c a l l y nothing to report here, since researchers have been preoccupied with e x p l a i n i n g discrepant performance by developing a l t e r n a t e competence models, rather than examining performance f a c t o r s which a f f e c t the formal l o g i c competence model. One notable exception i s the ' i n s i g h t ' model (Johnson-L a i r d & Wason, 1970a) which i s discussed below. I t i s d i f f i c u l t to describe e f f e c t s of age since almost a l l experiments have been c a r r i e d out with c o l l e g e students or a d u l t s . However, the f o l l o w i n g s t u d i e s were not confined to these populations. O'Brien (1975) found almost e x c l u s i v e p,q card s e l e c t i o n s i n elementary school c h i l d r e n . Jansson (1978) found c l e a r developmental trends for c o r r e c t responses over grades 8, 10 and 12, but h i s sc o r i n g system appeared to be somewhat i d i o s y n c r a t i c , so d i r e c t comparisons with the standard task with a d u l t s may not be p o s s i b l e . Further, Lunzer, Harrison and Davey (1972) used grade 12 g i r l s i n t h e i r study, but t h e i r procedures were a l s o non-standard, so d i r e c t comparisons with other stud i e s are a l s o tenuous. The only other study which has been found which i n v e s t i g a t e d between-subject f a c t o r s was Wason (1969) who 116 found no sex d i f f e r e n c e s . F o l l o w i n g precedent in the d i s c u s s i o n of s y l l o g i s m performance, the t h i r d and f i n a l group of f a c t o r s i s those which may a f f e c t the v a l i d i t y of i n t e r p r e t a t i o n s of s e l e c t i o n task performance. Two f a c t o r s appear to cause most d i f f i c u l t y . These are, ambiguity of the task and ambiguity of the r u l e . I t i s argued below that both ambiguities i n v a l i d a t e experimental conclusions for the 4-card s e l e c t i o n task. The 4-card s e l e c t i o n task i s e s s e n t i a l l y one which may be l i k e n e d to the e f f i c i e n t t e s t i n g of a hypothesis (the given r u l e ) . The purpose i n s e l e c t i n g cards has been v e r b a l i z e d to respondents i n the f o l l o w i n g ways (among other minor v a r i a t i o n s ) : "... i n order to f i n d out whether the sentence i n f r o n t of you i s true or f a l s e . " (Wason, 1969) "... to prove t h i s . " (Wason & Johnson-Laird, 1969) "... i n order to f i n d out whether they v i o l a t e the r u l e or not", (Van Duyne, 1974) "... i n order to f i n d out, once and f o r a l l , whether the sentence i s untrue." (O'Brien, 1975) I t may be seen that some d i r e c t i o n s emphasize e s t a b l i s h i n g the v a l i d i t y of the r u l e , others emphasize e s t a b l i s h i n g the f a l s i t y of the r u l e , and others (the most common) i n d i c a t e that e i t h e r case w i l l s u f f i c e . I t may be claimed that these two cases amount to the same t h i n g , since any v e r i f i c a t i o n of a r u l e i s "vacuous" (Wason & Evans, 1975), thereby l e a v i n g f a l s i f i c a t i o n as the only l e g i t i m a t e s t r a t e g y . However, i t appears that any cla i m that i t i s impossible to v e r i f y a c o n d i t i o n a l r u l e (confirm a hypothesis) r e s t s on e i t h e r (or both) of two assumptions. One assumption i s that some members 1 1 7 of the set of instances where a r u l e may or may not apply, are i n a c c e s s i b l e . For example, an experiment may be c a r r i e d out i n f i n i t e l y many times, and hence a f i n i t e number of confirming instances i s i n s u f f i c i e n t to prove that a hypothesis i s always t r u e . The other assumption i s that the necessary absence of d i s c o n f i r m i n g instances i s s u f f i c i e n t to v a l i d a t e a r u l e (a c r i t e r i o n of formal l o g i c ) . I t may be argued however, that n e i t h e r of these assumptions obtain in a t y p i c a l s e l e c t i o n task. In the f i r s t case, the set of instances of the given r u l e i s confined to the ( f i n i t e ) set of cards i n f r o n t of the respondents. Hence, the domain of the r u l e i s e x h a u s t i b l e . In the second case, respondents may require the necessary  presence of the element p.q to v a l i d a t e the r u l e ' I f p then q' (Strawson, 1952). I f t h i s - i s the case, these respondents would need to s e l e c t both the p and q cards i n order to t e s t the v a l i d i t y of the r u l e . Hence, v a l i d a t i o n i s c e r t a i n l y p o s s i b l e on a t h e o r e t i c a l l e v e l at l e a s t (and hence, may not be "vacuous"), and i t may in f a c t be necessary. This e s t a b l i s h e s the ambiguity of i n s t r u c t i o n s f o r the task, since the four purposes quoted above may not be perceived as i d e n t i c a l in i n t e n t . I t may be countered that t h i s ambiguity may be inconsequential on a p r a c t i c a l l e v e l . However, Johnson-Laird and Wason (1970b) gave two sets of i n s t r u c t i o n s f o r t h e i r s e l e c t i o n t a s k s . One set required s e l e c t i o n of cards which proved the r u l e t r u e . The other set required s e l e c t i o n s of cards which proved the r u l e f a l s e . Very d i f f e r e n t patterns of s e l e c t i o n s were found between these two sets of i n s t r u c t i o n s . 118 This s t r o n g l y suggests that d i f f e r e n t r e s o l u t i o n s of the above task ambiguity y i e l d s d i f f e r e n t r e s u l t s . The second f a c t o r which may a f f e c t v a l i d i t y i s ambiguity of the r u l e . A t y p i c a l r u l e i s , ' I f a card has an A on one side then i t has a 3 on the other s i d e ' . The respondent would u s u a l l y have four cards i n view, with faces (say) A, D, 2 and 3 showing. I f the respondent i n t e r p r e t s the term 'one s i d e ' as the side i n view, and 'the other s i d e ' as the side which i s not i n view, the only card for which the antecedent of the r u l e i s (or can be) s a t i s f i e d , i s the f i r s t (p) card. Hence, the f i r s t card would be the only card which may e i t h e r v a l i d a t e or f a l s i f y the r u l e (assuming a c o n d i t i o n a l i n t e r p r e t a t i o n of the r u l e ) . In g e n e r a l , i f r u l e ambiguity i s i n f a c t resolved i n the d i r e c t i o n of the above one side / other side d i s t i n c t i o n , greater frequencies of s e l e c t i o n s would be expected for s e l e c t i o n of the p card only. In f a c t , almost a t h i r d of respondents t y p i c a l l y respond i n t h i s way (see Table 9, page 111). Wason and Johnson-Laird (1970) tes t e d the e f f e c t s of t h i s ambiguity by comparing the standard form of the task with one i n which the one side / other side d i s t i n c t i o n was l e s s l i k e l y to hold. This was accomplished by having a l l symbols on only one face of a l l cards i n the experimental c o n d i t i o n . They compared frequencies of ' c o r r e c t ' responses for each task and concluded that there was no d i f f e r e n c e between the two tasks. U n f o r t u n a t ely, t h i s i s a mistake, since they d i d not t e s t the s e l e c t i o n s which the one side / other side hypothesis p r e d i c t s . The ' c o r r e c t ' responses which they used (p,2[) are 1 19 not the same as the p r e d i c t i o n of the hypothesis.(p o n l y ) . Hence, the one side / other side hypothesis was not tested by t h e i r procedures, contrary to t h e i r claims otherwise. Further, the c r i t i c a l p s e l e c t i o n s were not reported, so i t was impossible for the present author to t e s t the one side / other side hypothesis with t h e i r data alone. However, Bree and Coppens (1976) reported a study which used the second of Wason and Johnson-Laird's (1970) tasks, and an examination of t h e i r r e s u l t s revealed a very low incidence of p cards only. This supported the above hypothesis. Of the studies which have used symbols on both sides of the cards, only Legrenzi (1970) has obtained low frequencies for only p s e l e c t i o n s , whereas Wason (1968, 1969), Evans and Lynch (1973) , G i l h o o l y and Falconer (1974), Bracewell and H i d i (1974) , Van duyne, (1974) and others a l l obtained frequencies comparable with that found i n Table 9, i n accordance with the one side / other side hypothesis. As f u r t h e r support for t h i s hypothesis, when the p card was removed from the array of cards presented to the respondents, performance improved markedly (Johnson-Laird & Wason, 1970b; Lunzer et a l . , 1972; Lunzer, 1975; Roth, 1979). Lunzer (1975) a l s o showed that subsequent performance by the same subjects d i d improve, when a l l four cards were a v a i l a b l e . The reduced frequencies for p card s e l e c t i o n s may be a t t r i b u t e d to the observation that under r e s t r i c t e d c o n d i t i o n s , there was no ambiguity to res o l v e . In consequence, i t must be concluded that the one side / other side hypothesis i s at worst unresolved, and at best, very compelling. 1 20 In c o n c l u s i o n , two sources of ambiguity have been i d e n t i f i e d which suggest that performance on the 4-card s e l e c t i o n task depends h e a v i l y upon how these ambiguities are resolved by respondents. Any adequate competence model must be based on r e s u l t s which can be as c e r t a i n e d to be free of these a m b i g u i t i e s . This concludes the d i s c u s s i o n of f a c t o r s which a f f e c t performance with the 4-card s e l e c t i o n task. As with s y l l o g i s m t a s k s , i n t e r v e n t i o n i s not considered here. Consideration of the above f a c t o r s have l e d to the f o l l o w i n g r a t i o n a l i z a t i o n s of behaviour with t h i s task. Rationales for Performance. The purpose of t h i s s e c t i o n i s to present current models f o r the 4-card s e l e c t i o n task. The dominant, model i s based upon formal l o g i c . Arguments are presented to weaken the c r e d i b i l i t y of t h i s model and to support c o n s i d e r a t i o n of va r i o u s i n t e r p r e t a t i o n s of premises. The most dominant model which has been used to r a t i o n a l i z e 4-card s e l e c t i o n s was postula t e d by Johnson-Laird and Wason (1970a) and supported by Johnson-Laird and Wason (1970b), Wason and Johnson-Laird (1970), Wason and Shapiro (1971), Johnson-Laird et a l . (1972), Lunzer et a l . (1972), G i l h o o l y and Falconer (1974), Bracewell and H i d i (1974), Van Duyne (1974), Roth (1979) and to some extent by Wason and Evans (1975). In t h e i r ' i n s i g h t ' model, three stages of i n s i g h t were suggested. Respondents with no i n s i g h t merely s e l e c t e d cards which v e r i f i e d the given r u l e (p and/or q ) . Respondents with ' p a r t i a l i n s i g h t ' s e l e c t e d a l l cards which e i t h e r v e r i f i e d the r u l e or f a l s i f i e d the r u l e (p, q and eg), 121 while ' t o t a l i n s i g h t ' enabled respondents to recognize that i t was s u f f i c i e n t to s e l e c t only f a l s i f y i n g cards (p and £[) . Within t h i s context, a number of st u d i e s have been conducted to determine f a c t o r s which increase i n s i g h t . Thus, Johnson-L a i r d and Wason (1970b), Lunzer et a l . (1972), Lunzer (1975) and Roth (1979), have claimed that a reduced array of cards for s e l e c t i o n w i l l a i d i n s i g h t . Legrenzi (1971) has claimed that having respondents discover the r u l e for themselves increases i n s i g h t (although i t was very rare that the discovered r u l e was v e r b a l i z e d as a c o n d i t i o n a l ) . Lunzer et a l . (1972) concluded that p r i o r f a m i l i a r i t y with the cards f a c i l i t a t e d i n s i g h t (thereby subsuming Legrenzi's r e s u l t somewhat). F i n a l l y , Wason and Shapiro (1971), Johnson-Laird et a l . (1972), Lunzer et a l . (1972), G i l h o o l y and Falconer (1974), Bracewell and H i d i (1974) and Van Duyne (1974) have demonstrated that 'thematic' m a t e r i a l s (that i s , everyday f a m i l i a r content and/or r e l a t i o n s h i p s ) d r a m a t i c a l l y improve i n s i g h t . The strongest r i v a l hypothesis i s Evans' (1972b) 'matching' hypothesis, which was p o s t u l a t e d using a d i f f e r e n t task (Evans, 1972c). This hypothesis suggests that values of cards are s e l e c t e d only on the b a s i s that they match the a t t r i b u t e s given i n the r u l e (that i s , p and/or q, independent of negation). I t may appear that t h i s hypothesis challenges the e n t i r e ' i n s i g h t ' model, since i t p r e d i c t s almost 80% of responses (see Table 9, page 111). However, i t may be seen that t h i s hypothesis challenges only the 'no i n s i g h t ' part of the ' i n s i g h t ' model, i n which i t i s supposed that respondents 122 are v e r i f y i n g the r u l e rather than matching values. Evans has suggested that the tendency to match a t t r i b u t e s i s most l i k e l y to occur when the subject i s i n doubt. In a f a i r l y d e f i n i t i v e t e s t of these two hypotheses, Evans and Lynch (1973) found more support for the matching hypothesis than for the v e r i f i c a t i o n part of the ' i n s i g h t ' model. In any case, most of the ' i n s i g h t ' model remains i n t a c t . Wason and Evans (1975) sought i n t r o s p e c t i v e evidence to determine which hypothesis was more v i a b l e , but Evans and Wason (1976) demonstrated that post hoc i n t r o s p e c t i o n s were mostly ad hoc r a t i o n a l i z a t i o n s of performance, and hence were u n r e l i a b l e i n d i c a t o r s of s t r a t e g i e s . Other models which have been suggested include a s t o c h a s t i c model (Evans, 1977b) and an information-processing model (Bree, 1973; Bree & Coppens, 1976), but they do not seem to have gained f u r t h e r support. Thus, the question remains whether the ' i n s i g h t ' model i s as v a l i d as i t appears. On an e m p i r i c a l l e v e l , i t s c r e d i b i l i t y has been weakened somewhat by two f i n d i n g s . F i r s t , Johnson-Laird (1970), Johnson-Laird and Wason (1970b), Goodwin and Wason (1972), and Van Duyne (1974) have found that complete i n s i g h t , once e s t a b l i s h e d , was not s t a b l e . Second, A d i , Karpulus, and Lawson (1980) found that even when ' c o r r e c t ' responses were given, they often could not be j u s t i f i e d by respondents. On a t h e o r e t i c a l l e v e l , the i n t e r n a l v a l i d i t y of the i n s i g h t model may be questioned i n two ways. F i r s t , i t i s based upon performance data which could be s i g n i f i c a n t l y contaminated by the two ambiguities c i t e d e a r l i e r . Second, i t i s assumed that 1 23 performance i s based upon a c o n d i t i o n a l i n t e r p r e t a t i o n of the r u l e ' I f p then q'. This i s a v i t a l c o n s i d e r a t i o n since a b i c o n d i t i o n a l i n t e r p r e t a t i o n (for example) renders a l l four cards (p, q, p and cj) capable of both v a l i d a t i o n and f a l s i f i c a t i o n . For such a respondent, the i n s i g h t model would p r e d i c t s e l e c t i o n s only of a l l four cards, f o r a l l l e v e l s of i n s i g h t . I f a s i g n i f i c a n t number of respondents do in f a c t have a b i c o n d i t i o n a l i n t e r p r e t a t i o n , the p r e d i c t i o n s of the i n s i g h t model would be f a r from consonant with e m p i r i c a l evidence. The next s e c t i o n demonstrates that t h i s i s indeed the case. This weakens the i n s i g h t model which i s based l a r g e l y upon formal l o g i c c r i t e r i a , and opens the way for models which are based on d i f f e r e n t i n t e r p r e t a t i o n s of premi ses. Legrenzi (1970), Evans (1972b), O'Brien (1975), Bree and Coppens (1976), Staudenmayer and Bourne (1978) and Roth (1979) have a l l a t t e s t e d to the c r u c i a l r o l e that the i n t e r p r e t a t i o n of premises plays i n the r a t i o n a l i z a t i o n of performance i n the 4-card s e l e c t i o n task. Bree (1973) and Bree and Coppens (1976) have taken account of the two main i n t e r p r e t a t i o n s of c o n d i t i o n a l sentences i n t h e i r model, as w e l l as attempting to accomodate r u l e ambiguity (as evinced by t h e i r 'Strategy A'). Thus, for both s y l l o g i s m tasks and the 4-card s e l e c t i o n t a s k s , models based on formal l o g i c have given way to models in which i n t e r p r e t a t i o n s of premises play a c r u c i a l r o l e . The means by which i n t e r p r e t a t i o n s have been measured, and the consequences of the r e s u l t s are discussed below. 1 24 I n t e r p r e t a t i o n s of Premises The d i s c u s s i o n above re v e a l s that d i s s a t i s f a c t i o n with norms of formal l o g i c has l e d to the p o s t u l a t i o n of a l t e r n a t e models i n which at l e a s t some norms of formal l o g i c have been suspended. The dominant theme i n these a l t e r n a t e schemes has been r e c o g n i t i o n that the same l o g i c a l expression may have d i f f e r e n t meanings (or understandings) f o r d i f f e r e n t people. However, even w i t h i n the same meaning, there i s strong evidence to suggest that l o g i c a l i t y 1 between people may d i f f e r . Hence, the use of a s i n g l e paradigm (such as syl l o g i s m s or a 4-card s e l e c t i o n task) i n e v i t a b l y confounds l o g i c a l i t y and "understandings (or meanings). In support of t h i s c l a i m , Smedslund (1970) has noted that Procedures for determining l o g i c a l i t y presuppose understanding, and procedures for determining understanding presuppose l o g i c . One can escape from t h i s c i r c l e only by presupposing l o g i c a l i t y i n agreement with common-sense t h i n k i n g . C e r t a i n l y , i t has been e s t a b l i s h e d that the a l t e r n a t i v e . (presupposing understanding) i s e m p i r i c a l l y untenable. Two responses are i n order. F i r s t , the present author has been unable to f i n d what i s meant by the above term ' l o g i c a l i t y ' , or even what i s meant by the term 'to be l o g i c a l ' ( T a p l i n et a l . , 1974). A c c o r d i n g l y , since formal l o g i c seems to be inadequate i n t h i s respect, both of these terms w i l l be considered to be synonymous with what has been defined i n the present study as 'concordance'. With t h i s meaning, l o g i c a l i t y may be assessed, independent of 1The meaning of the term ' l o g i c a l i t y ' i s c l a r i f i e d i n the next paragraph. 125 understanding of premises. Second, the present author does not b e l i e v e that i t i s necessary to presuppose l o g i c a l i t y i n order to determine meanings of p r o p o s i t i o n s . This i s the purpose of the present s e c t i o n - to describe the procedures and r e s u l t s of the task known as the Eva l u a t i o n task. I t i s a means by which meanings of p r o p o s i t i o n s may be determined d i r e c t l y , without presupposing l o g i c a l i t y . The E v a l u a t i o n task has been b r i e f l y discussed above, w i t h i n the context of r a t i o n a l e s for s y l l o g i s m performance. Respondents are presented with representations ( u s u a l l y on cards) of each of the four elements p.q, p.q", p.q and p.q", and a l o g i c a l expression ( u s u a l l y a c o n d i t i o n a l ) . Each card i s then evaluated i n t u r n , to determine whether the respondent considers that the card makes the p r o p o s i t i o n true or f a l s e , or whether the card i s i r r e l e v a n t to the t r u t h or f a l s i t y of the expression. This task appears to have f i r s t been used by Wason (1966, 1968) and Wason and Johnson-Laird (1970), but i t was only at an informal l e v e l , and intended as a form of i n t e r v e n t i o n i n the 4-card s e l e c t i o n task. I t appears to have been f i r s t f o rmalized by Johnson-Laird and Tagart (1969). They required 24 undergraduates to evaluate cards f o r each of four l o g i c a l l y equivalent forms of the c o n d i t i o n a l sentence 'I f p then q'. The r e s u l t s showed that the assumption of l o g i c a l equivalence was completely untenable (see a l s o Legrenzi, 1970), but they are not elaborated here. Of more immediate concern was that 19 of 24 students evaluated the p.q, p.cj, p.q and p.g elements as True, F a l s e , I r r e l e v a n t and 1 26 I r r e l e v a n t r e s p e c t i v e l y , for the c o n d i t i o n a l sentence ' I f p then q'. Only one student used the formal l o g i c e v a l u a t i o n s of True, F a l s e , True, True. In a l l reported cases, a l l students considered that only the element p.g f a l s i f i e d the r u l e . These and f u r t h e r r e s u l t s for the e v a l u a t i o n task which were obtained by Legrenzi (1970), and Bree and Coppens (1976) are reported i n Table 10. Other eva l u a t i o n tasks have been Table 10 Evaluations of binary elements for ' I f p then q' Authors Binary Element 1 p.q p.g p.q p.g Johnson-Laird & Tagart (1969) Legrenz i (1970) Bree and Coppens (1976) T T T T T I r r I r r T T F T F I r r F F Other 19 1 5 0 22 0 3 0 9 _ 2 1 1 4 T o t a l 24 30 24 1T = True, F = F a l s e , I r r = I r r e l e v a n t 2'-' = Not reported c a r r i e d out (Ward, 1972; P a r i s , 1973; Evans & Newstead, 1977; Moshman, 1977, 1979; Damarin, 1977b; Marcus & Rips, 1979) but d i f f e r e d e i t h e r because of i n t e r v e n t i o n before e v a l u a t i o n , or because d i f f e r e n t response options were used, or because i n t r a s u b j e c t performance was not reported i n the format used i n Table 10. The three s t u d i e s i n t h i s t a b l e r e v e a l wide d i s c r e p a n c i e s i n r e s u l t s , at l e a s t i n part a t t r i b u t a b l e to d i f f e r e n t content. S i m i l a r content e f f e c t s were found by Marcus and Rips (1979). However, i t i s c l e a r that many 1 27 respondents considered that only the element p.cj f a l s i f i e d ' I f p then q', while many others considered that the element p.q a l s o f a l s i f i e d t h i s p r o p o s i t i o n . This l a t t e r f i n d i n g renders the ' i n s i g h t ' model untenable i n i t s present form, since i t p r e d i c t s the s e l e c t i o n of a l l four cards f o r a l l l e v e l s of i n s i g h t , f o r most of these respondents. A s i m i l a r task (the Construction task) has been reported by Wason and Shapiro (1971) a l s o as a form of i n t e r v e n t i o n i n the 4-card s e l e c t i o n task. I t was formalized by Evans (1972a), who required respondents to construct (representations of) elements which made the given r u l e True or F a l s e . I t was t h i s task for which Evans proposed h i s o r i g i n a l 'matching' hypothesis, and l a t e r t e s t e d i t on the 4-card s e l e c t i o n task (Evans & Lynch, 1973). The 'matching' hypothesis suggests a performance strategy which i s independent of the meaning of the p r o p o s i t i o n . I f i t i s t r u e , i t i n v a l i d a t e s the r e s u l t s of the Eva l u a t i o n task for a l l respondents who use a 'matching' s t r a t e g y . However, the present author has not found any evidence that t h i s s t r a t e g y has been t e s t e d with an Evaluati o n task. I t i s mentioned here because the Construction task and the E v a l u a t i o n task are q u i t e s i m i l a r and are used for the same purpose (and they may e a s i l y be confused). I t was considered important to e s t a b l i s h that the 'matching' hypothesis does not appear to have been shown to i n v a l i d a t e the Eva l u a t i o n task (as Evans, 1977a, apparently claimed). Within the context of the present study, the Eva l u a t i o n task may be considered to be a R e f e r e n t i a l Test. The s e r i a l 1 28 presentation of s i n g l e cards i n the Ev a l u a t i o n task corresponds ( i n the present study) to the presentation of r e f e r e n t i a l sets which correspond to binops 1, 2, 4 and 8 (see Table 4, page 25), with one exemplar of each binary element for each s e t . Hence, the R e f e r e n t i a l Test i n the present study i s a d i r e c t and n o n - t r i v i a l extension of the Evaluation task, i n c o r p o r a t i n g what Piaget c a l l s the 's t r u c t u r e d whole' of the binary operations. I t i s rare that any c r i t e r i a have been a p p l i e d in previous research i n the Eva l u a t i o n task, to enhance v a l i d i t y and r e l i a b i l i t y . However, P a r i s (1973) has incorporated an a t t r i b u t e i d e n t i f i c a t i o n task as a screening procedure, and Moshman (1979) and Marcus and Rips (1979) used s t a t i s t i c a l consistency c r i t e r i a as p r o t e c t i o n against random responding and systematic b i a s . The present study incorporates these and other procedures to optimize v a l i d i t y and r e l i a b i l i t y (see Chapter 3). The v i a b i l i t y of the Evaluati o n task r e s t s with the inf l u e n c e that i n f e r r e d meanings of l o g i c a l expressions have upon each i n d i v i d u a l ' s performance on some r e l a t e d task, such as with s y l l o g i s t i c reasoning or the 4-card s e l e c t i o n task. This i s the r o l e of the C o m p a t i b i l i t y c o n s t r u c t s used i n the present study. Two important c r i t e r i a need to be met i n order to evaluate C o m p a t i b i l i t y . F i r s t , an Ev a l u a t i o n task and a reasoning task both need to be administered to the same population sample. Second, i n t r a s u b j e c t (as opposed to i n t r a t a s k ) performance needs to be reported and c l a s s i f i e d . This means that a framework i s required whereby each subject's r e s u l t s with both the Ev a l u a t i o n task and the reasoning task 129 may be compared. In the present study, C o m p a t i b i l i t y has been c l a s s i f i e d i n t o four cases, l a b e l l e d Case 1, Case 2, Case 3 and Case 4 (see Chapter 1). These are i n order, from the most compatible to the l e a s t compatible. The i n f l u e n c e that meanings of p r o p o s i t i o n s have upon reasoning tasks i s as f o l l o w s . The present author has found two studies in which an Eva l u a t i o n task has been administered with a 4-card s e l e c t i o n task. O'Brien (1975) administered three tasks to respondents. The f i r s t was a Construction task for True cases only. The second was an Evaluat i o n task for False cases only. The t h i r d was a 4-card s e l e c t i o n task i n which each of p and q could be f a l s i f i e d i n two d i f f e r e n t ways. The l i m i t e d requirements of the f i r s t two tasks rendered them d i f f i c u l t to g e n e r a l i z e , but the most u s e f u l i n t r a s u b j e c t responses were reported i n h i s Experiment I I I . An examination of h i s Table I I I (p. 33) revealed that the Ev a l u a t i o n task e l i c i t e d f i v e d i f f e r e n t response patterns of cards which f a l s i f i e d the given c o n d i t i o n a l r u l e . I f i t i s assumed that each respondent used a l l of his / h e r s p e c i f i e d f a l s i f y i n g cards only, and was attempting only to f a l s i f y the given r u l e , f u r t h e r examination of Table I I I revealed that 6 out of 35 undergraduates gave compatible responses, while the remainder d i d not. The data are summarized i n Table 11. They suggest that most respondents are not attempting to only f a l s i f y the r u l e . I t i s c l e a r that performance i n the s e l e c t i o n task i s influenced by i n t e r p r e t a t i o n of the r u l e , as measured by f a l s i f y i n g instances only. This i s seen from the d i f f e r e n t p r o f i l e s of 1 30 Table 11 Comparison of 4-card s e l e c t i o n s with cards chosen as f a l s i f y i n g S e l e c t i o n s F a l s i f y i n g cards P p,q,g p,g q q,p,g A l l None To t a l p.g 4 7 4 3 1 1 0 0 0 19 p.g,p.q 2 0 1 0 0 0 O1 0 3 p.q 1 1 0 0 0 0 0 0 0 1 p.q,p.g,p.g 5 1 0 0 0 1 2 1 1 10 p.q,p.g 0 0 0 0 1 O1 0 0 1 None 0 0 0 0 0 0 1 O1 .1 T o t a l 1 2 8 5 3 2 1 3 1 35 C o m p a t i b l e , assuming a f a l s i f i c a t i o n strategy s e l e c t i o n s f or d i f f e r e n t ' f a l s i f y i n g ' i n t e r p r e t a t i o n s . Bree and Coppens (1976) a l s o administered an Evaluati o n task and a 4-card s e l e c t i o n task to the same popu l a t i o n . In the E v a l u a t i o n task, only two patterns of responses were reported. They were what they c a l l e d " i l l a t i v e i m p l i c a t i o n " (True, F a l s e , I r r e l e v a n t , I r r e l e v a n t for p.q, p.g, p.q and p.g r e s p e c t i v e l y ) and the "converse i m p l i c a t i o n " ( a c t u a l l y a mis-named type of equivalence, rather than the converse of i m p l i c a t i o n , with True, F a l s e , F alse and I r r e l e v a n t responses, r e s p e c t i v e l y ) . These were roughly equivalent to the f i r s t and second rows i n Table 11 above. Bree and Coppens' Table 2 (p. 583) i s summarized i n Table 12, using more standard names for E v a l u a t i o n s . An examination of t h i s t a b l e reveals that the i n t e r p r e t a t i o n of the r u l e s t r o n g l y i n f l u e n c e d performance on the 4-card s e l e c t i o n task. Thus, the inf l u e n c e of i n d i v i d u a l i n t e r p r e t a t i o n of premises has been e s t a b l i s h e d at l e a s t f or the 4-card s e l e c t i o n task. Less may be s a i d about s y l l o g i s m t a s k s . Kuhn 131 Table 12 Comparison of 4-card s e l e c t i o n s with e v a l u a t i o n s of cards S e l e c t i o n s Evaluation p,q p p,q,q" p,q" P,P T o t a l C o n d i t i o n a l 11 0 0 0 1 0 11 ( T , F , I r r , I r r ) Equivalence 2 1 2 3 1 9 (T,F,F,Irr ) Other 4 0 0 0 0 4 To t a l 17 1 2 3 1 24 'Compatible, assuming f a l s i f i c a t i o n s t rategy (1977) conducted an unusual E v a l u a t i o n task i n her Experiment 3. A r e f e r e n t i a l set which corresponded to binop 11 was presented to c h i l d r e n from grades 2, 4, 6 and 8. This i s the only study the present author has found i n which the h o l i s t i c nature of any of the binops was used i n a R e f e r e n t i a l Test. In the present study, r e f e r e n t i a l sets which correspond to a l l 16 binops were presented. Kuhn's s y l l o g i s m t e s t was conducted i n the usual way, but although the format and s t y l e were patterned a f t e r T a p l i n et a l . (1974), i n t r a s u b j e c t response patterns were not der i v e d . Instead, s c o r i n g was computed according to norms of formal l o g i c . C o r r e l a t i o n s (a q u a n t i t a t i v e measure of i n t r a s u b j e c t c o m p a t i b i l i t y ) between the e v a l u a t i o n task and the s y l l o g i s m task revealed a s i g n i f i c a n t c o r r e l a t i o n at grade 8 only. Q u a l i t a t i v e r e s u l t s were not reported. Marcus and Rips (1979) appear to have conducted the only other study i n which both an Ev a l u a t i o n task and a s y l l o g i s m task were administered to the same su b j e c t s . Three d i f f e r e n t contents were used, and the pervasive i n f l u e n c e of 1 32 content was apparent for both t a s k s . Both p and q could be f a l s i f i e d i n two d i f f e r e n t ways. In the Evaluat i o n task, only two response options were o f f e r e d , but i t was s t i l l p o s s i b l e to c l a s s i f y responses according to a c o n d i t i o n a l or a b i c o n d i t i o n a l i n t e r p r e t a t i o n . I n t r a s u b j e c t response patterns were reported for the s y l l o g i s m t e s t using a weakened form of the c r i t e r i a developed by T a p l i n et a l . (1974). However, i n t r a s u b j e c t c o m p a t i b i l i t y was not reported, nor was i t r e a d i l y able to be i n f e r r e d from the data. For example, i n Experiment 1, 'machine' content, 40.7% of subjects gave a c o n d i t i o n a l response i n the E v a l u a t i o n task, and 27.8% of subjects gave a c o n d i t i o n a l response i n the s y l l o g i s m t e s t . I t was impossible to determine how many subjects gave both responses. The present author has been unable to f i n d any studies i n which i n t r a s u b j e c t c o m p a t i b i l i t y between an Evaluat i o n task ( i n t e r p r e t a t i o n ) and a s y l l o g i s m task (performance) has been reported. This i s one purpose of the present study. I n t e r p r e t a t i o n of premises was measured by the R e f e r e n t i a l t e s t . Reasoning performance was measured by the Sy l l o g i s m t e s t . I n t r a s u b j e c t c o m p a t i b i l i t y was a l s o reported. This concludes the c e n t r a l d i s c u s s i o n of the present chapter. The overview which f o l l o w s i s designed to provide f u r t h e r t h e o r e t i c a l support for the constructs of the present study, and to place them i n a model-building p e r s p e c t i v e . 1 33 Overview The assessment of l o g i c a l reasoning has been discussed above w i t h i n two common paradigms. They are, s y l l o g i s m tasks and the 4-card s e l e c t i o n task. Discussion has been r e s t r i c t e d mainly to c o n d i t i o n a l p r o p o s i t i o n s . Within both paradigms, d i s s a t i s f a c t i o n with the current competence model (formal l o g i c ) has l e d to a suspension of at l e a s t some of the norms of t h i s model, and a re-examination of some of i t s underlying assumptions. In both cases, each i n d i v i d u a l ' s i n t e r p r e t a t i o n of the given l o g i c a l expressions has emerged as a fundamental determinant of performance, thereby supporting the model i n the present study on an e m p i r i c a l l e v e l . Falmagne (1975) c a l l e d f o r t h i s type of model when she observed that ... a more d i a l e c t i c scheme seems c a l l e d f o r , i n which language comprehension and l o g i c a l reasoning are seen as a l t e r n a t i v e p erspectives on the same object rather than complementary ways of s l i c i n g i t . The r e s u l t a n t experimental paradigm to measure i n t e r p r e t a t i o n (the Evaluat i o n task) has been supported on a t h e o r e t i c a l l e v e l by the d i s t i n c t i o n between i n t e r p r e t a t i o n a l f a c t o r s and o p e r a t i o n a l f a c t o r s (Evans, 1972b, 1977a; Staudenmayer, 1975; Evans & Newstead, 1977; Roberge, 1978; Roth, 1979). ' I n t e r p r e t a t i o n a l ' r e f e r s to i n t e r p r e t a t i o n s of premises or r u l e s , while ' o p e r a t i o n a l ' r e f e r s to operations required by the task (Evans, 1977a). Staudenmayer (1975) c a l l e d the o p e r a t i o n a l f a c t o r s ' e v a l u a t i o n ' , meaning ev a l u a t i o n of conclusions i n sy l l o g i s m s and not to be confused with the Ev a l u a t i o n task. The main i n t e r p r e t a t i o n a l f a c t o r s which have been i s o l a t e d w i t h i n the framework of t h i s 134 d i s t i n c t i o n are content and semantic r e l a t i o n of the l o g i c a l connective (Evans, 1972b; Staudenmayer, 1975), syntax (Staudenmayer, 1975; Evans, 1977a; Evans & Newstead, 1977) and temporal order of component p r o p o s i t i o n a l f unctions (Evans & Newstead, 1977). The main o p e r a t i o n a l f a c t o r s which have e i t h e r been i s o l a t e d or discussed are response bias (Evans, 1972b, 1977a; Staudenmayer, 1975; Evans & Newstead, 1977), nature and/or i n t e r p r e t a t i o n of the task (Staudenmayer, 1975; Evans & Newstead, 1977; Roth, 1979) and negation (Evans, 1977; Evans & Newstead, 1977). Roberge (1978) appears to have considered negation to be an i n t e r p r e t a t i o n a l f a c t o r . Other o p e r a t i o n a l f a c t o r s have a l s o been mentioned, such as response p r e d i s p o s i t i o n (Staudenmayer, 1975) and information-processing l i m i t a t i o n s (Roth, 1979). The d i s t i n c t i o n between i n t e r p r e t a t i o n a l f a c t o r s and op e r a t i o n a l f a c t o r s i s not an e m p i r i c a l one but a semantic one. However, Evans (1977a) has suggested that each set of f a c t o r s may be i n d e n t i f i e d e m p i r i c a l l y by g e n e r a l i z i n g i t s e f f e c t s over v a r i a t i o n s of the other set of f a c t o r s . I t i s tempting to draw an analogy between the i n t e r p r e t a t i o n a l / o p e r a t i o n a l d i s t i n c t i o n above, and the competence / performance d i s t i n c t i o n espoused i n l i n g u i s t i c s (Chomsky, 1965) and more r e c e n t l y in genetic epistemology ( F l a v e l l & W o h l w i l l , 1969; Neimark, 1975, 1979; Neimark & Chapman, 1975; Moshman, 1977). Neimark (1975) has c h a r a c t e r i z e d t h i s d i s t i n c t i o n as 135 ... the d i s t i n c t i o n between fundamental d e f i n i n g p r o p e r t i e s and c h a r a c t e r i s t i c s of a state (competence) versus the d e f i n i n g p r o p e r t i e s being i n f e r r e d from the e f f e c t s of s p e c i f i c experimental c o n d i t i o n s (performance). Piaget's Formal Operations are considered to be a competence model, abstracted from many contexts and tasks, and not n e c e s s a r i l y e m p i r i c a l l y demonstrable free of performance c o n s t r a i n t s . Examples of performance f a c t o r s which Neimark (1975) suggested are content, c l a r i t y of i n s t r u c t i o n s , and i n d i v i d u a l d i f f e r e n c e s such as sex, mental age and c o g n i t i v e s t y l e . S i m i l a r l y , Chomsky (1965)- has c h a r a c t e r i z e d a grammar of a language as ... a d e s c r i p t i o n of the i d e a l speaker-hearer's i n t r i n s i c competence. The sa n c t i o n i n g of the performance construct provides a buffer between competence and performance which renders a competence model very d i f f i c u l t to r e f u t e . In f a c t , i t may be argued that the formal l o g i c model i s a very good example of a competence model which has thereby survived countless l i n g u i s t i c and e m p i r i c a l a t t a c k s . • The r e l a t i o n s h i p between the i n t e r p r e t a t i o n a l / o p e r a t i o n a l d i s t i n c t i o n and the competence / performance d i s t i n c t i o n may be described as f o l l o w s . The above co n s i d e r a t i o n s suggest that the o p e r a t i o n a l f a c t o r s of the i n t e r p r e t a t i o n a l / o p e r a t i o n a l d i s t i n c t i o n are i n f a c t performance f a c t o r s i n the competence / performance d i s t i n c t i o n . This i s supported by noting the nature of the o p e r a t i o n a l f a c t o r s described above. I t may by analogy be considered that i n t e r p r e t a t i o n a l f a c t o r s are p e r t i n e n t to competence. However, the nature of the i n t e r p r e t a t i o n a l 136 f a c t o r s above reveals that they are t a s k - s p e c i f i c , and as such, they should properly a l s o be considered to be performance f a c t o r s . A very important point would be missed i f the i n t e r p r e t a t i o n a l / o p e r a t i o n a l d i s t i n c t i o n was seen only as a refinement of performance f a c t o r s w i t h i n the competence / performance dichotomy (as the arguments above, suggest). The point i s that the emergence of the i n t e r p r e t a t i o n a l f a c t o r s has been a d i r e c t consequence of d i s s a t i s f a c t i o n with formal l o g i c as a competence, model. An a l t e r n a t i v e competence model which i s comparable i n scope, does not yet seem to have been developed, but a completely d i f f e r e n t approach may not be necessary. The requirement for l o g i c a l consistency ( l o g i c a l i t y ) i s d i f f i c u l t to r e l i n q u i s h , but a suspension of the assumption of a one-to-one mapping between l o g i c a l expressions and l o g i c a l meanings (see Staudenmayer, 1975), would be consonant with the i n c r e a s i n g l y compelling evidence of the impact that the i n t e r p r e t a t i o n of premises has upon performance i n reasoning ta s k s . A c c o r d i n g l y , i t seems more appropriate that a competence model which i s based upon e m p i r i c a l evidence should not ignore task i n t e r p r e t a t i o n s upon which performance i s predicated. 1 37 CHAPTER 3 Design The sample s t r u c t u r e of t h i s study was a 4 x 4 f a c t o r i a l design. The two f a c t o r s were grade l e v e l (Grades 6, 8, 10 and 12) and p r o p o s i t i o n (see the four p r o p o s i t i o n s below). The response s t r u c t u r e was a c o l l e c t i o n of both nested and crossed v a r i a b l e s which were derived from two measures for each respondent. These two measures were the r e s u l t s of a Syl l o g i s m Test and of a R e f e r e n t i a l Test of one p r o p o s i t i o n for each respondent. Each of these two sets of r e s u l t s r e q u i r e d two a d m i n i s t r a t i o n s of a subtest of each t e s t . Each subtest included one of two d i f f e r e n t content areas (Numerical and Geometric). Thus, each respondent encountered four subtests. The f i r s t two subtests were a Syl l o g i s m Test subtest followed immediately by a R e f e r e n t i a l Test subtest with the same content. The second two subtests followed the same routine with the other content. To i n v i g i l a t e against p o s s i b l e l e a r n i n g e f f e c t s , the Sy l l o g i s m Test subtests preceded the R e f e r e n t i a l Test subtests for each content. Gender has been a popular v a r i a b l e to include i n many research s t u d i e s . A dominant f i n d i n g of research i n l o g i c a l t h i n k i n g has been that there are no s i g n i f i c a n t d i f f e r e n c e s between the sexes. This suggests that gender should be excluded from the present study. I t may be counter-argued that the co n s t r u c t s of the present study have not been examined i n s u f f i c i e n t d e t a i l to suppose that gender would not 1 38 be a v i a b l e f a c t o r which a f f e c t s these c o n s t r u c t s . However, gender i s not at t h i s stage a s i g n i f i c a n t f a c t o r i n the model which u n d e r l i e s t h i s study, so i t was not included i n the data analyses. Instruments and Controls The four l o g i c a l expressions which were examined i n t h i s study were the p r o p o s i t i o n s of the f o l l o w i n g form. P r o p o s i t i o n 1: I f p then q P r o p o s i t i o n 2: p i f and only i f q P r o p o s i t i o n 3: I f p then q and i f q then p P r o p o s i t i o n 4: I f p then q and i f not p then not q. P r o p o s i t i o n 1 i s u s u a l l y c a l l e d a ' c o n d i t i o n a l ' sentence, and P r o p o s i t i o n s 2, 3 and 4 are u s u a l l y c a l l e d ' b i c o n d i t i o n a l ' sentences. The two types of instruments i n t h i s study were the S y l l o g i s m Test and the R e f e r e n t i a l Test. Four ve r s i o n s of each t e s t were produced. Each v e r s i o n was designed to examine e x a c t l y one of the above four p r o p o s i t i o n s , but the r e s u l t s of each t e s t -were e s t a b l i s h e d by the a d m i n i s t r a t i o n of two subtests of each t e s t . This was accomplished by using one of two d i f f e r e n t types of content for each subtest. One type had a Numerical content, with p = ' i t i s l a r g e ' , and q = ' i t ends with a zero' i n reference to sets of numbers. 139 The other type had a Geometric content, with p = ' i t i s black', and q = ' i t i s a square' i n reference to sets of black shapes and white shapes. Thus, each respondent encountered two Sy l l o g i s m Test subtests (one f o r each content) and two R e f e r e n t i a l Test subtests. Content of the subtests was administered i n reverse order for h a l f of the c l a s s e s at each grade l e v e l to enable s t a t i s t i c a l t e s t s to be made to detect p o s s i b l e content e f f e c t s , order e f f e c t s and i n t e r a c t i o n s . For each subtest, a p r e l i m i n a r y t e s t was administered immediately preceding the main part of the subtest. The purpose of the p r e l i m i n a r y t e s t was to e s t a b l i s h an understanding of t e s t i n g procedures, format and expectations, and to detect those respondents who f a i l e d to s a t i s f y a number of c r i t e r i a ( l i s t e d below) which must be s a t i s f i e d before responses could be deemed to be usable. An understanding of the nature of each t e s t was promoted by reading aloud a l l i n s t r u c t i o n s and procedures. This was a l s o intended to minimize e f f e c t s due to d i f f e r e n c e s i n reading l e v e l s . Appendix A contains four of the s i x t e e n subtests which were used i n t h i s study. These are the S y l l o g i s m Test subtests f o r P r o p o s i t i o n 1, Geometric content and P r o p o s i t i o n 3, Numerical content, and the R e f e r e n t i a l Test subtests for P r o p o s i t i o n 2, Geometric content and P r o p o s i t i o n 4, Numerical content. The nature of a l l other subtests may be surmised from t h i s sample. Appendix B contains the i n s t r u c t i o n s which were used f o r the a d m i n i s t r a t i o n of S y l l o g i s m Test Geometric 140 content, and R e f e r e n t i a l Test Numerical content. In order to obviate ambiguity when values of content a t t r i b u t e s were negated, only two values were used for each a t t r i b u t e . These a t t r i b u t e s are l i s t e d below, with t h e i r values i n parentheses. Numerical content: Size ( l a r g e , small) Ending (zero, one) Geometrical content: Colour (black, white) Shape (square, c i r c l e ) . Thus ( f o r example), 'not-square' means ' c i r c l e ' and no other shape. I t may be seen i n Appendix B that respondents were informed of these r e s t r i c t i o n s before each corresponding set of questions. In a d d i t i o n , d e s c r i p t i o n s of these r e s t r i c t i o n s remained i n f u l l view of each student during t e s t a d m i n i s t r a t i o n . In the R e f e r e n t i a l Tests, r e f e r e n t i a l sets which contained e i t h e r 3 or 4 binary elements were composed of two r e f e r e n t s which v a l i d a t e d each element. R e f e r e n t i a l sets which contained 2 (1) binary elements were composed of three (four, r e s p e c t i v e l y ) r e f e r e n t s which v a l i d a t e d each element. Design of the Tests The S y l l o g i s m Test. The two Sy l l o g i s m Test subtests were i d e n t i c a l i n nature and s t r u c t u r e , except for t h e i r d i f f e r e n c e s i n r e f e r e n t i a l content. For each respondent, the responses from these two subtests were pooled i n order to derive three nested response c a t e g o r i e s . These three cat e g o r i e s are shown i n Figure 2, page 53. The f i r s t category d i s t i n g u i s h e d between respondents who had usable Syllogism Test r e s u l t s and those who d i d not. C r i t e r i a f or u s a b i l i t y 141 are described i n d e t a i l below. They were designed to d i s c r i m i n a t e between responses which were v a l i d and r e l i a b l e , and those which were not. These c r i t e r i a replace the usual t e s t s of v a l i d i t y and r e l i a b i l i t y of t e s t s of o r d i n a l v a r i a b l e s . The second category dichotomized a l l usable responses only. A measure of l o g i c a l consistency of reasoning with s y l l o g i s m s was constructed, which enabled a d i s t i n c t i o n to be made between concordant and discordant usable respondents. This c o n s t r u c t was introduced i n Chapter 1. C r i t e r i a f or concordance are described i n d e t a i l below. The t h i r d category was a polytomous c l a s s i f i c a t i o n of a l l concordant respondents only. Each concordant respondent was c l a s s i f i e d as having used e x a c t l y one of the binops which have been discussed i n Chapter 1. There are 16 p o s s i b l e binops a l t o g e t h e r . In t h i s study, the binop which was used by each concordant respondent i n the Sy l l o g i s m Test has been c a l l e d the s y l l o g i s t i c binop. The c l a s s i f i c a t i o n procedures are described below. In order to minimize the confounding e f f e c t s of negation, minimal use was made of i t i n the form of the sy l l o g i s m s , except where necessary i n key sentences. The exceptions were made i n order to c o n t r o l more important f a c t o r s . For example, Johnson-Laird and T r i d g e l l (1972) have shown that statements were more l i k e l y to be considered to be negated i f they were negated e x p l i c i t l y rather than i m p l i c i t l y . Hence, i n order to negate the statement ' i t i s black', the statement ' i t i s not black' was used rather than 142 the statement ' i t i s white'. Factors which were not c o n t r o l l e d i n these t e s t s include • r e v e r s a l of order of premises (since research has i n d i c a t e d that t h i s has no e f f e c t ) • the meanings of l o g i c a l connectives (since these were measured, rather than c o n t r o l l e d ) • caution (since t h i s was a l s o measured, rather than c o n t r o l l e d ) . I t would be manifest i n the accentuation of the number of binary elements which were included i n the s y l l o g i s t i c binops of concordant responses. • some atmosphere e f f e c t (since the use of negation took higher p r i o r i t y i n some cases, i n order to maintain c o n t r o l over acquiescence, random or f r i v o l o u s responding, and c l a r i t y of negated statements). The R e f e r e n t i a l Test. The two R e f e r e n t i a l . T e s t subtests were a l s o i d e n t i c a l i n nature, apart from the d i f f e r e n c e i n content. Again, responses were pooled f o r each respondent. Two nested c a t e g o r i e s were d e r i v e d . These two cat e g o r i e s are shown i n Figure 2, page 53. The f i r s t category d i s t i n g u i s h e d between respondents who had usable R e f e r e n t i a l Test r e s u l t s and those who d i d not. The c r i t e r i a f or R e f e r e n t i a l Test u s a b i l i t y were d i f f e r e n t from the corresponding c r i t e r i a f o r the S y l l o g i s m Test. They are described i n d e t a i l below. The second category was an 81 x 3 f a c t o r i a l c l a s s i f i c a t i o n of a l l usable R e f e r e n t i a l Test respondents. The f i r s t f a c t o r designated which one of 81 p o s s i b l e r e f e r e n t i a l c l a s s e s each respondent used i n the R e f e r e n t i a l 143 Test. The second f a c t o r designated which one of 3 types of s a t u r a t i o n l e v e l s the r e f e r e n t i a l c l a s s a t t a i n e d . Both of these c o n s t r u c t s have been discussed i n Chapter 1. C r i t e r i a for each of these two f a c t o r s are discussed below. C r i t e r i a f or the Dependent Measures The c r i t e r i o n measures f o r t h i s study were deri v e d by r e s t r u c t u r i n g the responses for a l l t e s t s according to the procedures which are described below. A PL/I computer program was w r i t t e n to execute these procedures, and to construct s i x contingency t a b l e s which were used for f u r t h e r analyses. The reader i s advised to use Appendices A and B i n conjunction with the f o l l o w i n g d e s c r i p t i o n s . The S y l l o g i s m Test. The f i r s t step was to determine which respondents had usable S y l l o g i s m Test r e s u l t s . Four c r i t e r i a had to be met i n order that responses could be deemed to be usable. They are as f o l l o w s . 1. That i n the p r e l i m i n a r y p o r t i o n of the Syl l o g i s m Test, respondents d i d not d i r e c t l y c o n t r a d i c t given information more than once over both contents. S p e c i f i c a l l y , a respondent was deemed to have c o n t r a d i c t e d information i f any of Questions A, B, C or D were answered 'no', 'yes', 'yes', or 'no' r e s p e c t i v e l y , for e i t h e r content. 2. That no questions were missed, which required an ev a l u a t i o n to be made of the p o s s i b i l i t y of each of four second premises of four s y l l o g i s m s , given that the f i r s t premise was tr u e . S p e c i f i c a l l y , a respondent was deemed to have missed such a question i f any of Questions E, F, G or H was omitted, for e i t h e r content. 3. That no evidence of response p r e d i s p o s i t i o n was found i n any of the answers which were given. S p e c i f i c a l l y , a respondent was deemed to e x h i b i t response p r e d i s p o s i t i o n i f any of Questions 1,2 or 3 was answered 'yes' or 'no' in e i t h e r content. Question 4 was not included i n t h i s c r i t e r i o n , since i t was intended only to draw a t t e n t i o n to the 'can't answer' response option, as d i s t i n c t from the 'maybe' response o p t i o n . The contents which were chosen f o r these t e s t s were b e l i e v e d by t h i s author to be n e u t r a l i n a l l conceivable ways. However, i t was deemed 1 44 prudent to t e s t t h e i r supposed n e u t r a l i t y f or each respondent. Other f a c t o r s such as acquiescence, other systematic biases and random responding were thereby a l s o c o n t r o l l e d . 4. That each s y l l o g i s m which should have been completed, was i n f a c t completed. S p e c i f i c a l l y , t h i s included any respondents who answered a f f i r m a t i v e l y to any of questions E, F, G or H, but e i t h e r d i d not answer at l e a s t one of the p a i r of questions which appeared on the same page, or responded 'can't answer' to e i t h e r q uestion. The option 'can't answer* was included i n order to d i s t i n g u i s h between 'maybe' as a l e g i t i m a t e (that i s , v a l i d ) response for indeterminate s y l l o g i s m s , and 'maybe' as an undecided response. For optimum c l a r i t y , a l l response options were v e r b a l l y elaborated upon i n the i n s t r u c t i o n s f o r each t e s t (see Appendix B). With the assumption that the t e s t d i r e c t i o n s were completely followed, but responses were random, the p r o b a b i l i t y of a respondent meeting the above four c r i t e r i a in turn i s as f o l l o w s . For c r i t e r i o n 1, .094 For c r i t e r i o n 2, 1.00 For c r i t e r i o n 3, .0156 For c r i t e r i o n 4, .37. These p r o b a b i l i t i e s were c a l c u l a t e d by enumerating a l l p o s s i b l e response options and determining which ones v i o l a t e d each r e s p e c t i v e c o n d i t i o n . I f t e s t d i r e c t i o n s were not followed, so that a n u l l response was as l i k e l y as any of the options s u p p l i e d , and responses were random, these p r o b a b i l i t i e s are as f o l l o w s . For c r i t e r i o n 1, .195 For c r i t e r i o n 2, .039 For c r i t e r i o n 3, .047 For c r i t e r i o n 4, .21. In order to have usable S y l l o g i s m Test responses, a 145 respondent must meet a l l four c r i t e r i a . In the f i r s t set of instances, the p r o b a b i l i t y of meeting a l l four c r i t e r i a on the b a s i s of chance alone, i s .000546. In the second set of instances, the p r o b a b i l i t y of meeting a l l four c r i t e r i a on the basi s of chance alone, i s .000076. Under e i t h e r set of assumptions, these c r i t e r i a would appear to be q u i t e s t r i n g e n t , andwould thereby ensure confidence i n the v a l i d i t y and r e l i a b i l i t y of subsequent responses. R e l i a b i l i t y was enhanced by the low p r o b a b i l i t i e s of Type I e r r o r s , and v a l i d i t y was enhanced by the q u a l i t a t i v e nature of the above c r i t e r i a . The next step was to determine i f a usable set of responses y i e l d e d a concordant response s e t . Two p o i n t s should be mentioned with respect to the S y l l o g i s m Test. F i r s t , i t was administered over two contents. Second, each s y l l o g i s m r e q u i r e d two e v a l u a t i o n responses. For e i t h e r of these reasons, i t was not appropriate to simply examine whether or not each response set corresponded to any of those which are shown i n Table 6 (page 61), since such sets d i d not e x i s t in the simple form of ordered quadruples. In order to i l l u s t r a t e the procedure for e s t a b l i s h i n g concordant response s e t s , one page from one S y l l o g i s m Test i s analysed below. A l l other relevant pages were t r e a t e d i n a s i m i l a r way. This procedure i s a g e n e r a l i z a t i o n of the procedure described near the end of Chapter 1. Each S y l l o g i s m Test subtest contained three questions on each page. Figure 3 shows page 5 of the Geometric Syllogism Test f o r P r o p o s i t i o n 1. The three questions are 146 * * Page 5 * * I F I T I S B L A C K T H E N I T I S A S Q U A R E Q u e s t i o n E ; C a n i t b e b l a c k ? C i r c l e Y E S o r NO o n y o u r a n s w e r s h e e t n e x t t o Q u e s t i o n E . I f t h e a n s w e r f o r Q u e s t i o n E i s N O , t u r n t o t h e n e x t p a g e ( p a g e 6 ) I f t h e a n s w e r f o r Q u e s t i o n E i s Y E S , y o u r t w o c l u e s now a r e * * I F I T I S B L A C K T H E N I T I S A S Q U A R E * * I T I S B L A C K Q u e s t i o n 5 ; I s i t a s q u a r e ? ( C i r c l e t h e b e s t a n s w e r b e s i d e Q u e s t i o n 5 o n t h e a n s w e r s h e e t ) Q u e s t i o n 6 : C a n i t b e a d i f f e r e n t s h a p e t h a n a s q u a r e ? ( C i r c l e t h e b e s t a n s w e r b e s i d e Q u e s t i o n 6 o n t h e a n s w e r s h e e t ) F i g u r e 3 . P a g e 5 o f t h e g e o m e t r i c S y l l o g i s m T e s t f o r P r o p o s i t i o n 1. 147 questions E, 5 and 6. Note that Question E t e s t e d the c o m p a t i b i l i t y between the f i r s t and second premises of the s y l l o g i s m i n the centre of the page. I n c o m p a t i b i l i t y between these premises precluded the a s s e r t i o n of the s y l l o g i s m . Note a l s o that Question 6 i s a negated form of Question 5. The added use of the negative form increased power against acquiescence, other response biases and random responding. For t h i s (and every) page, two 2 x 2 matrices were used to store the information on the page. The rows of each matrix represent p and p r e s p e c t i v e l y . The columns represent q and g r e s p e c t i v e l y . The non-empty c e l l s contained e i t h e r a zero or a one, which i n d i c a t e d e i t h e r the absence or the presence ( r e s p e c t i v e l y ) of the corresponding binary element. For example, a 1 i n c e l l (1,2) of one matrix i n d i c a t e d the presence of the binary element p.g. Table 13 shows how these two matrices were f i l l e d , for each combination of response options. The procedure was as f o l l o w s . I f the answer given for question E was 'no', row 1 of the f i r s t matrix was O's and c e l l (1,2) of the second matrix was 0. The second row was not considered to be r e l e v a n t , and the matrices were considered to be complete. I f the answer for question E was 'yes', the e n t r i e s were made in the f o l l o w i n g way. I f question 5 was answered 'yes', c e l l -(1,1) of the f i r s t matrix was a 1. C e l l (1,2) was a 0. I f the answer was 'no', c e l l (1,1) was a 0 and c e l l (1,2) was a 1. I f the answer was 'maybe', both c e l l s were 1, and the f i r s t matrix 148 Table 13 M a t r i x e n t r i e s for each set of answers on Page 5 of the S y l l o g i s m Test Responses for ques t ions M a t r i x for Question E NO YES YES YES Question No. Questions 5,6 N/A YES NO MAYBE q g q g q g q g P 0 0 p 1 0 p 0 1 p 1 1. P P p q g q g q g q g 0 p 1 p 0 p 1 P P P was cons idered to be complete . I f ques t ion 6 was answered ' y e s ' , or 'maybe' , c e l l (1,2) of the second matr ix was a 1. I f the answer was ' n o ' , c e l l (1,2) was a 0, and the second matr ix was cons idered to be complete . In a l l cases on Page 5, c e l l (1,1) of the second matr ix was cons idered to be i r r e l e v a n t . T h i s was because i t s ent ry would be unknown i f the answer to Question 6 was ' y e s ' or 'maybe' . Thus, two matr i ce s s tored the in format ion on each page of each S y l l o g i s m T e s t . There were four pages i n each of two t e s t s ( for each respondent) , and hence s i x t e e n matr ice s i n a l l . They were indexed as f o l l o w s : In the expres s ion ' M a t r i x ( i , j , k , 1 ) ' , Index i represents content (1 = Geometr ic , 2 = Numerical ) Index j represents the form of the second premise (1 = a f f i r m a t i v e , 2 = negat ive) 149 Index k r e p r e s e n t s the p r o p o s i t i o n a l f u n c t i o n f o r the second premise (1 = p, 2 = q) Index 1 r e p r e s e n t s the type of q u e s t i o n asked (1 = 'Is i t ...?', 2 = 'Can i t be another . . . ? ' ) . An a s t e r i s k i s used to i n d i c a t e both values of any index. Hence the two matrices which were used i n Table 13 above are represented by the e x p r e s s i o n 'Matrix(1,1,1,*)'. The matrices were used i n the f o l l o w i n g ways. The four matrices which co n t a i n e d a l l i n f o r m a t i o n f o r c e l l (1.1) (the element p.q) were M a t r i x ( * , 1 , * , 1 ) . The s i x m a t r i c e s which contained a l l i n f o r m a t i o n f o r c e l l (1.2) (the element p.cj) were M a t r i x ( * , 1 , 1 ,*) and Matrix(*,2,2,1 ). The s i x m a t r i c e s which co n t a i n e d a l l i n f o r m a t i o n f o r c e l l (2.1) (the element p.q) were Matrix(*,1,2,*) and M a t r i x ( * , 2 , 1,1). The e i g h t matrices which co n t a i n e d a l l i n f o r m a t i o n f o r c e l l (2.2) (the element p.g) were M a t r i x ( * , 2 , * , * ) . The number of matrices f o r each c e l l was d i f f e r e n t because of the d e c i s i o n to minimize the use of negation i n the t e s t q u e s t i o n s . In order to determine concordance, i t was r e q u i r e d that the c e l l s f o r the above ma t r i c e s reach the f o l l o w i n g c r i t e r i a f o r agreement. For c e l l (1,1), the c r i t e r i o n was that a l l 4 out of 4 c e l l s agree. The p r o b a b i l i t y of Type I e r r o r , based on random e n t r i e s , i s 1/8 (.125). For c e l l (1,2), the c r i t e r i o n was that 5 out of 6 c e l l s agree. The p r o b a b i l i t y of Type I e r r o r , based on random e n t r i e s , i s 7/32 (.2185). For c e l l (2,1), the c r i t e r i o n was that 5 out of 6 c e l l s agree. The p r o b a b i l i t y of Type I e r r o r , based on random e n t r i e s , i s 7/32 (.2185). 150 For c e l l (2,2), the c r i t e r i o n was that 6 out of 8 c e l l s agree. The p r o b a b i l i t y of Type I e r r o r , based on random e n t r i e s , i s 37/128 (.289). The c o n d i t i o n for concordance was.that a l l four c e l l s reached each r e s p e c t i v e c r i t e r i o n for agreement. The p r o b a b i l i t y of Type I e r r o r for concordance, based on random e n t r i e s , i s .00173. The t h i r d c r i t e r i o n measure f o r the S y l l o g i s m Tests was the s y l l o g i s t i c binop of concordant respondents. In the case that a respondent was concordant, s/he was assigned the dominant e n t r i e s from each c e l l above. If ( f o r example) the e n t r i e s thus obtained were 1 for c e l l s (1,1) and (2,2) and 0 otherwise, the respondent was assigned s y l l o g i s t i c binop 9. This i s the binop p.q v[p.g]v[p.q]v p.g. This concludes the d e s c r i p t i o n of the c r i t e r i a for the dependent measures of the S y l l o g i s m Tests. Corresponding c r i t e r i a for the R e f e r e n t i a l Tests are described below. The R e f e r e n t i a l Test. As with the S y l l o g i s m Tests, the f i r s t step i n the R e f e r e n t i a l Test analyses was to determine which respondents had usable R e f e r e n t i a l Test r e s u l t s . The four c r i t e r i a which had to be met i n order that responses could be deemed to be usable are as f o l l o w s . 1. That each respondent recognized a l l a t t r i b u t e s i n each R e f e r e n t i a l Test subtest. S p e c i f i c a l l y , a respondent was deemed to f a i l t h i s c r i t e r i o n i f questions 1, 2, 3 and 4 were not answered 'yes', 'no', 'no', and 'yes' r e s p e c t i v e l y . Half of these questions included negation. No binary elements were v a l i d a t e d by any exemplars for these questions. 2. That each respondent answered a l l questions i n each R e f e r e n t i a l Test subtest. This a p p l i e s to Questions 5 to 20 f o r both contents. 151 The pre s e n t a t i o n of the s i x t e e n r e f e r e n t i a l sets (pages 3 to 10 i n each R e f e r e n t i a l Test subtest) was randomly ordered for each content, subject to two c o n d i t i o n s . The f i r s t c o n d i t i o n was that each expected 'yes' response was flanked on each side by at l e a s t one expected 'no' response, to i n h i b i t response p e r s e v e r a t i o n . The second c o n d i t i o n was that the order of the r e f e r e n t i a l sets for each content was the reverse of the order for the other content (see Table 14, p. 153). For each content, respondents were required to answer e i t h e r 'yes' or 'no' to the same question for each r e f e r e n t i a l s e t . The remaining two c r i t e r i a are as f o l l o w s . 3. That each respondent gave answers which agreed f o r each type of r e f e r e n t i a l s e t , at l e a s t 13 out of 16 times. 4. That there was at l e a s t one r e f e r e n t i a l set for which both answers were 'yes'. With the assumption that t e s t d i r e c t i o n s were completely followed but responses were random, the p r o b a b i l i t i e s of a respondent meeting each of the above four c r i t e r i a i n turn are as f o l l o w s . For c r i t e r i o n 1, .0039 For c r i t e r i o n 2, 1.00 For c r i t e r i o n 3, .0106 For c r i t e r i a 3 and 4 together, .0104. If t e s t d i r e c t i o n s were not followed, so that a n u l l response was as l i k e l y as any of the options s u p p l i e d , and responses were random, a l l p r o b a b i l i t i e s would be decreased. The p r o b a b i l i t y of meeting c r i t e r i o n 2 for example, would be .0015. Hence, t h i s second set of assumptions i s considered_ to be covered by the f i r s t set of assumptions. The p r o b a b i l i t y 152 of meeting a l l four c r i t e r i a under the f i r s t set of assumptions i s .00004. I t would appear then, that i t i s reasonable to c l a i m a high degree of v a l i d i t y and r e l i a b i l i t y of responses which meet a l l four c r i t e r i a . F i n a l l y , a l l usable R e f e r e n t i a l Test r e s u l t s were used to determine the r e f e r e n t i a l sets which confirmed the given l o g i c a l expression for each respondent. An example i s shown in Table 14. Only those r e f e r e n t i a l sets f o r which the responses were both a f f i r m a t i v e were considered to confirm the l o g i c a l expression. For example, the answers for Question 5 i n Geometric content and Question 20 i n Numerical content would need to be both a f f i r m a t i v e i n order to assign an a f f i r m a t i v e response to binop [p.q]v p.g v p.q v p.g (number 7). The i n v a r i a n t binary elements of the binops of the a f f i r m e d r e f e r e n t i a l sets were then determined. From these, the r e f e r e n t i a l c l a s s was determined and i n consequence, whether the respondent had a saturated, c o n d i t i o n a l l y saturated or unsaturated c l a s s of r e f e r e n t i a l sets which confirmed the given l o g i c a l expression. Table 14 shows how a t y p i c a l set of responses would y i e l d an unsaturated r e f e r e n t i a l c l a s s i n which p.q i s n e c e s s a r i l y present, p.g i s n e c e s s a r i l y absent, and both p.q and p.g are p o s s i b l y present, p o s s i b l y absent. C o m p a t i b i l i t y . The t e s t r e s u l t s of only those respondents whose Syl l o g i s m Test and R e f e r e n t i a l Test r e s u l t s were both usable, and who had concordant S y l l o g i s m Test r e s u l t s , were used to determine c o m p a t i b i l i t y (see Figure 2, page 53). For each such respondent, the degree of c o m p a t i b i l i t y of the 153 Table 14 Procedure f o r d e r i v i n g r e f e r e n t i a l c l a s s e s Answers given Binop Geom Num L o g i c a l Binary elements expression Binop Q Ans Q.Ans confirmed? No. p.q p.g p.q p.g 5 No 6 Yes 7 No 8 No 9 Yes 10 No 1 1 No 1 2 Yes 13 No 14 No 15 No 16 No 17 No 18 Yes 19 No 20 No 20 No 1 9 Yes 18 No 17 No 16 Yes 1 5 No 14 No 13 No 1 2 No 1 1 No 10 No 9 Yes 8 No 7 Yes 6 No 5 No No Yes No No Yes No No No No No No No No Yes No No 7 9 2 1 5 8 5 13 10 12 1 4 0 3 1 1 1 4 6 1 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 1 0 0 0 1 1 0 R e f e r e n t i a l c l a s s 1 {p.q v[p.gjv(p.q)v(p.g)} Note. Binops used to determine r e f e r e n t i a l c l a s s are enclosed in a box. Uns a t u r a t e d r e f e r e n t i a l c l a s s , since binop 10 was not included / 154 / S y l l o g i s m Test with the R e f e r e n t i a l Test was evaluated, according to the norms for c o m p a t i b i l i t y which have been described i n Chapter 1. C o m p a t i b i l i t y i s e s s e n t i a l l y a means of determining whether the respondent has generated a l l binary elements i n the Syllogism Test. To do t h i s , the maximal binop of the respondent's r e f e r e n t i a l c l a s s (found by removing a l l round parentheses) i s compared with the respondent's s y l l o g i s t i c binop. F a i l u r e to generate a binary element would be manifest i n having at l e a s t one element missing from the s y l l o g i s t i c binop, which i s present i n the r e f e r e n t i a l c l a s s binop. I f the binops are i d e n t i c a l , Case 1 c o m p a t i b i l i t y occurs. I f the r e f e r e n t i a l c l a s s binop 'contains' the s y l l o g i s t i c binop (as i n the l a t t i c e s t r u c t u r e shown in Figure 1, page 39), Case 2 or 3 occurs. Otherwise Case 4 occurs -that i s , there i s at l e a s t one binary element which i s denied existence i n the R e f e r e n t i a l Test, but whose existence i s permitted i n the Syl l o g i s m Test. The A n a l y s i s A n a l y t i c a l Procedures The s t a t i s t i c a l methods which were used to analyse the data i n t h i s study, are based upon procedures for the a n a l y s i s of contingency t a b l e s , which may be found i n Goodman (1970, 1971, 1973), Brown (1976), and Dixon and Brown (1979). The s t a t i s t i c s program BMDP3F was used for most of the a n a l y s i s . A d e s c r i p t i o n of the nature of t h i s program may be found i n the BMD Reference manual (Dixon & Brown, 1979). The i n t e n t of a n a l y s i s of each table was to f i n d a 1 55 ( l o g - l i n e a r ) model which combined both parsimony and goodness  of f i t to the observed data. These two co n s t r u c t s oppose each other i n the sense that the model which i s the most parsimonious (the e q u i p r o b a b i l i t y model) has the worst f i t to the data of a l l models considered, and the model which f i t s the data best (the saturated model) i s the l e a s t parsimonious. For example, the saturated model f i t s the data e x a c t l y , but i t contains a l l e f f e c t s and a l l i n t e r a c t i o n s . Hence, the procedures which were used to obtain the best compromise between these two f a c t o r s are explained below. A model for a given contingency t a b l e c o n s i s t s b a s i c a l l y of a set of main e f f e c t s and a set of i n t e r a c t i o n e f f e c t s . These e f f e c t s are used to generate a ta b l e of values for each c e l l of the observed t a b l e . The generated values are those which would be expected i f the only e f f e c t s which e x i s t e d i n the population sample, were those which are included i n the model. As an example, suppose that an a n a l y s i s i s to be made of a 3-way t a b l e which has f a c t o r s A, B and C. The f i r s t l e t t e r of the f a c t o r s of each designated t a b l e w i l l be underscored. Thus, t h i s t a b l e i s denoted by A by B by C. In the present study, the 3-way t a b l e , Grade by P r o p o s i t i o n by s y l l o g i s m U s a b i l i t y , i s an example of a 3-way contingency t a b l e . The three main e f f e c t s f o r the A by B by C ta b l e are denoted by the symbols A, B and C. The three 2-way i n t e r a c t i o n e f f e c t s are denoted by the symbols A x B, A x C and B x C. The 3-way i n t e r a c t i o n i s denoted by A x B x C. Hence, there are 7 e f f e c t s a l t o g e t h e r , for a 3-way t a b l e . Each f i t t e d contingency t a b l e was computed, using an 156 i t e r a t i v e process described by Haberman (1972). The numerical value of each (lambda) e f f e c t i n a model i s i t s value i n the l o g - l i n e a r model which corresponds to the contingency t a b l e which i s f i t t e d by the above i t e r a t i v e process (see Goodman, 1971; Dixon & Brown, 1979). Unless otherwise i n d i c a t e d , a l l c e l l frequencies of the observed t a b l e were increased by .5 before a model was f i t t e d , i n accordance with recommendations of the authors of the BMD program. By doing so, the asymptotic bias of the estimates of the lambda e f f e c t s (see Goodman, 1970, 1971) and of the variance of the lambda e f f e c t s were decreased (Gart & Z w e i f e l , 1967). The term h i e r a r c h i c a l model i s used to r e f e r to a model which has the property that i f an e f f e c t i s included i n the model, then a l l e f f e c t s which are subsets of the included e f f e c t , are a l s o included in the model. For example, i f the 2-way i n t e r a c t i o n A x B i s included i n a h i e r a r c h i c a l model, then the main e f f e c t s A and B w i l l a l s o be included. The program BMDP3F f i t s only h i e r a r c h i c a l models to contingency t a b l e s . Hence, the term 'model' i n t h i s study, r e f e r s only to h i e r a r c h i c a l models. The i n c l u s i o n of an e f f e c t i n a model for a t a b l e of observed values, i s tantamount to f i x i n g the marginal t a b l e which corresponds to that e f f e c t to be the same as the marginal t a b l e i n the t a b l e of observed values. Hence, the i n c l u s i o n of the e f f e c t A x B i n a model, means that the marginal t a b l e s of the t a b l e of expected values, for the f a c t o r s A and B, and the A by B marginal t a b l e , are the same as the corresponding marginal t a b l e s of the t a b l e of observed 157 values for these three e f f e c t s . Any e f f e c t s which are not included i n a model, i n d i c a t e that the c e l l frequencies which correspond to excluded e f f e c t s are d i s t r i b u t e d as i f the f a c t o r s contained i n the excluded e f f e c t s are independent. For example, the usual t e s t of independence i n a 2-way ta b l e i s the same as f i t t i n g a model which excludes only the 2-way i n t e r a c t i o n e f f e c t . For the 3-way t a b l e described above, there are 19 d i f f e r e n t h i e r a r c h i c a l models ( i n c l u d i n g the e q u i p r o b a b i l i t y model) which may be used as candidates to f i t the data. These models are formed by i n d i c a t i n g e i t h e r the presence or absence of each of the 7 e f f e c t s l i s t e d above. In the present study, a model for a contingency t a b l e i s denoted as i n the f o l l o w i n g diagram. This not-ation means that the model i s designed to Model n for A by B by C (p = .39) 1 1 2 2 f i t the A by B by C contingency t a b l e , i t i s the 'n'th such model, and that there are two e f f e c t s included i n the model (denoted by the numerals 1 and 2 above). The included e f f e c t s are A x B and B x C (and hence, A, B and C). The e f f e c t s which are excluded are A x C and A x B x C. The value of p i s the p r o b a b i l i t y of obta i n i n g the observed r e s u l t s (or worse) under the hypothesis that the e f f e c t s shown i n the t a b l e , are the only e f f e c t s which are present i n the population sample. A ta b l e of expected frequencies may be compared with a t a b l e of observed frequencies, by using the p r o b a b i l i t i e s 158 as s o c i a t e d with e i t h e r of two s t a t i s t i c s which are c a l c u l a t e d by the BMDP3F program. The L i k e l i h o o d Ratio (LR) s t a t i s t i c i s used i n the present study, i n preference to the Pearson goodness-of-fit s t a t i s t i c . In the model above, the expresion 'p = .39' i n d i c a t e s that the p r o b a b i l i t y of o b t a i n i n g the value of the LR s t a t i s t i c , or greater (and hence, the observed data or worse), on the assumption that the model was t r u e , i s .39. Two h i e r a r c h i c a l models are nested i f a l l e f f e c t s i n one model are included i n the e f f e c t s of the second model. In the four examples given i n Table 15, model 1 i s nested w i t h i n models 2 and 4. Models 2 and 3 are both nested w i t h i n model Table 15 Examples of some nested models Model 1 f o r A by B by C 1 2 2 Model 2 f o r A by B by C 1 1 2 2 Model 3 f o r A by B by C 1 1 2 2 Model 4 f o r A by B by C 1 1 2 2 3 3 159 4. Models 1 and 3 are not nested and models 2 and 3 are not nested. The LR s t a t i s t i c has been chosen for use i n t h i s study, since i t i s a d d i t i v e f o r nested models, while the Pearson goodness-of-fit s t a t i s t i c i s not. This means that the d i f f e r e n c e between the LR s t a t i s t i c s for each of two nested models may be used as a s t a t i s t i c to determine whether one model i s a s t a t i s t i c a l improvement on the other model. The degrees of freedom of the deri v e d s t a t i s t i c i s simply the d i f f e r e n c e between the degrees of freedom of the LR s t a t i s t i c s of each model. The c o n d i t i o n s under which the d i s t r i b u t i o n of the LR s t a t i s t i c may be approximated by a chi-square d i s t r i b u t i o n are important in- the a n a l y s i s of contingency t a b l e s . Cochran (1954) has i n d i c a t e d that i t i s safe to use a chi-square approximation i f no more than 20% of the expected values of a ta b l e are l e s s than 5. In the present study, a ta b l e of expected values for which over 20% of expected values are l e s s than 5, i s c a l l e d a sparse t a b l e . In a d d i t i o n , a table of observed values i s a l s o c a l l e d sparse i f each v i a b l e model f o r the t a b l e , generates a t a b l e of expected values which i s sparse. Note that the a d d i t i o n of .5 to a l l c e l l frequencies of a t a b l e of observed values (as described above), could improve the status of a b o r d e r l i n e sparse t a b l e . In order to analyse a sparse t a b l e , s e v e r a l options are open. One option i s to c a l c u l a t e the p r o b a b i l i t y of o b t a i n i n g a LR value (or Pearson goodness-of-fit value) or worse, based on an underlying multinomial (or hypergeometric, for small populations) d i s t r i b u t i o n . However, t h i s process i s 160 l a b o r i o u s , and no appropriate computer programs were found by the author which could be used f o r t h i s purpose. Hence, a combination of the f o l l o w i n g three procedures was adopted for each sparse t a b l e . 1. A subtable of the observed t a b l e was analysed, provided that i t represented at l e a s t 80% of the population sample. 2. Marginal t a b l e s were analysed, provided that they were not a l s o sparse. This i s s i m i l a r to Brown's (1976) t e s t of marginal a s s o c i a t i o n . 3. Values of some f a c t o r s were c o l l a p s e d . In each case, conclusions which may be made were r e s t r i c t e d by these c o n s t r a i n t s , but the a l t e r n a t i v e computational procedures were not considered to be v i a b l e . The procedure which was used to s e l e c t the most appropriate model for each contingency t a b l e i n t h i s study, was as f o l l o w s . F i r s t , a l l e f f e c t s which were design e f f e c t s (that i s , w i t h i n the c o n t r o l of the experimenter) were noted. In t h i s study, Grade, P r o p o s i t i o n , and Grade x P r o p o s i t i o n , were three design e f f e c t s . A l l design e f f e c t s were included i n a l l f i t t e d models. They are denoted by the term 'Design e f f e c t ' i n the notation for the model. Next, p r o b a b i l i t y values for Brown's (1976) t e s t s of marginal a s s o c i a t i o n and p a r t i a l a s s o c i a t i o n were examined. For each of these two t e s t s , i f e i t h e r value f e l l below .05, the corresponding e f f e c t was noted. Models were f i t t e d to the observed data, by choosing combinations of e f f e c t s which were noted by the above two procedures. The only models which were r e t a i n e d were models for which a non-sparse t a b l e of expected values was generated which d i d not d i f f e r (p > .05) from the t a b l e of 161 observed values , using the LR s t a t i s t i c . For each model which was r e t a i n e d , the goodness of f i t of a l l models which d i f f e r e d from i t by only one e f f e c t ( e i t h e r more, or l e s s ) was noted. For the four models above, i f model 2 was to be t e s t e d , then model 1 would be one of the models which would be obtained from model 2, by the d e l e t i o n of one e f f e c t (the e f f e c t A x B). Model 4 would be one of the models which would be obtained from model 2 by the a d d i t i o n of one e f f e c t (the e f f e c t B x C). I f the d e l e t i o n of an e f f e c t r e s u l t e d i n a (subordinate) model which was s t a t i s t i c a l l y worse (p < .05) than the o r i g i n a l model, the e f f e c t was r e t a i n e d . Otherwise, i t was d e l e t e d , provided i t was not a design e f f e c t . I f the i n c l u s i o n of an e f f e c t r e s u l t e d i n a (superordinate) model which was a s t a t i s t i c a l improvement (p < .05).than the o r i g i n a l model, then the e f f e c t was added to the model. Otherwise i t was ignored. I t was p o s s i b l e to s t a t i s t i c a l l y compare p a i r s of models i n t h i s way, since the a d d i t i o n or d e l e t i o n of an e f f e c t produced a nested p a i r of models. The nested p a i r could be compared s t a t i s t i c a l l y , because the LR s t a t i s t i c i s a d d i t i v e for nested models. I f the above procedures d i d not produce a unique model, a l l p a i r s of nested models which were produced by the above procedures were compared. For each p a i r , a unique model was chosen on the b a s i s of the s t a t i s t i c a l d i f f e r e n c e between the two models. I f t h i s e l i m i n a t i o n procedure s t i l l f a i l e d to produce a unique model, a l l models remaining were compared, and a s u b j e c t i v e d e c i s i o n was made on the b a s i s of parsimony, goodness of f i t , and the nature of the models themselves. 1 62 In the present study, the model which was s e l e c t e d on the b a s i s of the above procedures was c a l l e d the most  appropriate model for the observed data. The s o l u t i o n s for most of the problems which are described i n Chapter 1, are presented i n Chapter 4 on the basis of e i t h e r the i n c l u s i o n or the e x c l u s i o n of s p e c i f i e d e f f e c t s i n the most appropriate model f o r each contingency t a b l e . The Tables In order to resolve the problems which have been described i n Chapter 1, s i x contingency t a b l e s were generated from the data. The dimensions of each t a b l e were chosen from the eight f a c t o r s which are shown i n Table 16. The Grade by P r o p o s i t i o n marginal t a b l e provided two of the dimensions for Table 16 Factors used i n contingency t a b l e s No. of Factor values Values Grade 4 6, 8, 10 and 12 P r o p o s i t i o n 4 1 , 2 , 3 and 4 U s a b i l i t y 2 Usable, unusable L o g i c a l consistency 2 Concordant, discordant S y l l o g i s t i c binop 16 0 to 15 (see Table 4, p. 25) R e f e r e n t i a l c l a s s 81 A l l combinations of e i t h e r the necessary presence, necessary absence, or p o s s i b l e presence or absence, of a l l four binary elements. S a t u r a t i o n 3 Saturated, c o n d i t i o n a l l y saturated, unsaturated C o m p a t i b i l i t y case 4 1 , 2 , 3 and 4. 163 each t a b l e . Each contingency t a b l e i s described below, together with a l i s t of the Chapter 1 research problems (pp. 54-56) to which i t i s r e l e v a n t . The Syllogism Test U s a b i l i t y Table. This i s the Grade by P r o p o s i t i o n by s y l l o g i s m U s a b i l i t y contingency t a b l e over the e n t i r e sample. This t a b l e was used for Questions S1 to S3 as f o l l o w s . Question S1. The relevant frequency of unusable responses was determined from the s y l l o g i s m U s a b l i l i t y marginal t a b l e . Their nature was c l a r i f i e d and discussed. Question S2. Does the P r o p o s i t i o n x s y l l o g i s m U s a b i l i t y i n t e r a c t i o n enter i n t o the most appropriate model for the table? Question S3. Does the Grade x s y l l o g i s m U s a b i l i t y i n t e r a c t i o n enter i n t o the most appropriate model for the table? The Syllogism Test L o g i c a l Consistency Table. This i s the Grade by P r o p o s i t i o n by L o g i c a l consistency contingency t a b l e over the usable Syllogism Test subpopulation. This t a b l e was used for Questions S4 to S6 as f o l l o w s . Question S4. The relevant frequency of concordant responses was determined from the L o g i c a l consistency marginal t a b l e . Question S5. Does the P r o p o s i t i o n x L o g i c a l consistency i n t e r a c t i o n enter i n t o the most appropriate model fo r the table? Question S6. Does the Grade x L o g i c a l consistency i n t e r a c t i o n enter i n t o the most appropriate model for the table? The Syllogism Test Binop Table. This i s the Grade by P r o p o s i t i o n by S y l l o g i s t i c binop contingency t a b l e over the usable, concordant Syllogism Test subpopulation. This t a b l e was used for Questions S7 to S10 as f o l l o w s . 1 64 Question S7. The relevant frequency of each s y l l o g i s t i c binop was determined from the S y l l o g i s t i c binop marginal t a b l e . Their nature was discussed w i t h i n the context of the expectations of formal l o g i c . Question S8. Does the P r o p o s i t i o n x S y l l o g i s t i c binop i n t e r a c t i o n enter i n t o the most appropriate model for the t a b l e ? Question S9. Does the Grade x S y l l o g i s t i c binop i n t e r a c t i o n enter i n t o the most appropriate model f o r the table? Question S10. Does the Grade x P r o p o s i t i o n x S y l l o g i s t i c binop i n t e r a c t i o n enter i n t o the most appropriate model f o r the table? The R e f e r e n t i a l Test U s a b i l i t y Table. This i s the Grade by P r o p o s i t i o n by r e f e r e n t i a l U s a b i l i t y contingency t a b l e over the e n t i r e population sample. This t a b l e was used f o r Questions R1 to R3 as f o l l o w s . Question R1. The relevant frequency of unusable responses were determined from the r e f e r e n t i a l U s a b l i l i t y marginal t a b l e . Their nature was c l a r i f i e d and discussed. Question R2. Does the P r o p o s i t i o n x r e f e r e n t i a l U s a b i l i t y i n t e r a c t i o n enter i n t o the most appropriate model for the t a b l e ? Question R3. Does the Grade x r e f e r e n t i a l U s a b i l i t y i n t e r a c t i o n enter i n t o the most appropriate model for the t a b l e ? The R e f e r e n t i a l Class Table. This i s the Grade by P r o p o s i t i o n by R e f e r e n t i a l c l a s s by Sat u r a t i o n contingency t a b l e over the usable R e f e r e n t i a l Test subpopulation. This t a b l e was intended to be used f o r Questions R4 to R11 i n the f o l l o w i n g ways. However, i t i s shown i n Chapter 4 that the afore-mentioned a n a l y t i c a l procedures required some m o d i f i c a t i o n s f o r a l l of these questions. 165 Question R4. The relevant frequency of each r e f e r e n t i a l c l a s s was determined from the R e f e r e n t i a l c l a s s marginal t a b l e . Their nature was discussed w i t h i n the context of the expectations of formal l o g i c . Question R5. Does the Grade x R e f e r e n t i a l c l a s s i n t e r a c t i o n enter i n t o the most appropriate model f o r the table? Question R6. Does the P r o p o s i t i o n x R e f e r e n t i a l c l a s s i n t e r a c t i o n enter i n t o the most appropriate model for the table? Question R7. Does the Grade x P r o p o s i t i o n x R e f e r e n t i a l c l a s s i n t e r a c t i o n enter i n t o the most appropriate model for the table? Question R8. The relevant frequency of occurrence of s a t u r a t i o n was determined from the S a t u r a t i o n marginal t a b l e . Question R9. Does the P r o p o s i t i o n x S a t u r a t i o n i n t e r a c t i o n enter i n t o the most appropriate model f o r the table? Question R10. Does the Grade x Saturation i n t e r a c t i o n enter i n t o the most appropriate model for the table? Question R11. Does the Grade x P r o p o s i t i o n x Saturation i n t e r a c t i o n enter i n t o the most appropriate model for the table? The C o m p a t i b i l i t y Table. This i s the Grade by P r o p o s i t i o n by S y l l o g i s t i c binop by R e f e r e n t i a l c l a s s by S a t u r a t i o n by C o m p a t i b i l i t y contingency t a b l e . This table was a c a t e g o r i z a t i o n of respondents who had usable S y l l o g i s m Tests and R e f e r e n t i a l Tests, and who were concordant i n the Syllogism. Tests. This t a b l e was intended to be used for Questions C1 to C4 and Question F i n the f o l l o w i n g ways. However, i s shown i n Chapter 4 that a n a l y t i c a l procedures were not able to be followed for Question C4. 166 Question C1. The relevant frequencies of attainment of each norm f o r c o m p a t i b i l i t y were determined from the C o m p a t i b i l i t y marginal t a b l e . Their nature was discussed w i t h i n the context of the a b i l i t y to generate a l l binary elements. Question C2. Does the P r o p o s i t i o n x C o m p a t i b i l i t y i n t e r a c t i o n enter i n t o the most appropriate model for the t a b l e ? Question C3. Does the Grade x C o m p a t i b i l i t y i n t e r a c t i o n enter i n t o the most appropriate model f o r the table? Question C4. Does the Grade x P r o p o s i t i o n x C o m p a t i b i l i t y i n t e r a c t i o n enter i n t o the most appropriate model f o r the table? Question F. This Question was resolved by r e f e r r i n g to s o l u t i o n s for Questions S5, S6 and C4. The reasons that questions S5, S6 and C4 were used for Question F, are presented below. I m p l i c a t i o n s for Formal Operational Thought This s e c t i o n argues that Formal Operational t h i n k e r s should be both concordant, and achieve Case 1 c o m p a t i b i l i t y i n the present study. Hence, usable respondents who f a i l e i t h e r c r i t e r i o n are not Formal O p e r a t i o n a l . I t i s f u r t h e r argued that concordance and Case 1 c o m p a t i b i l i t y are necessary but not s u f f i c i e n t c o n d i t i o n s f o r Formal Operational thought. Hence, the r e s u l t s of t h i s study may be used to i n f e r the absence but not the presence of Formal Operational thought. Some of the d i f f e r e n c e s between Formal Operational thought and s y l l o g i s t i c reasoning are as f o l l o w s . (1) M a n i f e s t a t i o n s of Formal Operational thought have been sought by Piaget i n the context of e m p i r i c a l experimentation by the respondent, and i t s associated hypothesis formation and t e s t i n g . P r o p o s i t i o n s are 1 67 n e i t h e r meaningless nor a r b i t r a r y , but meaningful, p l a u s i b l e and often c a u s a l l y connected. This i s u s u a l l y not true i n t e s t s of s y l l o g i s t i c reasoning ( f o r a rare counter example, see,Ennis & Paulus, 1965). (2) In t e s t s of Formal Operational thought, hypotheses are s e l f - g e n e r a t e d , the use of second-order operations (the INRC group) allows complete f l e x i b i l i t y i n the generation of a l t e r n a t e hypotheses. Further, the t e s t s of these hypotheses are s e l f - c o n s t r u c t e d , whereas i n S y l l o g i s m Tests, the f i r s t and second premises are both suppl i e d by the experimenter. (3) In t e s t s for Formal Operational thought, i t i s the binary operations (or some l i n g u i s t i c counterpart) which are t e s t e d as hypotheses rather than t h e i r component simple p r o p o s i t i o n a l f u n c t i o n s . For example, i n t e s t s of Formal Operations, a t e s t may be constructed of the v a l i d i t y of binary operation 11 by attempting to construct cases of p.q and of p.Cf, whereas i n s y l l o g i s t i c reasoning, a t e s t may be made of the v a l i d i t y of the simple p r o p o s i t i o n a l f u n c t i o n 'p', given that a l i n g u i s t i c counterpart of binary operation 11 i s t r u e , and that q i s true (or f a l s e ) . Although the d i f f e r e n c e s appear to be vast, i t may be argued that Formal Operational t h i n k i n g e n t a i l s important s k i l l s found i n s y l l o g i s t i c reasoning. One v i t a l aspect of Formal Operational thought i s that 168 ... the thi n k e r has a complete co m b i n a t o r i a l system for generating a l l p o s s i b l e combinations of p r o p o s i t i o n s ... (Neimark & Chapman, 1975). I t f o l l o w s that such t h i n k e r s should be able to a t t a i n Case 1 c o m p a t i b i l i t y as described i n the present study. The reasons are as f o l l o w s . F i r s t , discordant S y l l o g i s m Test responses i n d i c a t e e i t h e r that the same binary elements were not a v a i l a b l e to the respondent throughout the Sy l l o g i s m Test, or that the respondent allowed d i r e c t c o n t r a d i c t i o n s to occur r e g u l a r l y i n h i s / her own reasoning (by simultaneously a s s e r t i n g and denying the existence of the same binary element). This c o o r d i n a t i o n of the binary elements requires that a Formal Operational respondent should be at l e a s t concordant, regardless of the meaning of the f i r s t premise. I f concordant, a s y l l o g i s t i c binop may be derived from the Syll o g i s m Test. This s y l l o g i s t i c binop should correspond e x a c t l y to the meaning of the f i r s t premise as measured by the R e f e r e n t i a l Test. Exact correspondence means that a l l binary elements which are denied existence i n the respondent's r e f e r e n t i a l c l a s s , must a l s o be absent from the s y l l o g i s t i c binop. I t a l s o means that a l l elements which e i t h e r may or must e x i s t i n the respondents' r e f e r e n t i a l c l a s s , must a l s o occur i n the s y l l o g i s t i c binop. These c r i t e r i a ensure that a l l binary elements are generated i n the Syl l o g i s m Test, and have the same status as they have i n the R e f e r e n t i a l Test. This i s p r e c i s e l y what i s measured by Case 1 c o m p a t i b i l i t y . Hence, i f a respondent f a i l s to achieve Case 1 c o m p a t i b i l i t y , then that person i s not Formal O p e r a t i o n a l . I t a l s o f o l l o w s from the above 1 69 d i f f e r e n c e s between Formal Operations and s y l l o g i s t i c reasoning, that evidence of Case 1 c o m p a t i b i l i t y i s i n s u f f i c i e n t to i n f e r the existence of Formal Operational thought. I f i t i s true that t e s t s of s y l l o g i s t i c reasoning with known meanings of f i r s t premises, e n t a i l necessary but not s u f f i c i e n t c o n d i t i o n s for the inference of Formal Operational thought as the above arguments suggest, then one may use these t e s t s only to i n f e r the absence of Formal Operational thought and not i t s presence. Nevertheless, there are a number of authors who have i n f e r r e d e i t h e r the absence or the presence of Formal Operational thought, u s u a l l y at an age during which i t i s supposedly incompatible with t h e i r conceptions of Piaget's t h e o r i e s . That i s , e i t h e r the c l a i m i s made that respondents can do what the experimenter claims that Piaget says they cannot do, or v i c e - v e r s a . Of the former ki n d , Burt (1919), H i l l (1961), Ennis and Paulus (1965), Ennis et a l . (1969), Ennis (1971), Brainerd (Note 1) and many others have c r i t i c i z e d Piaget by c i t i n g evidence that young c h i l d r e n can s u c c e s s f u l l y perform on at l e a s t some l o g i c a l p r i n c i p l e s . These c r i t i c i s m s are based on assumptions which the present author has claimed are untenable (such as the s u f f i c i e n c y of any s u c c e s s f u l reasoning only, and the s u i t a b i l i t y of formal l o g i c as a norm f o r success). Of the l a t t e r k i n d , Wason (1968, 1969), Johnson-Laird and Tagart (1969), Johnson-Laird and Wason (1970a), Lunzer et a l . (1972), Lunzer (1975) and many others have c r i t i c i z e d Piaget on the grounds that t h e i r (College-age) respondents 1 70 could not perform as r e q u i r e d . I t has been mentioned e a r l i e r though, that performance with the 4-card problem (the instrument which was used by the above re s e a r c h e r s ) , could r e f l e c t how task ambiguity i s resolved, rather than r e f l e c t i n g the reasoning processes which were intended to be measured. C e r t a i n l y Jansson's (1978) discrepant r e s u l t s between the 4-card s e l e c t i o n task and a . s y l l o g i s m task suggest at l e a s t a lack of concurrent v a l i d i t y . A strong a s s o c i a t i o n between Formal Operations and formal l o g i c appears to be the most s i g n i f i c a n t f a c t o r c o n t r i b u t i n g to conclusions of research studies which are c r i t i c a l of P i a g e t . Consequently, a strong case may be made against these c o n c l u s i o n s . On the other hand, not a l l c r i t i c s have taken t h i s stance. Jo-hnson-Laird and Wason (1970a) for example, have suggested that a Formal Operational combinatorial a n a l y s i s should be s u f f i c i e n t to ' c o r r e c t l y ' solve the 4-card s e l e c t i o n task. In r e p l y , Lunzer (1975) and Roth (1979) have argued that even i f t h i s were so, performance i s r e s t r i c t e d not because of competence d e f i c i e n c i e s , but because the respondent does not think to use a c o m b i n a t o r i a l a n a l y s i s . The question then becomes whether spontaneity i n the use of a c o m b i n a t o r i a l a n a l y s i s i s required of Formal Operational t h i n k e r s . I f i t i s not, r e s u l t s u l t i m a t e l y support Piaget. On the other hand, i f spontaneity i s require