Quantitative Psychology: Some Chosen Problems and New Ideas (Advances in Psychology)

QUANTITATIVE PSYCHOLOGY: Some Chosen Problems and New Ideas ADVANCES IN PSYCHOLOGY 1s Editors G . E. STELMACH P. A ...

Author: Maria Nowakowska

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QUANTITATIVE PSYCHOLOGY: Some Chosen Problems and New Ideas

ADVANCES IN PSYCHOLOGY 1s Editors

G . E. STELMACH P. A . VROON

NORTH-HOLLAND A M S T E R D A M . NEW YORK . O X F O R D

QUANTITATIVE PSYCHOLOGY: Some Chosen Problems and New Ideas

Maria NOWAKOWSKA Institute of Philosophy and Sociology Polish Academy of Sciences and

Machine Intelligence Institute Iona College

1983

NORTH-HOLLAND AMSTERDAM. NEW YORK . OXFORD

Elsevier Science Publishers B.V., 1983 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, o r transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner

ISBN: 0 444 86708 2

Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O.Box 1991 1000 B Z Amsterdam The Netherlands

Sole distributors for the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52, Vanderbilt Avenue NewYork,N.Y. 10017 U.S.A.

PRINTED IN T H E NETHERLANDS

TO THE MEMOR Y OF STANISLAW KO WAL

This Page Intentionally Left Blank

Vii

PREFACE

I n t h e p e r i o d of deepening s p e c i a l i z a t i o n and l e s s e n i n g , o f communication, n o t o n l y between d i s c i p l i n e s , b u t o f t e n a l s o b e t w e e n d o m a i n s o f t h e same d i s c i p l i n e , t h e book a t t e m p t s t o show n o t o n l y t h e c h o s e n p r o b l e m s o f q u a n t i t a t i v e psychology and i t s e x t e n s i o n s , b u t a l s o o u t l i n e some w i d e r p e r s p e c t i v e s . I t u n i f i e s t h e p s y c h o l o g i c a l t o p i c s , among o t h e r s by t h e o r i e s

o f observab-

i l i t y , s e m i o t i c s a n d knowledge s t r u c t u r e s , as w e l l as a c t i o n theory, connecting psychological, l i n g u i s t i c , l o g i c a l and d e c i s i o n a l a s p e c t s o f b e h a v i o u r . Q u a n t i t a t i v e psychology i s understood h e r e a s statistical

and m o d e l i n g a p p r o a c h e s . A s r e g a r d s t h e f i r s t ,

t h e book shows t h e o r y o f m e n t a l t e s t s c o r e s , w i t h t h e

s t r e s s p u t on new d e v e l o p m e n t s o f t h i s t h e o r y ( e . g . g e n e r a l i z a b i l i t y t h e o r y ) , a n d i t s p s y c h o l o g i c a l and m e t h o d o l o g i c a l problems. A s r e g a r d s t h e second approach, t h e book shows b a s i c p r o b l e m s o f measurement t h e o r y , and a s e r i e s o f new m o d e l s , e . g .

of r i s k , d e c i s i o n s ,

p e r c e p t i o n o f time, e t c . O r i g i n a l m a t e r i a l c o n s t i t u t e s o v e r one h a l f o f t h e b o o k . The r e f e r e n c e s a r e l i m i t e d t o t h o s e w o r k s , w h i c h

were a c t u a l l y u s e d . While t h e b o o k i s n o t a t e x t b o o k i n t h e s t r i c t s e n s e ,

i t may b e p r o f i t a b l y u s e d a s c o m p l e m e n t a r y r e a d i n g f o r

viii

PREFACE

s t u d e n t s i n t h e s o c i a l , e d u c a t i o n a l , m a n a g e r i a l , comp u t e r and i n f o r m a t i o n s c i e n c e s ( i n t e r e s t e d i n a r t i f i c i a l i n t e l l i g e n c e and p a t t e r n r e c o g n i t i o n ) , as w e l l as p h i l o s o p h y , l i n g u i s t i c and l o g i c . It w i l l a l l o w them t o deepen t h e view o f t h e i r own a r e a s , and devel o p new i n t u i t i o n s and i d e a s , The book, however, i s i n t e n d e d p r i m a r i l y f o r r e s e a r c h e r s i n t h e a r e a s menti o n e d . It assumes t h e e l e m e n t a r y knowledge o f s e t t h e o r y , f u z z y s e t t h e o r y , l o g i c , a s w e l l as p r o b a b i i t y and s t a t i s t i c s .

Maria Nowakowska

Houston, A p r i l 1983

CONTENTS

Preface Introduction Chapter 1: AN OUTLINE OF THE CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS 1. Object of Measurement in Psychology 2. Test as a Measurement Tool 3. Conceptual Foundations of the Classical Theory of Tests 3.1. The axiom system of Gulliksen 3.2. Discussion of the axioms 3.3. A critique of Gulliksen's approach 4. Conceptual Foundations of the Contemporary Theory of Tests 4.1. Propensity distribution, true and error scores 4.2. Linear experimental independence 4.3. The concept of parallel measurements

5. 6. 7. 8. 9.

Reliability Reliability of Composite Tests Homogeneity of a Test Validity Prediction 9.1. Models of regression 9.2. Construction of battery of predictors b y screening 9.3. The expected and actual quality of prediction 10. Some Problems of the Construction of Tests

vi i xvi i

1

3

4 4 8 9 11 11 17 21 24 32 41 43

47 47

54 57 59

X

CONTENTENTS

11. Further Directions of Development of Test Theory: Theory of Generic True Scores and Theory of Generalizability 11.1. Introductory remarks 11.2. Basic concepts of the theory of generic true scores 11.3. Schemes of experiments 12. Bias in Selection Procedures 13. Main Directions of Research in Test Theory

102

Bibliography to Chapter 1

109

Chapter 2: FACTOR ANALYSIS: ARBITRARY DECISIONS WITHIN MATHEMATICAL MODEL 1. Some Multivariate Techniques in Psychology 1.1. Factor analysis: the mathematical model 1.2. Discussion 1.2.1. Arbitrary decisions in applications of factor analysis 1.2.2. The problems of invariance 1.2.3. Interpretation of results of factor analysis 1.2.4. Cognitive limitations of factor analysis -- the subjective bias 1.2.5. The status of variables underlying factor analysis 1.3. Multidimensional scaling 1.3.1, Conceptual foundations 1.3.2. The main directions of development of the theory of multidimensional scaling 1.3.3. Multidimensional scaling and factor analysis 2. An Example of Application of Factor Analysis: Personality Theory of Cattell 2.1. Descriptive approach to personality theory 2.2. General characteristics of Cattell's approach 2.3. The basic hypothesis of Cattell 2.4. Some problems of research strategy of Cattell 2.4.1. Personality structure from &-data 2.4.2. Verification of the basic hypothesis of Cattell

67 67 70 76 93

115 115 121 121 127 130 131 135 138 138 142 144 145 145 148 152 153 153

156

CONTENTS

2.4.3. Cattell's explanation of difficulties in verifying his hypothesis 2.5. Discussion of hypothesis of indifference of media 2.6. General remarks about Cattell's approach Bibliography to Chapter 2 Chapter 3: SOME TSYCHOLOGICAL PROBLEMS OF CONSTRUCTION AND APPLICATION OF QUESTIONNAIRES 1. Models of Response to a Questionnaire Item 1.1. Variability and ambiguity 1.2. A fuzzification of the model 1.3. Psychometric paradox 1.3.1. Discriminativeness 1.3.2. Analysis of the paradox 1.4. Factors determining the answer to a questionnaire item and its variability 1.5. Analysis of the data 1.5.1. Factor analysis 1.5.2. Model of answering to a questionnaire item 1.5.3. Perception of items and variability of answers 1.6. Question answering as a multiple-criterion decision making: model MASIA 1.6.1. The model 1.6.2. The trial answers 1.6.3. The finaJ answer 1.6.4. Theoretical analysis 1.6.5. Some results 2. Some Alternative Ideas in Testing 2.1. Contextual tests 2.1.1. The foundations 2.1.2. The principle of estimation 2.1.3. Construction of the test 2.1.4. Some preliminary simulation results 2.2. Dynamic questionnaire 2.2.1. The formal scheme in general case 2.2.2. Random walk model

xi

160 161 163 174

183 184 191 193 194 197 205 207 207 210 213 224 225 226 229 232 240 242 243 243 247 249 252 256 256 259

xii

CONTENTS

3. Applications of Test Theory: Methodological Problems of Measurement of Fuzzy Concepts 3.1. Fuzzy sets and fuzzy identity 3.2. Structure of concepts 3.3. Measurement of concepts Bibliography to Chapter 3 Chapter 4: FORMAL SEMIOTICS: REPRESENTATIONS, OBSERVABILITY AND PERCEPTION 1. Introduction 2. Multimedia1 Communication 2.1. Formal approach to syntax of multimedia1 communication languages 2.1.1. Definition of multimedia1 languages 2.1.2. Probabilistic interpretation 2.2. Semantics of multimedia1 languages 2.3. Pragmatic semantics 2.4. Semantic languages 2.5. Graph representation of a unit 3. Control Processes in Perception 3.1. Process of spontaneous inspection 3.2. Purposeful inspection 3.3. Generation of meaning under inspection 3.4. Mechanism of transportation in comparisons 3.5. Events representation of an object: pre-events and memory 4. Observability and Change 4.1. Joint observability 4.2. Masks and filters 4.3. Operations on masks and filters 4.4. Observability of change 4.5. Semantic aspects of representations 4.5.1. Algebra of descriptions 4.5.2. Assertions and queries 4.5.3. Fuzzy attributes and verbal copies 5. Formal Semiotic Systems 5.1. Representations 5.2. Multiple representations 5.3. Fuzzy meanings 5.4. Relationships between structures

262 263 268 276 283

285 287 304 305 307 313 315 323 331 334

343 343 349 357 360

372 377 377 384

388 391 395 399 402

404 412 412 414 417 420

CONTENTS

5.4.1. Structure of represented objects 5.4.2. Homomorphisms 5.5. Signs and information 5.6. Statistical semiotics 5.7. Comments Bibliography to Chapter 4 Chapter 5: SELECTED TOPICS IN MEASUREMENT THEORY 1. Basic Procedures of Measurement 1.1. Ordinal measurement 1.2. Standard sequences 1.3. Solving inequalities 2. An Abstract Presentation of Measurement Problem 3. The Role and Character of Axioms 4. Extensive Measurement in Psychology 4.1. Measurement of risk 4.2. Qualitative scale of probability 5. Difference Measurement 5.1. The structure of algebraic differences 5.2. Bisections 5.3. The scale of just noticeable differences (jnd) 5.4. Semiorders 6. Conjoint Measurement 6.1. Necessary conditions 6.2. Sufficient conditions 6.3. Generalization: multinomial conjoint measurement 7. Utility and Subjective Probability 7.1. Primitive notions and their interpretation 7.2. The form of representation 7.3. The axiom system of Luce and Krantz 8. Some Problems in Utility Theory 8.1. Certain paradoxes of utility theory 8.2. A model of choice behaviour 8.2.1. General postulates 8.2.2. The general model 8.2.3. Some special cases 8.2.4. Discussion

xiii

421 425 428

434 437 440

443 445 445 449 455 456 461 466 468 475 482

484 486 488 490 493

497 504 505 516 516 518 519 531 532

535 540 542 552

557

x iv

CONTENTS

8.3. A decision model for social behaviour 8.3.1. The model 8.3.2. Analysis of the model 8.3.3. Discussion. Extension of the model

9. A New Theory of Time

561 561

563 567 575 575 577

9.1. The basic formal scheme 9.2. The objective cardinal and ordinal time 9.3. Subjective time and distortions of time perception 582 9.4. Time and frequency of events 590 10. Problems of Scaling 59 7 10.1. Introductory remarks 597 10.2. Relations to theories of measurement 598 10.3. Classification of scaling techniques 600 10.3.1. Pair Comparison 601 10.3.2. Magnitude scaling 613 10.3.3. Rating scales, and classification into equally spaced categories 619 10.4. Perception of scale of social power 10.4.1. The concept of power 10.4.2. Voting power 10.4.3. The hypotheses 10.4.4. The experimental material 10.4.5. The results 10.4.6. Discussion 11. Linguistic Measurement 11.1. Classification schemes 11.2. Dynamic classifications 11.3. Linguistic measurement 11.4. Construction of linguistic scales: analogies and metaphors 11.5. Dimensionality of descriptions 11.6. Measurement by analogy Bibliography to Chapter 5

625

626 628 631

637 653

660

661 661

667 672 680

683 685 689

Chapter 6: FORMAL THEORY OF ACTIONS

698 1. The Basic Scheme 1.1. The deterministic case 698 1.2. Language of actions 705 1.3. Algebraic approach to goal structure 707 1.4. A fuzzification of admissibility of actions718 1.5. Stochastic transitions 722

CONTENTS

xv

2. Temporal Relations between Events 727 2.1. The temporal truth systems 728 2.2. Events 729 2.3. Relative necessity of events 732 2.4. Knowledge representation 733 2.5. Consistency and adequacy of knowledge 740 3. Development 743 4. Multiple Goals 756 4.1. General structure of a goal 758 4.1.1. The case of one criterion. Additivity 761 4.1.2. Goals and their ideals 764 4.1.3. The effect of multidimensionality 765 4.2. The problems of aggregation 767 4.3. An algebra of criteria. Conflict theory 770 5. Verbal and Nonverbal Actions: Motivation and Choice 780 5.1. Linguistic representation of motivational space 782 5.2. Motivational calculus 789 5.3. Linguistic representation of time in motivational calculus 793 5.4. Motivational consistency and structural properties of sets of strings of actions which fulfill or break a promise 805 6. Structure of Sets of Time-Events 811 7. Group Actions 821 8. An Application to System Synthesis 834 9. Preference Systems and Relations to Decision Theory 845 10. Planning Actions and' Generative Grammars 851 11. An Application to Organization Theory 853 12. Ethical Valuations 859 13. Aggregation of Valuations 863 14. Applications: A Theory of Social Change 881 14.1. Structure of the society 882 14.2. Social change 885 14.3. A theory of freedom 888 14.4. Alienation and dynamics of social change 891 14.5. Sub,iective DerceDtion of one's l e v e l of alienatibn 896

xvi

CONTENTS

14.5.1. The model 14.5.2. Analysis 14.6. Some hypotheses 14.7. Some other mechanisms of social change 14.7.1. Group pressures and group choice 14.7.2. Communication networks 14.8. Concluding remarks. Some further perspectives 15. Examples of Application to Developmental Psychology 15.1. Stadia1 languages of development 15.2. Education and psychological development: modeling of the process Bibliography to Chapter 6

897 900 902 906 907 918 922 928 928

933 940

xvii

INTRODUCTION

T h i s book a t t e m p t s t o go beyond some s t e r e o t y p e s of q u a n t i t a t i v e p s y c h o l o g y . The q u a n t i t a t i v e a p p r o a c h e s a r e o f t e n r e g a r d e d as t o o n a r r o w , n o t i n s p i r i n g , and s u p e r f i c i a l i n t h e i r p s y c h o l o g i c a l c o n t e n t . The s t a t i s t i c a l t h e o r y o f t e s t s e n c o u n t e r e d even some r e s i s t a n c e and c e r t a i n l o s s o f s o c i a l c o n f i d e n c e , b e c a u s e o f

i t s application t o various selection procedures, t h a t b o t h i n t h e view o f t h e r e s e a r c h e r s and o f t h e u s e r s a r e b i a s e d , i . e . u n f a i r t o some s o c i a l g r o u p s . Moreover, t h e r e i s a c o n v i c t i o n t h a t t h e c o g n i t i v e weakness of t e s t s c a n n o t be overcome by even b e s t p s y c h o m e t r i c p a r a m e t e r s . The t e s t t h e o r y paradigm i s p e r c e i v e d , by many p s y c h o l o g i s t s , as s c i e n t i f i c a l l y i n f e r i o r . T h i s c a u s e d some l e s s e n i n g o f a t t r a c t i v e n e s s

o f t h i s a p p r o a c h , d e s p i t e t h e f a c t t h a t r e s e a r c h on f o r m a l models o f t e s t t h e o r y i s c o n t i n u i n g

w i t h new

i n t e r e s t i n g r e s u l t s . It seems v e r y d i f f i c u l t t o go beyond some i n h e r e n t c o g n i t i v e bounds o f t h i s a p p r o a c h . The q u e s t i o n a r i s e s how one can b r i n g some new i d e a s t o t h e domain and a p p l i c a t i o n of t e s t t h e o r y . I t seems t h a t t h e o r y may be d e v e l o p e d t o w a r d s dynamic a n a l y s i s and changes o f c o n c e p t s t r u c t u r e , o r more g e n e r a l l y

--

knowledge s t r u c t u r e . I n t h i s i n t e r p r e t a t i o n , o n e would have t o assume t h a t t h e b a s i c c o n s t r u c t o f t h e t h e o r y

xviii

INTRODUCTION

( t r a i t ) i m p l i e s some way o f v i e w i n g t h e w o r l d , or s t a t e o f knowledge a b o u t t h e w o r l d . The a n a l y s i s would conc e r n development o f v a r i o u s t y p e s o f r e p r e s e n t a t i o n of knowledge, t h e i r i n t e r a c t i o n , s u b s t i t u t a b i l i t y , d i s t o r t i o n s , e t c . It would b e i m p o r t a n t t o a n a l y s e knowledge s t r u c t u r e , as a c q u i r e d w i t h c o o p e r a t i o n w i t h a compute r a w h i c h i n some s e n s e i m p o s e s s e q u e n t i a l i z a t i o n a n d instructionalisrn i n thinking. Also, c o m p u t e r i z e d t e s t i n g o p e n s up p o s s i b i l i t i e s of new t y p e o f t e s t r e s e a r c h , i n c l u d i n g more c o n t r i b u t i o n o f e x p e r i m e n t a t i o n , more s t r e s s on i n t e r m e d i a l t e s t s , based on t h e i d e a o f s e m a n t i c e q u i v a l e n c e , and g e n e r a l l y , more c o n t r i b u t i o n o f l i n g u i s t i c and p e r c e p t u a l a s p e c t s . I n a r t i f i c i a l i n t e l l i g e n c e and computer s c i e n c e s , i n connection w i t h developing the so-called e x p e r t s y s t e m s or f u t u r e "smart c o m p u t e r s " , t h e r e i s g r e a t n e e d o f knowledge a b o u t knowledge s t r u c t u r e a n d i t s dynamics. E n g i n e e r s and computer s c i e n t i s t s n o t o n l y w a i t for new r e s u l t s i n t h i s d i r e c t i o n , b u t h a v e t h e i r own i d e a s ; t h e i d e a s o f p s y c h o l o g i s t s a r e o f t e n t o o s l o w i n c o m i n g , or b u i l t on i d e a s f o r m u l a t e d e a r l i e r i n computer s c i e n c e s . This concerns i n p a r t i c u l a r i n f o r m a t i o n p r o c e s s i n g , and has some n e g a t i v e c o n s e q u e n c e s for d e v e l o p m e n t o f new p s y c h o l o g i c a l i n t u i t i o n s , c o n c e p t s and t h e o r i e s . The m a i n s t r e s s o f p s y c h o l o g i c a l r e s e a r c h s h i f t e d some

time ago frqm measurement o f t r a i t s t o models of psy-

c h o l o g i c a l p r o c e s s e s . P a r t i c u l a r l y s u c c e s s f u l here a r e d e c i s i o n m o d e l s . However, i t a p p e a r s t h a t i d e a l i z a t i o n s o f t h e s e p r o c e s s e s a r e t o o s t r o n g , and t h e i n t u i t i o n s on w h i c h t h e m o d e l s a r e b a s e d a r e t o o f a r from t h e r e a l e x p e r i e n c e o f d e c i s i o n m a k e r s . For i n s t a n c e ,

INTRODUCTION

xix

i n p r o b l e m s o f r i s k , t o o much s t r e s s was p u t on i n t u i t i o n s o f "homo e c o n o m i c u s " , w i t h o u t t a k i n g i n t o a c c o u n t e t h i c a l v a l u a t i o n s o f a c t i o n s a n d d e c i s i o n s . One c a n show t h a t w i t h t h e s e f a c t o r s i n c l u d e d i n t h e r i s k mod e l , o n e g e t s new t y p e s o f i n t e r p r e t a t i o n a n d p r e d i c t i o n o f b e h a v i o u r . Also s t u d y o f s e q u e n t i a l d e c i s i o n s i n g r o u p c o n t e x t o p e n s new r e s e a r c h p o s s i b i l i t i e s ( s e e Chapter 5 , S e c t i o n 8 ) . Modeling o f i n t e r n a l d e c i s i o n s i n t h e p r o c e s s o f a n s w e r i n g q u e s t i o n n a i r e items g i v e s

a b r i d g e between d e c i s i o n t h e o r y and t e s t t h e o r y , i . e . i n t r o d u c e s d e c i s i o n m o d e l s i n t o t h e l a t t e r . Models o f i n t e r n a l d e c i s i o n s a l l o w t o c a p t u r e some c o g n i t i v e biases i n these d e c i s i o n s , and t o r e c o n s t r u c t t h e o r e t i c a l l y t h e s e b i a s e s , as o b s e r v e d i n a c t u a l r e s p o n s e s ( s e e C h a p t e r 3 , S e c t i o n 1 . 6 ) . It a p p e a r e d , m o r e o v e r , t h a t it i s e s s e n t i a l t o b u i l d i n t e r a c t i o n a l d e c i s i o n models for some c l a s s e s o f s i t u a t i o n s ( s e e C h a p t e r 6 , S e c t ion 14). Finally,

one can b u i l d a r i c h t h e o r y d e s c r i b i n g t h e

s t r u c t u r a l a s p e c t s o f s i t u a t i o n s and d e c i s i o n s , which i s a n i m p o r t a n t complement o f d e c i s i o n a l a p p r o a c h e s (Chapter 6 ) . P r o g r e s s i n s p e c i a l i z a t i o n i n psychology , and slow d i f f u s i o n o f c o n c e p t s f r o m o n e domain t o a n o t h e r , c a u s e s for i n s t a n c e , t h a t i n e x p e r i m e n t a l r e s e a r c h on s u b j e c t i v e p r o b a b i l i t y , t h e r e a r e a l m o s t no s t u d i e s on s t a b i l i t y and v a r i a b i l i t y o f a s s e s s m e n t s . T h i s i s j u s t o n e o f t h e many e x a m p l e s o f n o t i n f r e q u e n t p s y c h o m e t r i c "under- i n s t r u m e n ta i t i o n " o f psycho l o g i c a l r e s e a r c h

.

Measurement t h e o r y s u p p l i e s c o g n i t i v e l y i m p o r t a n t models , o f f o u n d a t i o n a l c h a r a c t e r f o r q u a n t i t a t i v e

INTROD UCTlON

xx

psychology.

Here a l s o w e have a c o g n i t i v e paradigm,

of c o n s i d e r a b l e m a t h e m a t i c a l e l e g a n c e , and v e r y v a l u a b l e r e s u l t s , n o t o n l y for q u a n t i t a t i v e p s y c h o l o g y , but also c o n s t i t u t i n g a c o n t r i b u t i o n t o m a t h e m a t i c s . The c o g n i t i v e s p e c t r u m of t h e m o d e l s o f c l a s s i c a l meas u r e m e n t t h e o r y i s n o t t o o w i d e . The t h q o r y d e t e r m i n e s m a i n l y t h e c o n d i t i o n s w h i c h must b e m e t , i n o r d e r f o r t h e e x i s t e n c e o f a g i v e n t y p e o f r e p r e s e n t a t i o n of a

s e t o f o b j e c t s a n d r e l a t i o n s on t h e m . I t h a s , h o w e v e r , a d e s c r i p t i v e r a t h e r t h a n e x p l a n a t o r y c h a r a c t e r . It does not i n t r o d u c e deeper c o g n i t i v e models. I n p a r t i c u l a r , it does not t a k e i n t o account c e r t a i n p e r c e p t u a l , memory o r n e u r o b i o l o g i c a l v a r i a b l e s . The c o r e o f measurement t h e o r y paradigm i s f i n d i n g a numerical r e p r e s e n t a t i o n for a s e t of e m p i r i c a l r e l a t i o n s . The p r o p e r t i e s of t h e l a t t e r allow t o determine t h e c l a s s o f t r a n s -

f o r m a t i o n s under which t h e r e p r e s e n t a t i o n ( t h e s c a l e v a l u e s ) remain i n v a r i a n t ; t h i s c l a s s determines a l s o t h e set of meaningful a s s e r t i o n s about s c a l e v a l u e s w h i c h o n e c a n make. T h i s seems t o c o v e r t h e e s s e n t i a l a s p e c t s o f m e a n i n g f u l n e s s ; however, t h e scope

of t h e

definition is rather limited. I n p h i l o s o p h y of measurement t h e o r y , p e r c e p t u a l b i a s e s are a n a l y s e d by v a r i o u s a s s u m p t i o n s a b o u t e r r o r s i n c l a s s i f i c a t i o n s and o r d e r i n g s . The c e n t r a l r o l e i s p l a y e d by d i s c r i m i n a t i v e a b i l i t y . I t a p p e a r s t h a t t h e s t u d y o f d i f f e r e n t f o r m s o f s u b j e c t i v e time a n d i t s d i s t o r t i o n s , t h e t o p i c t h u s f a r unexplored i n measurement t h e o r y ,

a l l o w s t h e a n a l y s i s o f new a s p e c t s o f

s t r u c t u r e and f u n c t i o n of c o g n i t i v e p r o c e s s e s ( s e e Chapter 5 , S e c t i o n

9).

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I n m e a s u r e m e n t t h e o r y , t h e o b j e c t s may b e a r b i t r a r y , and t h e o n l y - r e q u i r e m e n t i s t h a t t h e e m p i r i c a l r e l a t i o n s s a t i s f y t h e a p p r o p r i a t e axioms , implying t h e e x i s t e n c e

of a s c a l e of t h e a t t r i b u t e i n question. I n conjoint m e a s u r e m e n t , t h e scheme i s t h e same, e x c e p t t h a t t h e r e l a t i o n a l s y s t e m i n v o l v e s o b s e r v a t i o n s o f two (or mor e ) a t t r i b u t e s . P e r h a p s some e x t e n s i o n s c o u l d b e obt a i n e d h e r e by c o n s i d e r i n g o b j e c t s o f complex s t r u c t u r e ( m u l t i d i m e n s i o n a l u n i t s or s t i m u l i ) . T h i s would req u i r e a n e x t e n s i o n and a new f o r m o f r e p r e s e n t a t i o n s , new f o r m s o f i n v a r i a n c e , and c h a n g e o f t h e c r i t e r i a of meaningfulness. An e x t e n s i o n o f m e a s u r e m e n t t h e o r y t o w a r d s w e a k e r f o r m s o f m e a s u r e m e n t was i n t r o d u c e d i n C h a p t e r 5 , S e c t i o n 11, d e a l i n g w i t h l i n g u i s t i c measurement, comprising a l s o m e a s u r e m e n t by a n a l o g y a n d by m e t a p h o r s . I n t h e same s e c t i o n , t h e dynamic c l a s s i f i c a t i o n was i n t r o d u c e d , where t h e c l a s s i f i c a t i o n schemes a r e s a m p l e d f r o m o c c a s i o n t o o c c a s i o n , a n d m e a s u r e m e n t i s made a c c o r d i n g t o t h e sampled scheme. It seems t h a t t h e o r y o f m e a s u r e m e n t c o n s i d e r s o n l y

o n e c l a s s o f c o n s t r a i n t s , i . e . t h o s e i n n u m e r i c a l dom a i n , i n t o w h i c h t h e e m p i r i c a l r e l a t i o n s a r e mapped. One c a n i m a g i n e a s t u d y o f o t h e r t y p e s o f c o n s t r a i n t s , c o n n e c t e d w i t h t h e mechanisms o f t r a n s p o r t a t i o n , v e r y b a s i c i n comparison o f a t t r i b u t e s o f s t i m u l i . The l a t t e r i s o n e o f t h e key mechanisms u s e d i n dy-

namic m o d e l s o f p e r c e p t i o n ( s e e C h a p t e r 4 , S e c t i o n 3 ) . It c o u l d p e r h a p s s e r v e as a new s t a r t i n g p o i n t f o r p o s s i b l e d e v e l o p m e n t s o f measurement t h e o r y . Maybe t h e o t h e r mechanisms w i l l h a v e s i m i l a r u s e f u l n e s s ,

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namely t h e mechanisms o f s p o n t a n e o u s and p u r p o s e f u l i n s p e c t i o n , a n d c o n n e c t e d w i t h them r a n d o m n e s s a n d p a r t i a l c o n t r o l o f p e r c e p t i o n . The l a t t e r i m p l i e s t h e e x i s t e n c e o f s t r a t e g i e s and o p t i m a l i z a t i o n o f p e r c e p t i o n p r o c e s s . The p e r c e p t u a l mechanisms c o u l d s e r v e a s a b a s i s o f new a x i o m s y s t e m s i n measurement t h e o r y , s i m i l a r l y a s masks a n d f i l t e r s ( s e e C h a p t e r 4) c o u l d a l l o w t h e s t u d y of new t y p e s o f c o n s t r a i n t s i n t h e o b s e r v a t i o n a l domain. A s may be s e e n from t h e a b o v e o u t l i n e , t h e a i m o f t h e

book was a c r i t i c a l p r e s e n t a t i o n o f some c h o s e n p r o b lems i n q u a n t i t a t i v e p s y c h o l o g y . T h u s , t h e book s t a r t s f r o m a p r e s e n t a t i o n o f t h e c o n t e m p o r a r y t h e o r y o f psyc h o l o g i c a l t e s t s . It covers a l l b a s i c concepts, includi n g t h e t h e o r y of g e n e r a l i z a b i l i t y . P a r t i c u l a r a t t e n t i o n h e r e i s d i r e c t e d a t t h e p r o b l e m s of u s a g e o f t e s t s for s e l e c t i o n , a n d b i a s e s i n s u c h p r o c e d u r e s . C h a p t e r 2 c o n c e r n s f a c t o r a n a l y s i s , as a n e x a m p l e o f

a model i n w h i c h a r b i t r a r i n e s s o f d e c i s i o n s a f f e c t s t h e r e s u l t s . The c h a p t e r a l s o shows p r o b l e m s o f m u l t i d i m e n sional scaling, i n particular i t s difficulties with t h e i n t e r p r e t a t i o n . A s a n e x a m p l e o f a t h e o r y b a s e d on

f a c t o r analysis, Cattell's theory of personality i s shown, and c o g n i t i v e beound imposed by t h e method a r e discussed. C h a p t e r 3 d e a l s w i t h p s y c h o l o g i c a l p r o b l e m s o f cons t r u c t i o n a n d a p p l i c a t i o n o f q u e s t i o n n a i r e s . Here main a t t e n t i o n i s c o n c e n t r a t e d on f a c t o r s d e t e r m i n i n g t h e an s wers , and p o s s i b l e b i a s e s and d i s t o r t i o n s o f t h e a n s w e r s . I n p a r t i c u l a r , it i s shown t h a t a m b i g u i t y of q u e s t i o n s a n d i t s e m p i r i c a l c o u n t e r p a r t -- v a r i a b i l i t y o f a n s w e r s -- i s o n e o f t h e most i m p o r t a n t f a c t o r s

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a f f e c t i n g t h e r e s p o n s e . It i s a l s o e x p l a i n e d why w e l l d i s c r i m i n a t i n g i t e m s a r e h i g h l y v a r i a b l e ( t h e phenomenon c a l l e d p s y c h o m e t r i c p a r a d o x ) . Next, t h e c h a p t e r e x p l o r e s an i m p o r t a n t c o n n e c t i o n between t e s t t h e o r y and t h e o r y o f fuzzy s e t s ; t h e f o r mer may namely p r o v i d e a n e m p i r i c a l a c c e s s t o measurement o f membership f u n c t i o n s f o r c o n c e p t s o f s p e c i a l structure. A t h e o r e t i c a l model o f a n s w e r i n g q u e s t i o n n a i r e i t e m s ,

based on e m p i r i c a l d a t a , i s shown. T h i s model a l l o w s f o r q u a l i t a t i v e p r e d i c t i o n o f v a r i a b i l i t y and s t a b i l i t y o f i t e m s , given t h e p s y c h o l o g i c a l c h a r a c t e r i s t i c s of these items.

Further

t h e o r e t i c a l e x t e n s i o n o f t h i s model is t h e

s t o c h a s t i c d e c i s i o n model M A S I A , which a l l o w s f o r a r e c o n s t r u c t i o n o f e m p i r i c a l l y observed f r e q u e n c i e s o f answers. F i n a l l y , t h i s c h a p t e r a l s o shows two a l t e r n a t i v e models o f t e s t i n g , namely c o n t e x t u a l t e s t s and dynamic q u e s t i o n n a i r e s . The f i r s t i s b a s e d on t h e i d e a o f u n f o l d i n g s c a l e s , w i t h s p e c i a l s t r u c t u r e o f complex s t i m u l i . The second c o n s i d e r s dynamic q u e s t i o n n a i r e s , where t h e dec i s i o n r e g a r d i n g t h e c h o i c e o f n e x t q u e s t i o n (or dec i s i o n about t e r m i n a t i o n o f t e s t ) i s r e a c h e d s e q u e n t i a l l y , depending on t h e answers t o t h e p r e c e d i n g i t e m s . C h a p t e r 4 p r e s e n t s a n o u t l i n e o f f o r m a l s e m i o t i c s . The s t a r t i n g p o i n t o f t h i s t h e o r y i s a r e l a t i o n between an o b j e c t ( r e a l o r imagined, random o r n o t ) , i d e n t i f i e d w i t h a c e r t a i n r e l a t i o n a l s t r u c t u r e , and i t s v a r i o u s v e r b a l , p i c t o r i a l , symbolic, n u m e r i c a l , e t c . r e p r e s e n t -

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a t i o n s . The main c o n c e p t o f t h e t h e o r y i s t h a t o f m u l t i d i m e n s i o n a l u n i t ( a composite s t i m u l u s ) , and multimed-

i a l languages, allowing p a r a l l e l description of s t r i n g s o f p h y s i c a l or m e n t a l a c t i o n s . Elements o f p r o b a b i l i s t i c i n t e r p r e t a t i o n o f u n i t s and m u l t i m e d i a 1 l a n g u a g e s a r e a l s o shown. T h i s a l l o w s t o i n t r o d u c e random o b j e c t s t o t h e t h e o r y o f o b s e r v a b i l i t y and change, and t o c o n s i d e r j o i n t o b s e r v a b i l i t y , as w e l l as t e m p o r a l t r a c e s o f o b s e r v a t i o n s , c a l l e d m a s k s . I n o t h e r words, o b s e r v a b i l i t y t h e o r y d e a l s w i t h v a r i o u s c o n s t r a i n t s on o b s e r v a b i l i t y , e x p r e s s e d by c o n c e p t s o f m a s k s and f i l t e r s . System o f models of v a r i o u s p e r c e p t i o n mechanisms p r e s e n t e d i n t h i s c h a p t e r was i n t r o d u c e d f o r e x p l a i n i n g the p a r t i a l control process of perception of objects. The s y s t e m d e p a r t s s u b s t a n t i a l l y from t h e m a t h e m a t i c a l a p p r o a c h e s i n p a t t e r n r e c o g n i t i o n and a r t i f i c i a l i n t e l l i g e n c e advanced t h u s f a r . The s e c t i o n on s e m i o t i c s y s t e m s , c o v e r i n g a l s o e l e m e n t s of s t a t i s t i c a l semiotics, closes t h e chapter. The s e t o f i n t e r r e l a t e d t h e o r i e s o f C h a p t e r 4 a l l o w s f o r t r e a t i n g i n a u n i f i e d way d i f f e r e n t p e r c e p t u a l , cognit i v e , l i n g u i s t i c and p h i l o s o p h i c a l problems. I t a l s o a l l o w s t o s e e t h e o r y o f t e s t s and t h e o r y o f measurement i n a wider p e r s p e c t i v e . The l a t t e r i s c o v e r e d i n C h a p t e r 5 , which c o n s i s t s o f 11 s e c t i o n s . The f i r s t e i g h t d e a l w i t h c l a s s i c a l problems: e x t e n s i v e , d i f f e r e n c e and c o n j o i n t measurements, u t i l i t y and s u b j e c t i v e p r o b a b i l i t y . S e c t i o n 8 shows some new problems o f u t i l i t y t h e o r y , namely a model o f

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c h o i c e d e c i s i o n u n d e r r i s k , a n d a model o f s o c i a l b e h a v i o u r i n g r o u p c o n t e x t s . A s r e g a r d s t h e f i r s t , it i s d e s i g n e d to c o v e r t h e s i t u a t i o n s i n w h i c h t h e SEU model d o e s n o t a p p l y ; i t t a k e s i n t o a c c o u n t some p s y c h o l o g i c a l f e a t u r e s of t h e m o t i v a t i o n model o f A t k i n s o n . The main r e s u l t o f t h i s g e n e r a l model o f c h o i c e c o n c e r n s t h e e x i s t e n c e o f non-normal r i s k a r e a s , w h i c h e x p l a i n some a p p a r e n t l y i r r a t i o n a l b e h a v i o u r of c e r t a i n d e c i s i o n makers. It a l s o e x p l a i n s t h e assymmetry b e t w e e n t h e r o l e s o f f e a r o f f a i l u r e a n d a c h i e v e m e n t m o t i v e i n dec i s i o n m a k i n g . I n t h i s model o n e a b a n d o n s a n a r r o w way of l o o k i n g a t s u c c e s s i n p u r e l y e c o n o m i c a l t e r m s , and t h i n k s o f r e w a r d s a l s o i n m o r a l , e t h i c a l or i d e o l o g i c a l terms. T h e s e c o n d model s t a r t s f r o m t h e a s s u m p t i o n t h a t i n

m a k i n g t h e c h o i c e o f a c t i o n , p e o p l e t e n d t o maximize b o t h t h e e x p e c t e d d i r e c t reward a n d a l s o i n d i r e c t r e w a r d ,

i n t e r m s of a c c e p t a n c e o r r e j e c t i o n by o t h e r s i n t h e r e f e r e n c e g r o u p . The a s s u m p t i o n s on u t i l i t y f u n c t i o n l e a d h e r e t o a s e r i e s of q u a l i t a t i v e h y p o t h e s e s and p r e d i c t i o n s o f b e h a v i o u r , d e p e n d i n g on t h e e v a l u a t i o n s . of t h e g r o u p . An e x t e n s i o n of t h e m a i n i d e a s o f t h i s model t o a g e n e r a l d e c i s i o n model ( n o t r e s t r i c t e d t o t h e c a s e of s o c i a l d e c i s i o n s a n d u t i l i t y i n d e x c o n n e c t e d w i t h t h e j u d g m e n t s ) was a l s o d i s c u s s e d . A p p l i c a t i o n s o f t h e i d e a s o f measurement t h e o r y t o cons t r u c t i o n of a t h e o r y of t i m e , b o t h o b j e c t i v e a n d subj e c t i v e , i s shown i n S e c t i o n 9 . The t h e o r y i s b a s e d on fuzzy r e l a t i o n d e s c r i b i n g t h e s u b j e c t i v e perception o f time. Some h y p o t h e s e s o f p s y c h o l o g i c a l c h a r a c t e r about t h e d i s t o r t i o n s o f p e r c e p t i o n o f t i m e are formulated.

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A f t e r s h o w i n g b a s i c p r o b l e m s o f s c a l i n g ( S e c t i o n lo), a p p l i c a t i o n s o f t h i s t h e o r y t o s c a l i n g s o c i a l power

a r e d i s c u s s e d . The a t t e n t i o n i.s c o n c e n t r a t e d on t h e f a c t o r s accounting f o r t h e d i s t o r t i o n s of t h e p e r c e p t i o n o f s o c i a l power. The c h a p t e r e n d s w i t h a n e x t e r . s i o n o f c o n c e p t i o n o f measurement t h e o r y t o i t s weak-est f o r m

--

linguistic

m e a s u r e m e n t . The l a t t e r seems t o b e t h e most n a t u r a l and f l e x i b l e f o r m o f measurement a t human d i s p o s a l . T h i s t h e o r y i s r e l a t e d i n a n e s s e n t i a l way w i t h t o p i c s o f p e r c e p t i o n , d e s c r i p t i o n s , v e r b a l c o p i e s and i d e n t i f -

i c a t i o n of Chapter of motivation i n

4, as w e l l a s t h o s e o f r e p r e s e n t a t i o n Chapter 6.

The book e n d s w i t h t h e c h a p t e r d e v o t e d t o s t r u c t u r a l p r o p e r t i e s o f d e c i s i o n s and a c t i o n s , and a g e n e r a l d e s c r i p t i o n o f human b e h a v i o u r i n a v e r y f l e x i b l e f o r m a l f r a m e w o r k . The b a s i c n o t i o n s h e r e a r e l a n g u a g e o f a c t i o n s , an a l g e b r a i c s t r u c t u r e of g o a l s , fuzzy admissib i l i t y o f a c t i o n s , and s t o c h a s t i c t r a n s i t i o n s . N e x t s e c t i o n a n a l y s e s a n important a t t r i b u t e o f an e v e n t , namely t h e t i m e o f i t s o c c u r r e n c e , a n d i n t r o d u c e s t h e c o n c e p t o f h i s t o r y a n d t e m p o r a l t r u t h s y s t e m s . The e v e n t s a r e d e f i n e d a s s e t s o f h i s t o r i e s , and t h e n o t i o n o f n e c e s s i t y o f an e v e n t i s i n t r o d u c e d .

The knowledge i s i d e n t i f i e d w i t h a s e t o f a s s e r t i o n s about t h e h i s t o r y of t h e s y s t e n . I t s t r u t h , b e l i e v a b i l i t y , c e r t a i n t y , e t c . are generalized t o t h e notion of 4-bility,

and t h e c o n d i t i o n s o f adequacy, c o n s i s t e n c y ,

t r u t h and p r e c i s i o n a r e d i s c u s s e d . S p e c i a l k i n d of h i s t o r i e s , r e p . r e s e n t i n g d e v e l o p m e n t , a r e t h e n i n t r o d u c e d and a n a l y s e d t'hrough t h e c o n c e p t o f

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p h a s e s and t h e i r l a n g u a g e . Section

4

d e a l s w i t h g o a l s o f a c t i o n s , e s p e c i a l l y mul-

t i p l e and d i v e r s e g o a l s , and t h e n w i t h t h e i r a g g r e g a t i o n , as w e l l as with t h e theory

of c o n f l i c t s a r i s i n g

i n c a s e o f j o i n t l y u n a t t a i n a b l e g o a l s . The l a t t e r t h e ory

i s o f p a r t i c u l a r importance f o r d e c i s i o n s i t u a t -

ions. R e l a t i o n s between v e r b a l and n o n v e r b a l a c t i o n s ( S e c t i o n

5 ) i s b a s e d on t h e e x p e c t a t i o n s o f c o n s i s t e n c y between u t t e r e d p l a n s , i n t e n t i o n s and a c t i o n s . The a n a l y s i s o f u t t e r a n c e s i n which a p e r s o n p r e s e n t s h i s p l a n s , j u s t i f i e s or e x p l a i n s h i s g o a l s , e t c . a l l o w s t o introduce a l i n g u i s t i c representation of motivation, while l o g i c a l a n a l y s i s of t h e s e u t t e r a n c e s l e a d s t o m o t i v a t i o n a l c a l c u l u s , b e i n g a s e t of i n f e r e n c e r u l e s b a s e d on t h e n o t i o n o f s e m a n t i c i m p l i c a t i o n . T h i s c a l c u l u s was l a t e r e n r i c h e d by i n c l u s i o n o f t e m p o r a l v a r i a b l e s , l e a d i n g , among o t h e r s , t o d e f i n i t i o n s o f some quantifiers. R e l a t i o n s between m o t i v a t i o n a l c o n s i s t e n c y and s t r u c t u r -

a l p r o p e r t i e s o f s e t s o f s t r i n g s o f a c t i o n s were cons i d e r e d on example o f p r o m i s e s . The t h e o r e m s h e r e r e f l e c t c e r t a i n g e n e r a l r e g u l a r i t i e s i n t r e a t m e n t o f norms of s o c i a l o r d e r , namely r e l a t i v e u n i v e r s a l i t y o f cont e x t s t o which t h e y a p p l y . These r e s u l t s may be o f some i n t e r e s t for deontic logic. E v e n t s o c c u r i n t i m e , and t h e t e m p o r a l p a t t e r s o f t h e i r o c c u r r e n c e , p e r i o d i c i t y , h o r i z o n e t c and d e a l t with i n Section 6 .

Next, t h e system i s extended t o

t h e c ase of simultaneous a c t i o n s o f s e v e r a l persons, s o

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INTRODUCTION

a s t o c a p t u r e two a s p e c t s : j o i n t p e r f o r m a n c e , a n d i n t e r a c t i o n s between s t a t e s o f t h e systems c o r r e s p o n d i n g t o d i f f e r e n t persons. T h i s g e n e r a l i z e s t h e concept o f m u l t i m e d i a 1 u n i t s o f communication language o f Ch a p te r

4,

and a l s o l e a d s t o g e n e r a l problems o f s y s t e m s s y n th e -

s i s ( S e c t i o n 8 ) , i . e . a g g r e g a t i o n of simple s y s t e m s t o c o m p o s i t e o n e s , u n d e r some c o n s t r a i n t s . Here t h e n o t i o n s of p o s i t i v e and n e g a t i v e i n t e r f e r e n c e and synergy a r e i n t r o d u c e d and a n a l y s e d . Connections w i t h d e c i s i o n t h e o r y and p r e f e r e n c e s y s t e m s a r e shown i n S e c t i o n 9; it i s a r g u e d t h a t a c t i o n t h e o r y

i s i n f a c t a theory of predecisional situations. In the s e q u e l i t i s shown how g e n e r a t i v e grammars may be u s e d f o r f o r m a l i z a t i o n o f t h e t h e o r y o f p l a n s and d e s i g n s ( S e c t i o n lo), a n d how a c t i o n t h e o r y may b e a p p l i e d t o o r g a n i z a t i o n t h e o r y ( S e c t i o n 11). Here t h e a c t i o n s y s -

t e m i s e n r i c h e d by t h e n o t i o n o f a n i n s t r u c t i o n , w h i c h m o d i f i e s t h e a d m i s s i b i l i t y of s t r i n g s o f a c t i o n s . E t h i c a l v a l u a t i o n s o f a c t i o n s a r e d i s c u s s e d i n n e x t two s e c t i o n s . Thus, S e c t i o n 1 2 concerns t h e problems o f e t h i c a l v a l u a t i o n s from m e a s u ~ e m e n t - t h e o r e t i c a l viewp o i n t . N o t i o n s o f l i b e r a l and p u r i s t i c a g g r e g a t i o n s

are introduced, generalized next t o aggregations of v a l u e s o b t a i n e d from some e v a l u a t i o n s w i t h r e s p e c t t o different criteria. The c h a p t e r e n d s w i t h v a r i o u s e x a m p l e s o f a p p l i c a t i o n o f some c o n c e p t s o f t h e t h e o r y ; i n p a r t i c u l a r , S e c t i o n

1 4 d e a l s w i t h a theory of s o c i a l change, w i t h b a s i c n o t i o n s o f f r e e d o m a n d a l i e n a t i o n , combined w i t h t h e c o n s i d e r a t i o n s of s u b j e c t i v e p e r c e p t i o n of o n e ' s a l i e n a t i o n . This p e r c e p t i o n changes w i t h s e q u e n t i a l l y acquire d knowledge o f t h e s i t u a t i o n of o t h e r s .

xxix

INTRODUCTION

To d e e p e n t h e a n a l y s i s , a model o f g r o u p p r e s s u r e a n d g r o u p c h o i c e i s shown. T h i s model c o n c e r n s s i t u a t i o n s i n w h i c h a g r o u p makes a d e c i s i o n f o r o r a g a i n s t a n i s s u e . The main r e s u l t a l l o w s t o o p t i m a l l y a l l o c a t e t h e propaganda t a s k s . An e s s e n t i a l complement o f t h e model o f s o c i a l c h a n g e i s a t h e o r y of communication n e t w o r k s , b e i n g models f o r s p r e a d and s c o p e o f d i s s e m i n a t i o n o f i n f o r m a t i o n i n a group. A n o t h e r example o f a p p l i c a t i o n o f

a c t i o n s t h e o r y con-

c e r n s d e v e l o p m e n t a l p s y c h o l o g y , namely m u l t i d i m e n s i o n a l s t a d i a 1 languages of development, e s p e c i a l l y convenient f o r d e s c r i p t i o n o f e a r l y s t a d i a o f development o f a child. The b o o k c l o s e s w i t h an o u t l i n e o f a p p l i c a t i o n o f a c t i o n t h e o r y t o e d u c a t i o n a l psychology, and mention numerous o t h e r a p p l i c a t i o n s , e . g .

of

t o theory of d i a l o -

gues, e t c .

It i s w o r t h t o s t r e s s t h a t t h e f o r m a l t h e o r y o f a c t i o n s i s t h e f i r s t i n t h e l i t e r a t u r e system t h a t connects p s y c h o l o g i c a l , l l n g u i s t i c , l o g i c a l and d e c i s i o n a l a s p e c t s of behaviour. The above s k e t c h o f t h e c o n t e n t o f t h e book shows t h e d i r e c t i o n s o f r e s e a r c h and s o l u t i o n s , b o t h

i n the

t r a d i t i o n a l p r o b l e m s o f q u a n t i t a t i v e p s y c h o l o g y , as

w e l l as i n t h e a t t e m p t s o f i t s e x t e n s i o n s . The g o a l o f t h e book w i l l be met, i f i t d i r e c t s t h e a t t e n t i o n o f p s y c h o l o g i s t s a n d s o c i a l r e s e a r c h e r s t o some new

p o s s i b i l i t i e s o f t h e development o f t h e domain, and i f i t s h o u l d s u p p l y new i n t u i t i o n s and m o d e l s t o r e -

xxx

INTRODUCTION

l a t e d d i s c i p l i n e s , such a s a r t i f i c i a l i n t e l l i g e n c e

and computer s c i e n c e s , management s c i e n c e s , as w e l l a s p h i l o s o p h y and l o g i c .

1

CHAPTER 1

AN OUTLINE OF THE CONTEMPORARY THEORY

OF PSYCHOLOGICAL TESTS

1. OBJECT OF MEASUREMENT I N PSYCHOLOGY

The o b j e c t o f any measurement a r e s u c h p r o p e r t i e s o f o b j e c t s , w h i c h t h e l a t t e r may p o s s e s s i n v a r y i n g d e g -

r e e ; s u c h a t t r i b u t e s a r e s a i d t o be m o d a l . E a c h measur e m e n t i s s u b j e c t t o c e r t a i n e r r o r s , and t h e b a s i c p r o b lem i n any m e a s u r e m e n t t h e o r y i s t h e i n f e r e n c e a b o u t t h e v a l u e s m e a s u r e d on t h e b a s i s o f t h e v a l u e s o b s e r v ed.

I n p s y c h o l o g y , w h e r e t h e main o b j e c t o f s t u d y a r e humans, o n e i s i n t e r e s t e d i n t h e v a l u e s o f t h e i r p s y c h o l o g i c a l t r a i t s . The l a t t e r n o t i o n l e d t o numerous c o n t r o v e r s i e s ; for t h e p u r p o s e o f m e a s u r e m e n t , h o w e v e r , i t

i s n o t n e c e s s a r y t o d w e l l on t h e s e c o n t r o v e r s i e s , and t a k e as a s t a r t i n g p o i n t o f t h e d e f i n i t i o n o f a t r a i t t h e f a c t of t h e e x i s t e n c e of s t a t i s t i c a l r e g u l a r i t i e s o f b e h a v i o u r . The v a l u e o f a t r a i t f o r a g i v e n p e r s o n

i s simply a c e r t a i n parameter of t h e p r o b a b i l i t y d i s t r i -

bution of c e r t a i n s p e c i f i c behaviours of t h i s person i n specific situations. Such an approach does n o t r e q u i r e d i s c u s s i n g t h e prob-

2

CHAPTER 1

l e m how " r e a l " i s t h e t r a i t , i . e . w h e t h e r i t h a s a n y p h y s i c a l or p h y s i o l o g i c a l c o u r t e r p a r t s . I n p s y c h o l o g y , and i n p a r t i c u l a r i n p s y c h o m e t r i c s w h i c h d e a l s w i t h measurements of v a l u e s o f t r a i t s

,

t h e only e s s e n t i a l

f a c t i s t h a t p e o p l e d o b e h a v e as i f t h e s e t r a i t s r e a l l y e x i s t e d , or more p r e c i s e l y , tl-at d i f f e r e n t p e r s o n s behave as i f t h e y had t h e s e t r a i t s i n d i f f e r e n t d e g r e e s ( p s y c h o m e t r i c s d e a l s o n l y w i t h modal t r a i t s ) . T h u s , from a f o r m a l p o i n t o f v i e w a t r a i t ( a n d more p r e c i s e l y , i t s v a l u e ) i s a c e r t a i n parameter of t h e f r e q u e n c y d i s t r i b u t i o n o f some b e h a v i o u r s . A s a cons e q u e n c e , l e a r n i n g t h i s v a l u e a l l o w s a s t a t i s t i c a l pred i c t i o n of such behaviours. From t h e p o i n t o f view o f p s y c h o l o g y , a t r a i t may b e d e f i n e d as a t e rm t h a t g e n e r a l i z e s s u c h s e t s o f b e h a v i o u r s which have a t tendency t o a p p e a r and vary j o i n t l y . I n o t h e r words, a t r a i t i s a n a b s t r a c t i o n o f conc r e t e behaviours ( o r other t r a i t s ) . The m a i n o b j e c t o f i n t e r e s t i n p s y c h o l o g y , h o w e v e r ,

are t h o s e t r a i t s which have e x p l a n a t o r y c h a r a c t e r : t h e y a r e c a l l e d c o n s t r u c t s , or t h e o r e t i c a l n o t i o n s . The e x p l a n a t i o n h a s t h e f o r m o f a c e r t a i n h y p o t h e s i s or a s e t o f h y p o t h e s e s , c o n c e r n i n g c a u s a l i n t e r p r e t a t i o n of j o i n t a p p e a r e n c e of t h o s e b e h a v i o u r s which a r e t h e d e s c r i p t i o n o f t h e t r a i t . One o f t h e m a i n g o a l s of such explanation i s t o provide t h e o r e t i c a l premises

f o r p r e d i c t i o n s . For i n s t a n c e , w i t h t h e term s u c h as " i n t r o v e r s i o n " one a s s o c i a t e s n o t o n l y a s e t o f behavi o u r s w h i c h a r e c h a r a c t e r i s t i c f o r i t , b u t a l s o some hypotheses which e x p l a i n t h e i n t e r r e l a t i o n s between these behaviours,

t h e i r o r i g i n , d y n a m i c s , and g e n e r a l

CONTEMPORARY THEORY OF PSYCHOLOGICAL TESTS

3

p r e d i c t i o n of b e h a v i o u r o f i n t r o v e r t i c p e r s o n s . The t h e o r e t i c a l n o t i o n s a r e c h a r a c t e r i z e d by t h e f a c t t h a t t h e i r c o n t e n t i s u s u a l l y f u z z y and n o t u n i q u e l y d e f i n e d ; i n o t h e r words, t h e same t e r m , w i t h t h e same e x p l a n a t o r y h y p o t h e s e s , may be a s s o c i a t e d w i t h sev e r a l s e t s of b e h a v i o u r s , e a c h c o n s t i t u t i n g a good des c r i p t i o n of t h i s terms. The f l e x i b i l i t y o f t h e c o r r e s pondence between t h e name of t h e t r a i t and i t s d e s c r i p t i o n i s c a l l e d s u r p l u s meaning. A s a consequence o f surp l u s meaning, w e may c o n s t r u c t s e v e r a l e q u a l l y good d e f i n i t i o n s o f t h e same t r a i t , and a l s o have s e v e r a l measurement t o o l s o f i t .

2 . TEST AS A MEASUREMENT TOOL

A f t e r d e f i n i n g t h e o b j e c t of measurement i n p s y c h o l o g y ,

i t i s now n e c e s s a r y t o d e t e r m i n e t h e methodology o f s u c h measurements, and a l s o t o p r o v i d e t h e r u l e s o f i n f e r e n c e about t r u e v a l u e s of t r a i t s from t h e o b s e r v e d v a l u e s o f measurement. The s o l u t i o n h e r e i s d e t e r m i n e d by t h e a c c e p t e d s t a t i s t i c a l d e f i n i t i o n of p s y c h o l o g i c a l t r a i t , i d e n t i f i e d w i t h a p a r a m e t e r of a f r e q u e n c y d i s t r i b u t i o n o f b e h a v i o u r s which s e r v e sa a d e s c r i p t i o n of t h e given t r a i t . This parameter, c a l l e d t h e t r u e v a l u e of a t r a i t ( f o r a g i v e n s u b j e c t ) i s estimat e d from a sample of b e h a v i o u r s . F o r m a l l y , a psychologi c a l t e s t i s j u s t a t o o l which a l l o w s u s t o c o l l e c t s u c h a sample o f b e h a v i o u r s (which a r e a d e s c r i p t i o n o f a g i v e n t r a i t ) , and a l s o a l l o w s us t o d e t e r m i n e t h e n u m e r i c a l v a l u e of t h e e s t i m a t e , g i v e n t h e b e h a v i o u r s . The n u m e r i c a l v a l u e i n q u e s t i o n i s o f t e n c a l l e d t h e t e s t score.

4

CHAPTER 1

To f o r m a l l y d e s c r i b e t h i s s i t u a t i o n , l e t A d e n o t e t h e p o p u l a t i o n u n d e r s t u d y , and l e t G be t h e s e t o f m e a s u r ement t o o l s u n d e r c o n s i d e r a t i o n . I n p a r t i c u l a r c a s e , G may c o n s i s t o f j u s t one t o o l . F o r a C A and g t. G , a n a p p l i c a t i o n of g t o a g i v e s a n u m e r i c a l r e s u l t , which The l a t t e r w i l l be r e g a r d e d ga’ as a v a l u e o f a c e r t a i n random v a r i a b l e X The d i s ga * t r i b u t i o n o f t h i s random v a r i a b l e ,

w i l l be d e n o t e d by x

i s c a l l e d the EropensiQ

d i s t r i b u t i o n of person a w i t h

r e s p e c t t o t e s t g. The p r o p e n s i t y d i s t r i b u t i o n i s n o t o b s e r v a b l e , s i n c e i n p r a c t i c e , t h e r e i s no way o f o b t a i n i n g a s u f f i c i e n t l y l o n g s e q u e n c e of i n d e p e n d e n t o b s e r v a t i o n s o f t h e ( e . g . b e c a u s e of l e a r n i n g e f f e c t ) . random v a r i a b l e X ga I t i s assumed t h a t t h e v a l u e s o f t h e random v a r i a b l e

a r e e x p r e s s e d on a s c a l e o f a t l e a s t i n t e r v a l t y p e , ga s o t h a t one may compute t h e e x p e c t a t i o n and v a r i a n c e .

X

3. CONCEPTUAL FOUNDATIONSS OF THE CLASSICAL THEORY OF TE:STS

3.1. The axiom s y s t e m o f G u l l i k s e n The f i r s t f o u n d a t i o n s o f t h e t h e o r y o f t e s t s were g i v e n i n 1950 by G u l l i k s e n ( G u l l i k s e n 1 9 5 0 ) ; t h i s t h e o r y was s u b s e q u e n t l y m o d i f i e d by N0vic.k ( 1 9 6 6 ) a n d Lord a n d


5

Novick ( 1 9 6 8 ) . G u l l i k s e n a d a p t e d t o psychology t h e model o f measurement i n p h y s i c a l s c i e n c e s . He assumed t h a t e a c h p e r s o n has, a t any g i v e n t i m e , some v a l u e s of p s y c h o l o g i c a l t r a i t s , i n analogy w i t h p h y s i c s , where e a c h o b j e c t h a s some w e l l d e f i n e d v a l u e s o f i t s a t t r i b u t e s , r e g a r d l e s s

of t h e a c t o f measurement and t h e c h o i c e o f measurement t o o l . F o r m a l l y , t o e a c h p e r s o n a 6 A and t e s t g C G t h e r e corresponds a value T equal t o t h e value of ga ' t h e t r a i t measured by g f o r p e r s o n a . Next, a g a i n i n a n a l o g y w i t h p h y s i c a l s c i e n c e s , G u l l i k s en assumed t h a t when a measurement i s performed, t h e n t h e observed v a l u e ' e q u a l s t o t h e t r u e v a l u e p l u s a r a n dom e r r o r . Thus, t h e measurement e r r o r i s t h e d i f f e r e n c e

or

(3.1) A s a c o n s e q u e n c e , b o t h t h e t r u e v a l u e and t h e e r r o r

o f measurement a r e n o t o b s e r v a b l e : i n formula ( 3 . 1 ) onl y t h e l e f t hand s i d e i s o b s e r v a b l e , w h i l e t h e compon e n t s o f t h e r i g h t hand s i d e a r e n o t . I n o r d e r t o enable t h e a p p l i c a t i o n s of s t a t i s t i c a l methods o f e s t i m a t i o n of t h e t r u e v a l u e on t h e b a s i s o f t h e o b s e r v e d v a l u e s , i t was n e c e s s a r y t o a c c e p t some a s s u m p t i o n s c o n c e r n i n g t h e randomness of t h e o b s e r v e d

6

CHAPTER 1

scores. It was namely assumed t h a t t h e randomness o f t h e o b s e r v -

ed s c o r e i s of a "double" c h a r a c t e r : i t s f i r s t s o u r c e i s t h e randomness o f t h e measurement p r o c e s s , and t h e second s o u r c e i s t h e randomness o f t h e c h o i c e of t h e p e r s o n from t h e p o p u l a t i o n . The f i r s t t y p e of randomness i s f o r m a l i z e d by assuming

i s a r e a l i z a t i o n of t h e ga with the propensity d i s t r i b u t i o n random v a r i a b l e X ga' F The c h a r a c t e r of t h e second s o u r c e of randomness ga' r e q u i r e s more d e t a i l e d e x p l a n a t i o n , because of t h e

t h a t t h e observed s c o r e x

f a c t t h a t i t i s s p e c i f i c f o r mental t e s t theory, with no c o r r e s p o n d i n g c o u n t e r p a r t i n t h e p h y s i c a l s c i e n c e s . I t i s namely assumed t h a t t h e p e r s o n i n q u e s t i o n i s s e l e c t e d a t random from t h e p o p u l a t i o n under s t u d y . The l a t t e r p o p u l a t i o n c o n s i s t s , roughly speaking, of a l l p e r s o n s who a r e i d e n t i c a l w i t h t h e s u b j e c t from t h e p o i n t o f view o f t h e c r i t e r i a which d e c i d e d on t h e c h o i c e o f t h i s p a r t i c u l a r p e r s o n . For i n s t a n c e , i f one c h o o s e s a s a i l o r ( s a y ) t o measure h i s t r a i t s u c h a s i n t r o v e r s i o n , t h e n i t i s assumed t h a t t h e t r u e s c o r e o f i n t r o v e r s i o n o f t h i s s a i l o r i s a v a l u e of a random v a r i a b l e , whose d i s t r i b u t i o n i s t h e d i s t r i b u t i o n o f t h e t r u e s c o r e s of i n t r o v e r s i c n i n t h e p o p u l a t i o n of all sailors. F o r m a l l y , i t i s assumed t h a t t h e s u b j e c t s from t h e pop u l a t i o n A a r e sampled a c c o r d i n g t o some p r o b a b i l i t y d i s t r i b u t i o n . It i s n o t n e c e s s a r y t o s p e c i f y t h i s d i s t r i b u t i o n ; g e n e r a l l y , one assumes ( i n most c a s e s ) t h a t i t i s uniform on a c e r t a i n s u b s e t of A . A t any r a t e , as t h e r e s u l t of t h i s a s s u m p t i o n , t h e i n d e x a i s random,


7

S y m b o l i c a l l y , i t w i l l be c o n v e n i e n t t o i n t r o d u c e t h e notation X for t h e random v a r i a b l e , whose r e a l i z a t i o n s g* a r e t h e v a l u e s i n t e s t g for a randomly s e l e c t e d p e r s o n . S i m i l a r l y , t h e t r u e s c o r e f o r a randomly s e l e c t e d p e r -

son w i l l be t r e a t e d a s a random v a r i a b l e T

g*

’

whose va-

l u e s are t h e t r u e s c o r e s i n t e s t g , and t h e e r r o r s of measurements are t h e r e a l i z a t i o n s of t h e random v a r i a b le I g*

.

Thus, e q u a t i o n ( 3 . 1 ) w i l l now t a k e t h e form X

g*

= T

g*

+

I

g*

.

(3.2)

The n e c e s s i t y o f s u c h a c o n s t r u c t i o n i s connected c l o s e l y w i t h t h e f a c t t h a t i n p r a c t i c e one cannot o b t a i n s u f f i c i e n t numbers of i n d e p e n d e n t o b s e r v a t i o n s for t h e same p e r s o n . The p o i n t i s , t h a t for a s i n g l e p e r s o n a , t h e i n f e r e n c e on h i s t r u e s c o r e i: i s based on t h e ga e s t i m a t i o n of e r r o r e For t h e l a t t e r , i n t u r n , one ga would have t o p e r f o r m many i n d e p e n d e n t o b s e r v a t i o n s of t h e random v a r i a b l e X (for f i x e d p e r s o n a ) . ga

.

S i n c e i n p r a c t i c e one can c o u n t o n l y f o r two (or a t there ga’ i s no way o f e s t i m a t i n g t h e v a r i a n c e of e Thus, t h e ga’ (which i s t h e e s t i m a t i o n c o n c e r n s t h e v a r i a n c e of I g* a v e r a g e v a r i a n c e of e r r o r s of measurements). b e s t s e v e r a l ) independent o b s e r v a t i o n s o f X

A s a r e s u l t , one g e t s t h e p o s s i b i l i t y of e s t i m a t i n g t h e

t r u e s c o r e o f a f i x e d p e r s o n on t h e b a s i s of h i s o b s e r v ed s c o r e . N a t u r a l l y , f o r s u c h an e s t i m a t i o n t o be p o s s i b l e , i t i s n e c e s s a r y t o make some p o s t u l a t e s a b o u t t h e errors I i n s u c c e s s i v e measurements, and about t h e g*

8

CHAPTER i

-

r e l a t i o n between G and t h e t r u e s c o r e T These posg* g* t u l a t e s , c o n s t i t u t i n g t h e axioms of t h e t e s t t h e o r y o f G u l l i k s e n , a r e as f o l l o w s : I . The e x p e c t e d e r r o r s c o r e i s z e r o : E(1

g*

) = 0;

11. The e r r o r s c o r e and t h e t r u e s c o r e a r e u n c o r r e l a -

ted: E(1

T

g* g*

) = 0;

111. E r r o r s c o r e s i n two d i s t i . n e t measurements a r e un-

correlated

3 . 2 . D i s c u s s i o n o f t h e axioms The f i r s t and t h e t h i r d axiom a r e t a k e n d i r e c t l y from t h e t h e o r y o f p h y s i c a l measurements. I n d e e d , axiom I s t a t e s t h a t measurements a r e u n b i a s e d , w h i l e axiom I11 i s a weakened v e r s i o n of t h e assumption t h a t e r r o r s i n s u c c e s s i v e measurements a r e i n d e p e n d e n t . However, axiom I1 has no c o u n t e r p a r t i n p h y s i c a l s c i e n c e s , s i n c e i n

physics

t h e t r u e v a l u e i s n o t a random v a r i a b l e .

By axiom 11, t h e v a r i a n c e of t h e observed s c o r e i s t h e

sum o f two components: v a r i a n c e of t r u e s c o r e and v a r i a n c e o f e r r o r ' ) ' The o b j e c t i s t o e s t i m a t e t h e second ') F o r m a l d e r i v a t i o n s w i l l be p r e s e n t e d l a t e r i n t h i s chapter.

CONTEMFORARY THEORY OFPSYCHOLOGIC4L TESTS

9

o f t h e s e v a r i a n c e s . The r e a s o n i n g l e a d i n g t o d e t e r m i n a t i o n o f t h e v a r i a n c e of e r r o r s c o r e i s a s f o l l o w s . I f t h e v a r i a n c e of t r u e s c o r e i s l a r g e i n comparison w i t h t h e v a r i a n c e o f t h e o b s e r v e d s c o r e (hence i f t h e v a r i a n c e o f e r r o r s c o r e i s small as compared w i t h t h e v a r i a n c e o f observed s c o r e ) , t h e n -- b e c a u s e of axiom I11 a s s e r t i n g l a c k o f c o r r e l a t i o n i n two s u c c e s s i v e measurements -- t h e observed s c o r e s i n two measurements w i l l d i f f e r l i t t l e , hence t h e y w i l l be h i g h l y c o r r e l a t e d . On t h e o t h e r hand, i f t h e e r r o r v a r i a n c e i s l a r g e i n comparison w i t h t h e v a r i a n c e o f o b s e r v e d s c o r e , t h e measurements i n two t r i a l s w i l l have low c o r r e l a t i o n . F o r m a l l y , i t f o l l o w s from t h e axioms o f G u l l i k s e n , t h a t t h e c o e f f i c i e n t of c o r r e l a t i o n between two s u c c e s s i v e measurements i s e q u a l t o t h e s o - c a l l e d r e l i a b i l i t y of measurements, d e f i n e d as t h e r a t i o of t h e v a r i a n c e o f t r u e s c o r e s t o t h e v a r i a n c e of o b s e r v e d s c o r e s . Thus, t h e knowledge o f r e l i a b i l i t y , a c q u i r e d by e s t i m a t i n g t h e c o r r e l a t i o n c o e f f i c i e n t between t h e r e s u l t s of two measurements o f t h e same group of s u b j e c t s by t h e same t e s t , combined w i t h t h e knowledge of t h e v a r i a n c e of t h e observed s c o r e , allows us t o e s t i m a t e t h e v a r i a n c e of t r u e s c o r e , hence a l s o t h e e r r o r v a r i a n c e . I n t h i s way, i f one a c c e p t s t h e axioms o f G u l l i k s e n , one g e t s a p o s s i b i l i t y of e s t i m a t i o n of e r r o r v a r i a n c e , hence also c o n s t r u c t i o n of c o n f i d e n c e i n t e r v a l s f o r t h e unknown t r u e s c o r e of a g i v e n p e r s o n .

3.3. A c r i t i q u e of G u l l i k s e n ' s approach The axiom s y s t e m of G u l l i k s e n was c r i t i c i z e d ; t h e main

10

CHAPTER i!

o b j e c t i o n was t h e assumption t h a t t h e t r u e s c o r e e x i s t s i n d e p e n d e n t l y o f t h e f a c t o f measurement. F o r i n s t a n c e , Thorndike (1964) claimed t h a t s i n c e t h e t r u e scores a r e never observable d i r e c t l y (they a r e s u b j e c t t o measurement e r r o r s ) , , t h e n o t i o n of t r u e scor e i s a m y t h , which c a n n o t have any t h e o r e t i c a l s i g n i f i c a n c e . On t h e o t h e r hand, Loevinger ( 1 9 5 7 ) claimed t h a t t h i s c o n c e p t cannot have any p r a c t i c a l s i g n i f i c a n c e . According t o h e r , i t makes s e n s e t o d e a l o n l y w i t h observed s c o r e s . W i t h t h e p r e s e n t s t a t u s of s t a t i s t i c a l knowledge

, both

t h e s e o b j e c t i o n s a r e u n t e n a b l e , , s i n c e t h e f a c t of i m p o s s i b i l i t y of l e a r n i n g t h e e x a c t v a l u e of a p a r a m e t e r s h o u l d n o t be an argument a g a i n s t i n t r o d u c i n g t h i s parameter. A more s e r i o u s o b j e c t i o n a g a i n s t t h e axioms o f G u l l i k -

s e n , and i n p a r t i c u l a r a g a i n s t t h e concept o f t h e t r u e s c o r e , l i e s i n t h e f a c t t h a t G u l l i k s e n was f o r c e d t o a c c e p t p o s t u l a t e s f o r which t h e r e a r e no e m p i r i c a l p r o c e d u r e s of t h e i r v e r i f i c a t i o n . T h i s may be s e e n as f o l l o w s . If we assume, as i n p h y s i c a l s c i e n c e s , t h a t t h e t r u e v a l u e o f a t r a i t of a g i v e n p e r s o n e x i s t s i n d e p e n d e n t l y o f t h e f a c t o f measurement , we may d e f i n e t h e e r r o r s c o r e , simply as t h e d i f f e r e n c e between t h e o b s e r v e d and t r u e s c o r e s . To c o n s t r u c t a t h e o r y which would l e a d t o methods of i n f e r a n c e , G u l l i k s e n was f o r ced t o make some a s s u m p t i o n s about t h e e r r o r s c o r e s ( s u c h as t h e i r u n b i a s e d n e s s and l a c k of c o r r e l a t i o n i n c o n s e c u t i v e measurements). ‘Phis amounts t o assuming t h a t t h e t o o l of measurement i s u n b i a s e d : i t s a v e r a g e e r r o r s c o r e i s z e r o . Now, i n p h y s i c a l s c i e n c e s , s u c h


11

t y p e o f assumption may b e v e r i f i e d e x p e r i m e n t a l l y , simpl y by m e a s u r i n g many t i m e s t h e same known q u a n t i t y . On

t h e o t h e r hand, i n t h e t h e o r y o f t e s t s , t h e r e e x i s t no procedure f o r v e r i f y i n g t h e unbiasedness o f t h e t o o l s of measurement. The same a p p l i e s a l s o t o o t h e r p o s t u l a t e s of G u l l i k s e n .

4. CONCEPTUAL FOUNDATIONS

OF THE CONTEMPORARY

THEORY OF TESTS

4 . 1 . P r o p e n s i t y d i s t r i b u t i o n , t r u e and e r r o r s c o r e s D e s p i t e t h e above o b j e c t i o n s , t h e r e i s c o n s i d e r a b l e i n t u i t i v e a p p e a l o f t h e c o n c e p t s o f t r u e and e r r o r scor e s , based on s t r o n g a n a l o g i e s w i t h t h e p h y s i c a l measur e m e n t , and a l s o on t h e u t i l i t y o f t o o l s based on t h i s t h e o r y . T h i s caused Lord and Novick (1968) t o c o n s t r u c t a new a x i o m a t i z a t i o n of t h e t h e o r y , s u c h t h a t a l l axioms of G u l l i k s e n would become t h e o r e m s .

The s t a r t i n g p o i n t o f t h e c o n s t r u c t i o n o f Lord and Nov i c k (1968) i s t h e s y s t e m

where A i s t h e s e t o f p e r s o n s ( p o p u l a t i o n of s u b j e c t s ) , G i s t h e c l a s s o f t e s t s under s t u d y , ' r i s a p r o b a b i l i t y d i s t r i b u t i o n on A , and X i s a f a m i l y of random vaga = t ] i s t o be i n t e r p r e t e d as r i a b l e s . The e v e n t $ X ga "the subject a obtained the score t i n t e s t g".

12

CHAPTER 1

It i s assumed t h a t i f a # a ' , t h e n f o r a r b i t r a r y g , g '

i n G ( d i s t i n c t o r n o t ) , t h e random v a r i a b l e s X

and gYa X a r e i n d e p e n d e n t . T h i s means t h a t t h e r e s u l t s of g ' ,a' measurements of d i f f e r e n t s u b j e c t s a r e i n d e p e n d e n t . Let u s d e n o t e

(4.2) t h i s distribution function i s called the propensity

d i s t r i b u t i o n of s u b j e c t a i n t e s t g . Thus, t h e t e s t s c o r e of p e r s o n a i n t e s t g i s t r e a t e d as a r e a l i z a t i o n of a random v a r i a b l e , whose p o s s i b l e values are a l l potentially possible r e s u l t s of t h i s t e s t f o r t h e g i v e n p e r s o n , and whose p r o b a b i l i t y d i s t r i b u t i o n i s c h a r a c t e r i s t i c f o r t h i s t e s t and p e r s o n . Because of t h e a l r e a d y mentioned i m p o s s i b i l i t y o f p e r f o r m i n g i n d e p e n d e n t o b s e r v a t i o n s of t h e same t e s t f o r t h e same p e r s o n , t h e p r o p e n s i t y d i s t r i b u t i o n i s n o t o b s e r v a b l e . Thus, i t i s a c e r t a i n t h e o r e t i c a l c o n s t r u c t . I t s a c c e p t a n c e i s m o t i v a t e d by t h e e m p i r i c a l f a c t o f v a r i a b i l i t y of t e s t r e s u l t s f o r t h e same p e r s o n ; t h u s , i t i s a much s t r o n g e r argwnent t h a n t h e a n a l o g i e s w i t h p h y s i c a l measurement, used by G u l l i k s e n . POSTULATE 1. F o r e v e r y a 6 A

distribution F 2

d (Xga)

ga

g

GG

the propensity

has f i n i t e v a r i a n c e

= Var

(xga) c

03

.

From P o s t u l a t e 1 i t f o l l o w s i n p a r t i c u l a r t h a t propens i t y d i s t r i b u t i o n s have a l s o f i n i t e e x p e c t a t i o n s , t o be d e n o t e d by


= E(Xga).

et.

13

(4.3)

ga

DEFINITION. The e x p e c t a t i o n ( 4 . 3 ) i s c a l l e d t h e t r u e s c o r e of person a i n t e s t g. Next, d e n o t e 1 ga

= x

(4.4)

- T

ga

ga'

We have t h e r e f o r e E(1

since E ( I

ga

ga

) = 0

) = E(X

ga

)

(4.5)

- E(Tga)

Thus, t r u e s c o r e i s d e f i n e d i n t e r m s of p r o p e n s i t y d i s t r i b u t i o n as i t s e x p e c t a t i , o n , w h i l e t h e e r r o r s c o r e ( 4 . 4 ) i s d e f i n e d as t h e d i f f e r e n c e between t h e o b s e r v e d and t r u e s c o r e s . Thus, e r r o r s c o r e i s s t i l l unobservabl e , as i n t h e t h e o r y of G u l l i k s e n ; however, f o r e v e r y person, t h e expected e r r o r s c o r e i s z e r o i n v i r t u e of t h e d e f i n i t i o n , and n o t as a p o s t u l a t e , as i n t h e t h e o r y of Gulliksen.

It i s w o r t h t o m e n t i o n t h a t P o s t u l a t e 1 o f f i n i t e n e s s of v a r i a n c e i s a u t o m a t i c a l l y s a t i s f i e d i n a p p l i c a t i o n s of t e s t t h e o r y , s i n c e as a r u l e , e v e r y t e s t i s c o n s t r u c t e d i n s u c h a way t h a t i t s p o s s i b l e r e s u l t s a r e bounded; t h i s i mplies f i n i t e n e s s o f v a r i a n c e . F o r s y s t e m ( 4 . 1 ) we a c c e p t t h e f o l l o w i n g CONSTRAINT ON OBSERVABILITY. F o r any a one may make o n l y one o b s e r v a t i o n o f X

A and g 6 G ga'

14

CHAPTER 1

T h i s c o n s t r a i n t i m p l i e s t h e i m p o s s i b i l i t y of e s t i m a t i n g

variance cf2(Xga)

f o r any f i x e d g and a .

I n c i d e n t a l l y , t h e e l e m e n t s of t h e s e t G a r e t o be i n t e r p r e t e d n o t as t e s t s , b u t as d i s t i n c t c a s e s o f use of a s p e c i f i c t e s t . I n t h e s e q u e l , t o d e s c r i b e a r e p e a t ed a p p l i c a t i o n o f t h e same t e s t , t o t h e same p e r s o n , w e s h a l l use n o t a t i o n s s u c h as X and Xha, and we s h a l l ga make some assumptions about t h e r e l a t i o n s between t h e s e random v a r i a b l e s . The c o n s i d e r a t i o n s t h u s f a r d i d n o t use t h e d i s t r i b u t i o n ?T a c c o r d i n g t o which t h e p e r s o n s a r e sampled from p o p u l a t i o n A . The randomness of' c h o i c e of p e r s o n a e A caiises t h e t r u e s c o r e 'T t o be a random v a r i a b l e . I n ga t h e s e q u e l , t h i s random v a r i a b l e w i l l be d e n o t e d b y T * i t s v a l u e s are, t h e r e f o r e , t h e unobservable t r u e g*' scores 7 and i t s d i s t r i b u t i o n depends on t h e a c c e p t ga' ed s a m p l i n g scheme, i . e . t h e distribution r. Similarly, X w i l l d e n o t e t h e random v a r i a b l e , whose g* v a l u e s a r e t h e observed s c o r e s o f t h e p e r s o n s sampled. Thus, randomness o f Xg, i s o f a "double" c h a r a c t e r : f i r s t one samples t h e p e r s o n a , hence ( f o r f i x e d g) h i s

propensity d i s t r i b u t i o n F and t h e n one samples X ga ' ga according t o t h i s propensity d i s t r i b u t i o n . I n a s i m i l a r way one d e f i n e s t h e e r r o r s c o r e I g* f o r e v e r y p e r s o n a c A we have

Since

t h e same r e l a t i o n h o l d s f o r any sampling scheme%; we

have t h e r e f o r e


15

THEOREM 4 . 1 .

X

g*

= T

g*

+ I

g*

For s u b s e q u e n t c o n s i d e r a t i o n s , i t i s n e c e s s a r y t o s t r e n g h t e n somewhat P o s t u l a t e 1, namely assume POSTULATE 1 ' .

Observe t h a t P o s t u l a t e 1' d o e s n o t f o l l o w from Postul a t e 1 of f i n i t e n e s s of v a r i a n c e for e v e r y a . However, s u c h an i m p l i c a t i o n h o l d s i n t h e most i m p o r t a n t c a s e when a l l v a r i a n c e s V a r (X ) a r e bounded b y a common c o n s t a n t , i . e . when C2(X K for a l l a E A . ga

7'

RARY THEORY OF PSYCHOLOGICAL TESTS

29

ed a t t e n u a t i o n f o r m u l a , which p r o v i d e s an answer t o a n i m p o r t a n t t h e o r e t i c a l l y and p r a c t i c a l l y q u e s t i o n a b o u t t h e r e l a t i o n between t h e t r u e s c o r e s i n two d i s t i n c t measurements. To d e r i v e t h i s f o r m u l a , l e t X and Y d e n o t e two d i s t i n c t

measurements, w i t h t r u e s c o r e s r e s p e c t i v e l y T and T y , X and e r r o r s c o r e s I x , I y . The problem l i e s i n d e t e r m i n i n g t h e c o r r e l a t i o n ; j ( T T ) X’ Y between t h e t r u e s c o r e s i n t e r m s of o b s e r v a b l e q u a n t i t i e s : c o r r e l a t i o n between o b s e r v e d s c o r e s , and r e l i a b i l i t i e s of measurements X and Y . L e t u s f i r s t d e t e r m i n e t h e c o v a r i a n c e between t h e observed scores: COV ( X , Y )

= E(XY)

- E(X)E(Y)

s i n c e a l l terms i n t h e penultimate expression, except t h o s e which e n t e r i n t o c o v a r i a n c e between TX and Ty a r e z e r o . Thus, we proved COROLLARY:

If X and

Y are distinct

measurements, t h e n

30

CHAPTER 1

To d e t e r m i n e t h e c o e f f i c i e n t of c o r r e l a t i o n between TX and T y , l e t us d i v i d e b o t h s i d e s o f ( 5 . 1 1 ) by & ( T X ) d ( T y ) and l e t u s w r i t e

We o b t a i n e d t h e r e f o r e THEOREM 5 . 3 .

where rx and Y.

and

( A t t e n u a t i o n formula)

ry a r e r e l i a b i l i t i e s of measurements X

I f XI and Y' a r e measurements p a r a l l e l t o X and Y , t h e n by theorem 5 . 3 we may w r i t e

(5.13)

The meaning o f t h e s e f o r m u l a s i s as f o l l o w s : t h e y a l l o w t o estimate t h e c o r r e l a t i o n between two ( u n o b s e r v a b l e ) c o n s t r u c t s TX and Ty, u s i n g t h e o b s e r v a b l e c o e f f i c i e n t s o f c o r r e l a t i o n between measures X and Y o f t h e s e const r u c t s , and r e l i a b i l i t i e s o f measurements X and Y.

CONTEMDORARY THEORY OF PSYCHOLOGICAL TESTS

31

The p r a c t i c a l importance o f t h e n o t i o n o f r e l i a b i l i t y i s c o n n e c t e d p r i m a r i l y w i t h two t h e o r e m s . Theorem 5 . 1 ( o r , more p r e c i s e l y , formula ( 5 . 9 ) ) c o n n e c t s t h e e r r o r s c o r e v a r i a n c e w i t h t h e v a r i a n c e of o b s e r v e d s c o r e and r e l i a b i l i t y ( w i t h r e s p e c t t o t h e same p o p u l a t i o n ) . Theorem 5 . 3 ( A t t e n u a t i o n f o r m u l a ) corinects t h e c o r r e l a t i o n between t r u e s c o r e s o f two t e s t s w i t h c o r r e l a t i o n b e t ween o b s e r v e d s c o r e s i n t h e s e t e s t s and t h e i r r e l i a b i lities. G e n e r a l l y s p e a k i n g , i n b o t h t h e s e theorems a p a r a m e t e r which i s i m p o r t a n t f o r i n f e r e n c e i s e x p r e s s e d t h r o u g h some o t h e r p a r a m e t e r s , f o r which one may p r o v i d e a n e m p i r i c a l method of e v a l u a t i o n . I n t h e f i r s t c a s e , t h e parameter i n q u e s t i o n i s t h e v a r i a n c e o f e r r o r s c o r e , needed f o r b u i l d i n g c o n f i d e n c e i n t e r v a l s f o r t r u e s c o r e , e t c . I n t h e second c a s e , t h e p a r a m e t e r i n q u e s t i o n i s t h e c o r r e l a t i o n between t r u e s c o r e s i n two t e s t s , hence a measure o f t h e e x t e n t t o which t h e s e t e s t s measure t h e same t r a i t . I n b o t h c a s e s , t h e main p o i n t l i e s i n e s t i m a t i o n of r e l i a b i l i t y . Under t h e assumption t h a t r e p e a t e d measure-

ments w i t h t h e same t e s t a r e p a r a l l e l , a n e s t i m a t e of r e l i a b i l i t y i s o b t a i n e d t h r o u g h t h e c o r r e l a t i o n between measurements. T h i s , however, r e q u i r e s two measurements o f t h e same p o p u l a t i o n , which may be sometimes i n c o n v e n i e n t . Moreover, t h e i n f l u e n c e o f f a c t o r s s u c h as l e a r n i n g , memory, p o s s i b l e d i f f e r e n c e s i n t e s t i n g cond i t i o n s , e t c . may c a u s e t h e a s s u m p t i o n o f p a r a l l e l i s m t o be i n a d e q u a t e . Thus, t h e r e a r i s e s t h e problem of f i n d i n g theorems which could s e r v e a s a b a s i s f o r cons t r u c t i o n o f e s t i m a t o r s o f r e l i a b i l i t y which u s e t h e r e s u l t s of one measurement o n l y . T h i s problem i s c l o -

32

CHAPTER 1

s e l y r e l a t e d t o other i s s u e s concerning r e l i a b i l i t y , of g r e a t p r a c t i c a l and t h e o r e t i c a . 1 i m p o r t a n c e , namely t h e r e l a t i o n s between t h e l e n g t h o f a t e s t (number of i t e m s ) and i t s r e l i a b i l i t y .

6 . R E L I A B I L I T Y OF COMPOSITE TESTS

I n t h e c o n s i d e r a t i o n s t h u s f a r we d i d n o t t a k e i n t o account t h e i n t e r n a l s t r u c t u r e c f t e s t s : t h e y were t r e a t ed as w h o l e s . I n p r a c t i c e , p s y c h o l o g i c a l t e s t s a r e u su a l l y composed o f a s e r i e s of s u b t e s t s ( i n p a r t i c u l a r , i t e m s ) , whose r e s u l t s a r e t r e a t e d a d d i t i v e l y , g i v i n g t h e f i n a l score. I n such cases it i s n a t u r a l t o a s k f o r t h e r e l a t i o n s between t h e p a r a m e t e r s o f t h e t e s t as a whole, and t h e c o r r e s p o n d i n g p a r a m e t e r s of s u b t e s t s . O f p a r t i c u l a r i m p o r t a n c e h e r e a r e t h e f o r m u l a s which e x p r e s s t h e r e l a t i o n between r e l i a b i l i t y o f t h e t e s t as

a f u n c t i o n o f r e l i a b i l i t i e s o f i t s components. We s h a l l c a r r y on t h e d e t a i l e d c a l c u l a t i o n s for t h e

s p e c i a l c a s e o f two s u b t e s t s ; t h e g e n e r a l f o r m u l a s f o r a n a r b i t r a r y number of s u b t e s t s w i l l be g i v e n w i t h o u t proof. L e t us t h e r e f o r e c o n s i d e r t h e s i m p l e s t s i t u a t i o n , when

1 + Y 2’ where Y1 and Y2 a r e d i s t i n c t ( l i n e a r l y e x p e r i m e n t a l l y i n d e p e n d e n t ) measurements, w i t h t r u e s c o r e s T and T 2 and e r r o r s c o r e s I1 and I*. T h e r e f o r e 1

X

= Y

Y1 and

= T1

+

11, Y2 = T2

+

I

2

33


X + T + I ,

where T = T1

+

T2,

I = I1

+

I

2'

L e t us f i r s t d e t e r m i n e t h e v a r i a n c e o f t h e o b s e r v e d ,

t r u e and e r r o r s c o r e s f o r X . We have h e r e

G 2 ( T ) = 62 (T1 = G

2

+

T2) = a2(Tl)

t

2( T 2 )

+

2Cov(T1,T2)

2 (T,) + G ( T 2 ) + 2p(T1,T2)r(T1)G(T2) ( 6 . 2 )

(6.3) where t h e l a s t formula f o l l o w s from t h e f a c t t h a t t h e e r r o r s c o r e s i n d i s t i n c t measurements a r e u n c o r r e l a t e d . I n a p a r t i c u l a r c a s e when Y1 and Y2 a r e p a r a l l e l mea2 2 2 s u r e m e n t s , we have d (Y,) = d ( Y 2 ) , T1 = T 2 , G (T,) = 2 2 2 ( I 2 ) .D e n o t i n g f o r s i m p l i c i t y Y, 0 ( T 2 ) and G (I1) = and Y2 by Y and Y', f o r m u l a s ( 6 . 1 ) - ( 6 . 3 ) assume a

-

s i m p l e r form

34

CHAPTER 1

(6.6) where t o o b t a i n t h e f o r m u l a ( 6 . 5 ) from ( 6 . 2 ) w e used t h e f a c t t h a t T1- T 2 , hence p ( T 1 , T 2 ) = 1. It i s t o be remarked t h a t t h e v a r i a n c e o f t r u e s c o r e i n c r e a s e d by t h e f a c t o r of 4, w h i l e t h e v a r i a n c e o f t h e e r r o r s c o r e only d o u b l e d . A s a r e s u l t , t h e r e l i a b i l i t y of t h e composite t e s t exceeds t h e r e l i a b i l i t y of e a c h o f i t s components. To s e e t h i s , i t s u f f i c e s t o d e t e r m i ne t h e r e l i a b i l i t y of t e s t X ( u n d e r t h e a s s u m p t i o n t h a t t h e components Y and Y ' a r e p a r a l l e l ) :

s i n c e C 2 ( T , ) / S 2 ( Y ) i s t h e r e l i a b i l i t y of e a c h o f t h e components Y , Y ' , hence e q u a l s P ( Y , Y ' ) by theorem 5 . 2 , t h e e q u a l i t y ( 6 . 7 ) may b e r e w r i t t e n as

(6.8)

If X and X '

a r e p a r a l l e l measurements, t h e l a s t formu l a may be w r i t t e n as

(6.9)

CONTEMPORARY THEORY OFPSYCHOLOGICAL TESTS

35

c e e d s 1, hence

which p r o v e s t h a t t h e r e l i a b i l i t y of a t e s t composed o f two p a r a l l e l s u b t e s t s e x c e e d s t h e r e l i a b i l i t y o f e a c h o f t h e components. Analogous c o n s i d e r a t i o n s a p p l y to t h e c a s e of a t e s t composed o u t o f n p a r a l l e l components, One may namely prove t h e f o l l o w i n g theorem.

THEOREM 6 . 1 . L e t t h e measurements. Y1, r a l l e l , and l e t

x

= Y

1

+

Y

2

Y2,

...,

Y

n be pa-

*

t

...

+ Yn;

moreover, l e t P d e n o t e t h e c o e f f i c i e n t o f c o r r e l a t i o n between p a r a l l e l measurements Y1, Y i . Then

(6.11) (6.12) A s may be s e e n , t h e v a r i a n c e o f t r u e s c o r e grows f a s t e r

than the variance of e r r o r score with t h e increase of t h e number of components n . A s a r e s u l t , t h e r e l i a b i l i t y of a t e s t i n c r e a s e s w i t h i t s ' l e n g t h . It i s e a s y t o compute t h e r e l i a b i l i t y of t e s t X composed o u t o f n p a r a l l e l components; i t e q u a l s t o t h e f o l l o w i n g expression:

36

CHAPTER 1

using t h e f a c t t h a t C2(T1)/Z2(Y

) =,4Y 1

we o b t a i n

THEOREM 6.2. (Spearman-Brown). The r e l i a b i l i t y of a

t e s t c o n s i s t i n g o f n p a r a l l e l components e q u a l s

(6.13)

T h i s formula c o n n e c t s t h e r e l i a b i l i t y of a t e s t w i t h

t h e r e l i a b i l i t i e s of i t s components, and a l s o shows t h a t w i t h t h e i n c r e a s e o f n ( i . e . w i t h t h e i n c r e a s e of t h e number o f components) t h e r e l i a b i l i t y of a t e s t i n c r e a s e s t o 1. Formula ( 6 . 1 3 ) i s a s p e c i a l c a s e of a more g e n e r a l t h e orem which g i v e s t h e r e l a t i o n between t h e r e l i a b i l i t y o f a t e s t and p a r a m e t e r s of i t s components ( w i t h o u t assumption of p a r a l l e l i s m ) . W e have namely t h e f o l l o w i n g theorem (proved by Guttman, 1 9 4 5 ) , which w i l l be p r e s e n t e d here w i t h o u t p r o o f : THEOREM 6 . 3 .

Let

Y2,..., Y n be d i s t i n c t measurements, and l e t X = Y1 t Y2 t ... t Y n . Then t h e r e l i a Y1,

b i l i t y of t h e t e s t X s a t i s f i e s t h e i n e q u a l i t y

(6.14) T h i s theorem a l l o w s u s t o e s t i m a t e from below t h e rel i a b i l i t y o f a t e s t , knowing o n l y t h e v a r i a n c e s of i t s

components and t h e v a r i a n c e o f t h e composite t e s t (hen-

37


ce knowing t h e v a l u e s a t t a i n a b l e on t h e b a s i s o f one measurement o n l y )

.

The q u a n t i t y which a p p e a r s on t h e r i g h t hand s i d e o f (6,14), namely

(6.15) i s called the coefficient

.J(

of Cronbach ( s e e Cronbach

1951). The c o e f f i c i e n t & may be e t e r m i n e d e m p i r i c a l l y on t h e b a s i s o f one a p p l i c a t i o n o f a t e s t ; n a t u r a l l y , i t s val u e depends on tHe p a r t i t i o n of t h e t e s t X i n t o compon e n t s Y1,.

. . , 'n

*

Let u s imagine t h a t t h e t e s t c o n s i s t s o f 2n items (subOn t e s t s ) ; one may t h e n d e t e r m i n e t h e v a l u e d = 6 2n ' t h e o t h e r hand, one may a l s o c o n s i d e r v a r i o u s p a r t i t i o n s of t h e t e s t X i n t o two components, e a c h comprisi n g n i t e m s ( s u b t e s t s ) . F o r e a c h s u c h p a r t i t i o n , one may d e t e r m i n e t h e v a l u e d = d 2 . The f o l l o w i n g i m p o r t a n t theorem p r o v i d e s r e l a t i o n between p o s s i b l e v a l u e s o f d and t h e v a l u e d 2 n . 2

THEOREM 6 . 4 .

If t h e two p a r t s of t h e t e s t a r e s e l e c t e d

a t random, t h e n

I n o t h e r words, under a random p a r t i t i o n i n t o two p a r t s of n i t e m s e a c h , t h e e x p e c t e d v a l u e o f t h e c o e f f i c i e n t o(

o!

computed f o r s u c h a p a r t i t i o n e q u a l s t h e c o e f f i c i e n t f o r t h e p a r t i t i o n i n t o 2n i t e m s .

38

CHAPTER 1

Theorem 6 . 3 a s s e r t s t h a t t h e r e l i a b i l i t y o f a t e s t i s

.

a t l e a s t as h i g h as t h e c o e f f i c i e n t d This leads t o t h e q u e s t i o n under which c o n d i t i o n s we have t h e e q u a l i t y s i g n i n formula ( 6 . 1 4 ) .

I t i s e a s y t o s e e t h a t t h i s i s t h e c a s e when t h e compoY a r e p a r a l l e l , s i n c e t h e n t h e Cronbach's nents Y1,..., n e q u a l s t h e r i g h t hand s i d e of t h e f o r m u l a of S p e a r -

man-Brown. T o prove i t , l e t us c a l c u l a t e t h e v a l u e

4 i n c a s e when

2

...

t h e t e s t s a r e p a r a l l e l . We have t h e n G ( Y , ) = = nc2(Y1). N e x t , d 2 (X) = n C 2 ( Y l ) r n [ 1 t ( n - l ) p l . Consequently ,

V 2 ( Y ) and Z G 2 ( Y i )

The a s s u m p t i o n t h a t Y1,

...,Y n

a r e p a r a l l e l i s n o t nece-

s s a r y for e q u a l i t y i n formula ( 6 . 1 4 ) . t h a t we have t h e f o l l o w i n g theorem: THEOREM 6 . 5 .

I t a p p e a r s namely

I n order t h a t t h e equality sign holds i n

( 6 . 1 4 ) i t i s n e c e s s a r y and s u f f i c i e n t t h a t t h e measurements Y

-

1'""

Y

n

be e s s e n t i a l 1 . y 9 - e q u i v a l e n t .

Thus, for t e s t s which a r e composed o u t o f n e s s e n t i a l l y c - e q u i v a l e n t components Y 1 , , , , Y n , t h e r e l i a b i l i t y

equals

.

39


I n t h e p a r t i c u l a r c a s e when t h e s u b t e s t s Y1, dichotomous i t e m s , t h e l a s t f o r m u l a becomes

where pi and "no" known as addition then

...,

Yn a r e

and qi a r e t h e p r o b a b i l i t i e s o f answers "yes" r e s p e c t i v e l y i n i - t h i t e m . T h i s formula i s t h e Kuder-Richardson f o r m u l a KR 2 0 . When i s a l l items have t h e same mean ( p r o b a b i l i t y p i ) ,

which i s t h e Kuder-Richardson formu s KR and R i c h a r d s o n 1937).

1 ( s e e Kuder

The above a n a l y s i s o f r e l i a b i l i t y o f composite t e s t s may b e summarized as f o l l o w s . By d e f j n i t i o n , t h e t r u e s c o r e of s u c h a t e s t i s t h e sum o f t r u e s c o r e s of' t h e components, and t h e same i s t r u e f o r t h e e r r o r s c o r e s . The main problem l i e s i n d e t e r m i n i n g t h e r e l i a b i l i t y of t h e t e s t , g i v e n i n f o r m a t i o n a b o u t t h e parameters o f s u b t e s t s . The main r e s u l t h e r e i s t h e i n e q u a l i t y ( 6 . 1 4 ) which a s s e r t s t h a t t h i s r e l i a b i l i t y i s not l e s s t h a n some q u a n t i t y which i n v o l v e s o n l y t h e number o f s u b t e s t s , t h e i r v a r i a n c e s and t h e v a r i a n c e o f t h e composite t e s t . This i n e q u a l i t y does n o t allow us t o , d e t e r m i n e t h e rel i a b i l i t y , b u t p r o v i d e s a lower bound f o r i t . However, a n o t h e r theorem ( o f Spearman-Brown) a l l o w s u s t o b u i l d an e s t i m a t o r of r e l i a b i l i t y ; t h i s formula i s a l s o c a l l e d sometimes t h e s p l i t - h a l f f o r m u l a , and i s based on t h e

40

CHAPTER 1

assumption t h a t t h e p a r t s a r e p a r a l l e l . C o n s e q u e n t l y , i f only one knows how t o p a r t i t i o n t h e t e s t i n t o two p a r a l l e l s u b t e s t s , one may e s t i m a t e t h e r e l i a b i l i t y , knowing o n l y t h e v a r i a n c e s o f s u b t e s t s and t h e v a r i a n c e o f t h e t e s t as a whole. The t r o u b l e l i e s , of course, i n obtaining p a r a l l e l subtests. I n p r a c t i c e , a n e s t i m a t e o f r e l i a b i l i t y by t h i s method i s o b t a i n e d by p a r t i t i o n i n g t h e t e s t a t random ( e . g . by t a k i n g odd numbered i t e m s as one s u b t e s t , and even numb e r e d i t e m s as a n o t h e r s u b t e s t ; o r a l t e r n a t i v e l y , one may group items i n p a i r s w i t h e q u a l means, and a l l o c a t e members o f p a i r s randomly t o s u b t e s t s ) . The i d e a u n d e r l y i n g random p a r t i t i o n i s t h a t whatever t r a i t i s measured by t h e t e s t , i t must a l s o be measured by i t s s u b t e s t s , i f t h e y a r e randomly chosen. The method of e s t i m a t i n g r e l i a b i l i t y based on t h i s i d e a i s c a l l e d s p l i t - h a l f r e l i a b i l i t y e s t i m a t i o n . N a t u r a l l y , i t has been g e n e r a l i z e d i n t o p a r t i t i o n i n t o more t h a n two subtests. Such an e s t i m a t i o n d o e s n o t r e q u i r e two measurements, and i s t h e r e f o r e f r e e from b i a s e s due t o l e a r n i n g , memory e f f e c t , changes of t e s t i n g c o n d i t i o n s , e t c . However, t h e d i s a d v a n t a g e l i e s i n t h e f a c t t h a t i f t h e c o n d i t i o n o f p a r a l l e l i s m i s n o t met, one o b t a i n s an underestimate of r e l i a b i l i t y . B e s i d e s t h e method o f e s t i m a t i o n , t h e theorem o f Spearman-Brown g i v e s a v e r y i m p o r t a n t t h e o r e t i c a l conseque n c e , namely i t i m p l i e s t h a t r e l i a b i l i t y of a composit e t e s t i n c r e a s e s t o 1 w i t h t h e i n c r e a s e o f t h e number o f s u b t e s t s . T h i s f o l l o w s from t h e f a c t t h a t t h e v a r -


41

ia nce of t r u e scor e i n c r e a s e s f a s t e r t h a n t h e variance of e r r o r s c o r e . G e n e r a l l y , r e l i a b i l i t y as a p a r a m e t e r depends on t h e p o p u l a t i o n under s t u d y , and i t r e p r e s e n t s p r e c i s i o n o f measurement w i b h r e s p e c t t o t h i s p o p u l a t i o n . More p r e c i s e l y , r e l i a b i l i t y depends on t h e s a m p l i n g scheme, i . e . on t h e p r o b a b i l i t y d i s t r i b u t i o n a c c o r d i n g t o which t h e s u b j e c t s are sampled. A s a r u l e , t h e s e p r o b a b i l i t i e s c o n s t i t u t e a uniform d i s t r i b u t i o n on some s e t of p e r s o n s , and a r e z e r o f o r all o t h e r p e r s o n s ; t h e r e s u l t s , however, do n o t depend on t h i s p a r t i c u l a r form of sampling d i s t r i b u t i o n . A s a t h e o r e t i c a l p a r a m e t e r , r e l i a b i l i t y i t s e l f needs

t o be e s t i m a t e d on t h e b a s i s o f a sample, u s i n g t h e a p p r o p r i a t e e s t i m a t o r s . I n s h o r t , r e l i a b i l i t y i s a parameter of a c e r t a i n e s t i m a t o r , and i t s e l f i t must be e s t i m a t e d . Both t h e p r e s e n t e d methods o f e s t i m a t i o n ( " s p l i t - h a l f " and " c o r r e l a t i o n a l " ) have t h e i r w e l l defined s t a t i s t i c a l properties, leading t o determination of b i a s and s t a n d a r d e r r o r of r e l i a b i l i t y e s t i m a t e s , and c o n s e q u e n t l y , a l l o w i n g t h e c o n s t r u c t i o n o f c o n f i dence i n t e r v a l s .

7 . HOMOGENEITY OF A TEST

T e s t s f o r which t h e i r components Y 1 , . . . ,

..

yn

a r e essen-

t i a l l y c;: - e q u i v a l e n t a r e c a l l e d homogeneous. I n t u i t i v e l y , a t e s t i s homogeneous, i f e a c h o f i t s components

measure t h e same t r a i t ( u p t o an a d d i t i v e c o n s t a n t ) , though p e r h a p s w i t h v a r y i n g a c c u r a c y .

42

CHAPTER I

Theorem 6 . 5 a s s e r t s t h a t a t e s t composed o f e s s e n t i a l l y T - e q u i v a l e n t components h a s t h e s a m a l l e s t p o s s i b l e rel i a b i l i t y , e q u a l t o t h e c o e f f i c i e n t .\ of Cronbach; conv e r s e l y , a t e s t whose r e l i a b i l i t y e q u a l s t h e c o e f f i c i e n t OC i s homogeneous. I n f o r m a l l y , homogeneity i s a concept which c o n c e r n s t h e r e l a t i o n between t h e t e s t as a whole, and i t s components (subtests), so t h a t it is r e l a t i v e with respect t o a p a 3 i t i o n o f the t e s t .

G e n e r a l l y , homogeneity means

t h a t e a c h o f t h e s u b t e s t s measures t h e same t r a i t . I n p r a c t i c e , homogeneity may be a t t a i n e d by c h o o s i n g i t e m s with appropriate content.

The t h e o r y o f t e s t s g i v e s a c r i t e r i o n f o r v e r i f y i n g whether a g i v e n t e s t i s homogeneous: t h e r e l i a b i l i t y of t h i s t e s t s h o u l d be e q u a l t o C r o n b a c h ' s

o(

,

hence

t o a q u a n t i t y which depends on t h e v a r i a n c e s o f subt e s t s and t h e v a r i a n c e o f t h e t e s t as a whole. The p r a c t i c a l a p p l i c a b i l i t y o f t h i s c r i t e r i o n r e l i e s on t h e p o s s i b i l i t y o f e s t i m a t i n g r e l i a b i l i t y and GA s e p a r a t e l y . The f i r s t may be e s t i m a t e d t h r o u g h t h e coe f f i c i e n t o f c o r r e l a t i o n i n two a p p l i c a t i o n s o f t h e

t e s t ( n a t u r a l l y , i n t h i s c a s e one s h o u l d a v o i d u s i n g t h e s p l i t - h a l f e s t i m a t o r s , s i n c e t h e y a r e based on t h e assumption o f e s s e n t i a l z - e q u i v a l e n c e , which i s t h e p r o p e r t y a b o u t which one t r i e s t o make i n f e r e n c e ) . The p o i n t i s t h a t homogeneity i s r e l a t i v e t o t h e p a r t i t i o n o f t e s t i n t o s u b t e s t s . To u n d e r s t a n d t h a t , i t i s enough t o r e a l i z e t h a t t h e r e may e x i s t t e s t s which a r e homogeneous under one p a r t i t i o n , and n o t homogeneous under a n o t h e r p a r t i t i o n . Imagine a t e s t c o n s i s t i n g o f


43

1 0 0 ( s a y ) i t e m s , which i s homogeneous under a p a r t i t i o n

i n t o 5 s u b t e s t s o f 20 i t e m s e a c h . I t i s e v i d e n t , howe v e r t h a t t h e s e 2 0 - i t e m s t e s t s need n o t be homogeneous under t h e i r p a r t i t i o n i n t o two 1 0 - i t e m t e s t s : t h e exp e c t a t i o n s i n t h e s e s u b t e s t s may be d i f f e r e n t , and onl y t h e i r sums c o r r e s p o n d i n g t o 2 0 - i t e m s u b t e s t s must be i d e n t i c a l . Thus, t h e o r i g i n a l 1 0 0 i t e m t e s t need n o t be homogeneous under t h e p a r t i t i o n i n t o 1 0 s u b t e s t s o f 10 items e a c h . F i n a l l y , i t i s w o r t h s t r e s s i n g t h a t homogeneity, unders t o o d a s "measuring t h e same" by s u b t e s t s may be f o r m a l l y e x p l i c a t e d i n v a r i o u s o t h e r w a y s , e . g . by r e q u i -

rement t h a t t h e s u b t e s t s be p a r a l l e l , e t c .

a.

VALIDITY

G e n e r a l l y s p e a k i n g , v a l i d i t y of a t e s t e x p r e s s e s t h e d e g r e e t o which t h e t e s t measures o r p r e d i c t s a g i v e n c r i t e r i o n . C o n s e q u e n t l y , v a l i d i t y of a t e s t ( o r a b a t t e r y of t e s t s ) i s r e l a t i v e t o a c r i t e r i o n , and a t e s t would t y p i c a l l y have many v a l i d i t i e s . Each o f t h e m , i n t u r n , may be e x p r e s s e d i n v a r i o u s w a y s as a c o e f f i c i e n t of some s o r t . G e n e r a l l y , one t a k e s h e r e t h e c o r r e l a t i o n c o e f f i c i e n t , which r e p r e s e n t s t h e d e g r e e o f l i n e a r p r e d i c t a b i l i t y of a g i v e n c r i t e r i o n by t h e r e s u l t s o f t h e t e s t or battery of t e s t s . The c r i t e r i a which a p p e a r h e r e may be p a r t i t i o n e d i n t o two c a t e g o r i e s : t h o s e c r i t e r i a which have t h e form o f t h e o r e t i c a l c o n c e p t s ( c o n s t r u c t s ) , and t h o s e which have t h e form o f o b s e r v a b l e v a r i a b l e s , connected w i t h t h e b e h a v i o r o f t h e s u b j e c t i n some t i m e i n t e r v a l .

44

CHAPTER 1

Thus, t h e knowledge of v a l i d i t i e s o f t h e t e s t w i t h r e s p e c t t o v a r i o u s c r i t e r i a a l l o w s u s t o d e t e r m i n e t h e val u e of t h e t e s t as a p s y c h o l o g i c a l t o o l . One can d i s t i n g u i s h two t y p e s of s t r a t e g i e s which l e a d t o d e t e r m i n i n g t h e psychologucal v a l u e of a t e s t . The f i r s t c l a s s of s t r a t e g i e s i s c o n s i s t e n t w i t h t h e

c o n c e p t of measurement, as e x p l i c a t e d i n t h e c l a s s i c a l t e s t t h e o r y . For a g i v e n t h e o r e t i c a l c o n c e p t , t h e cons t r u c t o r of a t e s t forms i t s d e s c r i p t i o n , i . e . b u i l d s a c e r t a i n t e s t . The b a s i c q u e s t i o n which a r i s e s i s whether or n o t t h e t e s t i n q u e s t i o n i s i n d e e d a t o o l for measurement o f t h e i n t e n d e d c o n s t r u c t ( i . e . whether i t i s v a l i d w i t h r e s p e c t t o t h e c o n s t r u c t which i t i s i n t e n d e d t o m e a s u r e ) . S i n c e c o n s t r u c t s are n o t d i r e c t l y m e a s ~ r a b 1 . e t~o f i n d t h i s v a l i d i t y ( c a l l e d c o n s t r u c t v a l i d i t y ) , one has t o use i n d i r e c t methods. The s u c c e s s i v e s t e p s of t h e p r o c e d u r e may b e r o u g h l y summarized as f o l l o w s . The f i r s t s t e p i s c o l l e c t i n g , on t h e ground of some p s y c h o l o g i c a l t h e o r y , o f a s e t o f a s s e r t i o n s about : ( a ) r e l a t i o n s o f t h e c o n s t r u c t under s t u d y w i t h some other constructs; ( b ) l a c k o f r e l a t i o n between t h e c o n s t r u c t under s t u d y

w i t h some o t h e r c o n s t r u c t s ;

( c ) r e l a t i o n s between t h e c o n s t r u c t under s t u d y w i t h observable v a r i a b l e s . The second s t e p c o n s i s t s o f t h e c h o i c e of t e s t s , which

are a c c e p t a b l e as measures o f c o n s t r u c t s which a p p e a r i n ( a ) and ( b ) .


45

The t h i r d s t e p c o n s i s t s o f f o r m u l a t i n g , on t h e b a s i s of

(a), (b) and ( c ) , of a s e t o f h y p o t h e s e s c o n c e r n i n g t h e measure o f t h e c o n s t r u c t ( t e s t ) . These h y p o t h e s e s a s s e r t t h a t i f t h e t e s t i s i n d e e d an a d e q u a t e measure o f t h e c o n s t r u c t which i t i n t e n d s t o measure, t h e n i t ought t o : ( A ) c o r r e l a t e h i g h l y w i t h t h e measures o f t h o s e c o n s t r u c t s which - a c c o r d i n g t o t h e t h e o r y -- s h o u l d be r e l a t ed t o t h e c o n s t r u c t measured ( t h i s t y p e o f v a l i d i t y i s r e f e r r e d t o as convergent v a l i d i t y ) ;

(B) n o t c o r r e l a t e w i t h t h e measures o f t h o s e c o n s t r u c t s which a r e -- a c c o r d i n g t o t h e t h e o r y -- n o t r e l a t e d t o the construct i n question ( t h i s i s called discriminant v a l i d i t y ) ;

(C) a l l o w a v a l i d p r e d i c t i o n o f t h o s e o b s e r v a b l e behavi o u r s which -- a c c o r d i n g t o t h e t h e o r y -- are r e l a t e d t o t h e construct i n question. Generally, t h e l a t t e r v a l i d i t i e s a r e r e f e r r e d t o as e m p i r i c a l , and d e p e n d i n g on w h e t h e r t h e y concern a c r i t e r i o n which i s r e l a t e d w i t h t h e p r e s e n t or f u t u r e b e h a v i o u r , t h e y are c a l l e d d i a g n o s t i c or p r o g n o s t i c . Finally, the fourth s t e p i s the empirical v e r i f i c a t i o n o f t h e s e t o f h y p o t h e s e s ( A ) , (B) and (C) by an approp r i a t e e x p e r i m e n t and s t a t i s t i c a l p r o c e d u r e s . It i s c l e a r , t h a t i n p a r t i c u l a r c a s e s some o f t h e s e t s ( A ) , ( B ) and ( C ) may be empty. A t any r a t e , a t l e a s t one o f t h e s e t s ( A ) and (C) must be nonempty. The reas o n i s t h a t a c o n s t r u c t about which one cannot a s s e r t a n y t h i n g p o s i t i v e c o n c e r n i n g i t s r e l a t i o n t o o t h e r con-

46

CHAPTER f

s t r u c t s o r o b s e r v a b l e v a r i a b l e s , ( o r about which one can make only n e g a t i v e a s s e r t i o n s o f t h e t y p e i n (B)) cannot be t h e o r e t i c a l l y f e r t i l e .

,

A p o s i t i v e v e r i f i c a t i o n of h y p o t h e s e s ( A ) ,

(B) and ( C ) i s a premise f o r t h e h y p o t h e s i s t h a t t h e t e s t i n q u e s t i o n i s a n a d e q u a t e measure of t h e c o n s t r u c t .

Consequently, t h e f i r s t t y p e o f s t r a t e g y l e a d s t o det e r m i n a t i o n of p s y c h o l o g i c a l v a l i d i t y of a t e s t t h r o u g h more o r l e s s p r e c i s e d e t e r m i n a t i o n of i t s s e m a n t i c s . The second t y p e o f s t r a t e g y i s o f p r i m a r i l y p r a g m a t i c c h a r a c t e r . The s t a r t i n g p o i n t h e r e i s t h a t a t e s t det e r m i n e s a c e r t a i n a p r i o r i unknown s e t of c r i t e r i a , namely t h o s e f o r which i t may p r o v i d e a good p r e d i c t i o n . T h i s t y p e of s t r a t e g y c o n s i s t s o f d e t e r m i n i n g t h i s s e t of c r i t e r i a , by method of s u c c e s s i v e t e s t i n g whether o r n o t a g i v e n v a r i a b l e may be a d e q u a t e l y p r e d i c t e d by t h e r e s u l t s of t h e t e s t . I n o t h e r words, one t r i e s t o det e r m i n e e m p i r i c a l v a l i d i t y ( d i a g n o s t i c or p r o g n o s t i c ) w i t h respect t o various c r i t e r i a . In t h i s case, the p s y c h o l o g i c a l v a l u e o f t h e test, depends on t h e t y p e o f c r i t e r i a which i t p r e d i c t s . It i s c l e a r , t h a t t h i s a p p r o a c h may a l s o lead t o det e r m i n i n g t h e s e m a n t i c s o f t h e t e s t , t h r o u g h t h e psychol o g i c a l a n a l y s i s o f t h e c o n t e n t of t h e set o f p r e d i c t e d criteria. I n p r a c t i c e , t h e c r i t e r i a a r e n o t chosen a t random from some " u n i v e r s e of c r i t e r i a " , b u t a r e s e l e c t e d s y s t e m a t i c a l l y , a c c o r d i n g to t h e g o a l o f r e s e a r c h and t h e assumed c o n t e n t o f t h e t e s t ( t h e s o - c a l l e d f a c e validity).

CONZEWORARY THEORY OF PSYCHOLOGICAL TESTS

47

To sum up, t h e two t y p e s of s t r a t e g i e s o u t l i n e d above

c o n s t i t u t e two complementary s o u r c e s o f i n f o r m a t i o n about t h e psychological value of t e s t s . I n b o t h c a s e s , t h e v a l u e of t h e t e s t o r b a t t e r y of t e s t s , i s a s s e s s e d t h r o u g h some c o e f f i c i e n t s which e x p r e s s t h e v a l i d i t i e s of t h e t e s t w i t h r e s p e c t t o v a r i ous c r i t e r i a .

9 . PREDICTION

9 . 1 . Models of r e g r e s s i o n The problem o f d e t e r m i n i n g t h e v a l i d i t y of a t e s t or b a t t e r y of t e s t s w i t h r e s p e c t t o a g i v e n c r i t e r i o n may be r e p r e s e n t e d i n an a b s t r a c t way as t h e q u e s t i o n o f t h e d e g r e e t o which a g i v e n random v a r i a b l e ( t e s t ) or a v e c t o r o f random v a r i a b l e s ( b a t t e r y ) a l l o w s t o p r e d i c t t h e v a l u e s of some o t h e r random v a r i a b l e ( c r i t e r i o n ) . It s h o u l d be s t r e s s e d , t h a t t h e t o p i c of v a l i d i t y t r e a t -

ed as "goodness" o f p r e d i c t i o n , i s c o n s i d e r e d h e r e for a f i x e d s e t of p r e d i c t o r s and a f i x e d c r i t e r i o n . The l o g i c a l f o u n d a t i o n f o r p r e d i c t i o n i s t h e e x i s t e n c e

of s t a t i s t i c a l r e l a t i o n s h i p s between t h e p r e d i c t o r s and t h e c r i t e r i o n . The c o n s t r u c t i o n of p r e d i c t o r s i s based on t h e knowledge of t h e s e r e l a t i o n s h i p s , and t h e a c t u a l q u a l i t y o f p r e d i c t i o n depends on whether t h e o b s e r v a t i o n s a r e s u b j e c t t o t h e same laws which s e r v e d as a basis f o r d e t e r m i n i n g t h e p r e d i c t o r s ( i . e . finding the optimal prognosis).

48

I n general, t o a s e t of T h i s number r i o n . Thus,

CHAPTER 1

p r e d i c t i o n c o n s i s t s o f a s s i g n i n g one number v a l u e s o f t h e o b s e r v e d random v a r i a b l e s . i s r e f e r r e d to as p r e d i c t o r f o r t h e c r i t e t h e problem i s t o d e t e r m i n e a f u n c t i o n ,

which c o u l d s e r v e as p r e d i c t o r . To f o r m a l l y d e s c r i b e t h e s i t u a t i o n , l e t us f i r s t con-

s i d e r t h e s i m p l e s t c a s e : l e t u s namely imagine t h a t t h e p r e d i c t i o n s a r e t o concern t h e v a l u e s o f a random v a r i a b l e Y , and a r e based on o b s e r v a t i o n s of a random v a r iable X. A s already stated, i n order f o r prediction t o b e p o s s i b l e , i t i s n e c e s s a r y t h a t X and Y a r e s t a tistically related. The p r o g n o s i s i s a c e r t a i n f u n c t i o n u: i f we o b s e r v e t h e v a l u e X = x, t h e n we t a k e u ( x ) as t h e p r e d i c t e d v a l u e of Y . S i n c e t h e c r i t e r i o n Y i s random, i t assumes v a r i o u s val u e s d e p e n d i n g on chance. C o n s e q u e n t l y , t h e p r e d i c t i o n is subject t o an e r r o r ( i . e . the predicted value u(x> d i f f e r s from t h e a c t u a l v a l u e t a k e n up by Y). To d e t e r m i n e t h e o p t i m a l p r e d i c t i o n , i t i s n e c e s s a r y t o

d e f i n e t h e s o - c a l l e d loss f u n c t i o n , which e x p r e s s e s num e r i c a l l y t h e consequences o f a s s e r t i n g u = u ( x ) i f i n reality Y = y. L e t u s d e n o t e by L t h e loss f m c t i o n ,

d e p e n d i n g on two v a r i a b l e s : t h e p r o g n o s i s u , and t h e v a l u e y o f t h e c r i t e r i o n . The problem of f i n d i n g t h e o p t i m a l p r e d i c t o r

may t h e n be f o r m u l a t e d as t h a t o f f i n d i n g t h e f u n c t i o n

u f o r which

EL[Y, u ( X ) ] = min.


49

Here t h e e x p e c t e d v a l u e is c a l c u l a t e d w i t h r e s p e c t t o t h e j o i n t d i s t r i b u t i o n o f X and Y . I n g e n e r a l , t h e loss f u n c t i o n depends on t h e e r r o r o f p r e d i c t i o n , t h a t i s , on t h e d i f f e r e n c e y - u ( x ) . The b e s t developed mathematically i s h e r e t h e p r e d i c t i o n t h e o r y i n which t h e l o s s f u n c t i o n i s d e f i n e d as t h e s q u a r e o f e r r o r ; t h e n t h e p a r a m e t e r which e x p r e s s e s t h e goodness of p r e d i c t i o n i s t h e mean s q u a r e e r r o r of p r o g n o s i s . T h i s t h e o r y i s known under t h e name r e g r e s s ion theory. Formally, l o s s f u n c t i o n

equals here

hence EL[Y,

u(X)]

= E[Y

-

u(X)I2.

R e g r e s s i o n t h e o r y i s based on t h e theorem which a s s e r t s t h a t i n p r e d i c t i n g a v a l u e o f a random v a r i a b l e , i n t h e s e n s e o f minimal s q u a r e e r r o r , t h e o p t i m a l p r e d i c t i o n i s a l w a y s t h e e x p e c t a t i o n of t h i s random v a r i a b l e . T h i s theorem a l l o w s t h e r e f o r e t o d e s c r i b e t h e f u n c t i o n which i s t h e o p t i m a l p r e d i c t i o n i n t h e c l a s s of a l l f u n c t i o n s ; t h i s function i s called t h e t r u e regression. For a given set of p r e d i c t o r s , t h e v a l u e of t h i s f u n c t i o n e q u a l s simply t h e e x p e c t e d v a l u e o f t h e c r i t e r i o n , g i v en t h e observations. Thus, t h e f u n c t i o n u f o r which t h e e x p e c t e d s q u a r e e r r o r i s minimal i s d e f i n e d by t h e r e l a t i o n u(x) = E(Y(X

=

x).

CHAPTER I

50

I n g e n e r a l c a s e , when t h e p r e d i c t i o n i s based on observa t i o n o f n random v a r i a b l e s X1,

.. ., X n ,

t h e problem i s

t o f i n d a f u n c t i o n u of n v a r i a b l e s s u c h t h a t ECY

-

u(X1,.

. ., X n ) l

2

=

miri.

I n a s i m i l a r w a y , t h e o p t i m a l p r o g n o s i s , which minimizes t h e mean s q u a r e e r r o r , has t h e form

The f u n c t i o n s o d e f i n e d i s c a l l e d t h e t r u e r e g r e s s i o n

.

( o f Y on (Xl,.. , X n ) ) .

A p p l i c a t i o n o f t r u e r e g r e s s i o n as a f u n c t i o n of p r e d i c t i o n l e a d s , n a t u r a l l y , t o t h e smallest mean s q u a r e e r r o r of p r e d i c t i o n , i n d e p e n d e n t l y o f t h e v a l u e s of p r e d i c t o r s ; because o f t h a t , t h e t r u e r e g r e s s i o n i s t h e b e s t p o s s i b l e p r e d i c t o r of t h e c r i t e r i o n . However, t o det e r m i n e t h e t r u e r e g r e s s i o n r e q u i r e s t h e knowledge o f t h e j o i n t d i s t r i b u t i o n of t h e v a r i a b l e s Xi and t h e c r i t e r i o n Y , and t h e f o r m u l a s a r e u s u a l l y r a t h e r cwnbersome. Because o f t h a t , i n p r a c t i c e one u s e s o f t e n t h e s o - c a l l e d l i n e a r r e g r e s s i o n , c a l l e d a l s o t h e regr e s s i o n of t h e second k i n d . It i s o b t a i n e d by r e s t r i c t i o n of t h e c l a s s o f p r e d i c t i o n f u n c t i o n s t o l i n e a r f u n c t i o n s , i . e . t o l i n e a r c o m b i n a t i o n s of t h e o b s e r v e d v a l u e s . By c h o o s i n g t h e o p t i m a l f u n c t i o n from t h i s c l a s s ( i . e . by c h o o s i n g t h e l i n e a r f u n c t i o n which give s t h e minimal mean s q u a r e e r r o r o f p r e d i c t i o n ) one o b t a i n s t h e l i n e a r r e g r e s s i o n f u n c t i o n . I n o t h e r words, l i n e a r r e g r e s s i o n i s b e s t i n t h e c l a s s of a l l l i n e a r p r e d i c t o r s . The w e i g h t s o f p r e d i c t o r s i n t h e r e g r e s s i o n


51

e q u a t i o n , and t h e f r e e t e r m , a r e c a l l e d t h e r e g r e s s i o n coefficients. F o r m a l l y , t h e c o n s i d e r a t i o n s are r e s t r i c t e d t o f u n c t i o n s o f t h e form

. ,xn)

u(xl,..

= a0

+ alxl +

... +

anxn.

The mean s q u a r e e r r o r e q u a l s t h e n E ( Y - a

0

- a X 1 1

... -

a X ) n n

2

and t h e problem r e d u c e s t o d e t e r m i n i n g t h e r e g r e s s i o n * * * c o e f f i c i e n t s , i . e . t h e v a l u e s a Q , al, ..., an which m i nimize t h e above e x p r e s s i o n . A s opposed t o t h e c a s e o f t r u e r e g r e s s i o n , d e t e r m i n i n g

t h e r e g r e s s i o n c o e f f i c i e n t s does n o t p r e s e n t t e c h n i c a l d i f f i c u l t i e s , and r e q u i r e s t h e knowledge o n l y o f t h e means, v a r i a n c e s and c o v a r i a n c e s between p r e d i c t o r s and between p r e d i c t o r s and t h e c r i t e r i o n (and n o t of t h e f u l l knowledge o f t h e ' j o i n t d i s t r i b u t i o n ) . F o r a n example, l e t u s c o n s i d e r t h e problem o f f i n d i n g t h e o p t i m a l l i n e a r p r e d i c t i o n o f Y on t h e b a s i s o f one random v a r i a b l e X . The p r e d i c t i o n f u n c t i o n i s i n t h i s c a s e of t h e form

a0

+ alX,

and t h e r e g r e s s i o n c o e f f i c i e n t s a r e t o be d e t e r m i n e d so that

E(Y

-

a.

-

alx)'

= min.

52

CHAPTER 1

One can c a l c u l a t e ( s e e e . g . Lord and Novick, 1 9 6 8 ) t h e o p t i m a l v a l u e s o f c o e f f i c i e n t s , as

a.* = E(Y)

-

E(X)

p(X,Y)-

6(X)

and

*

-

"1 -

5(Y) (j-0

JXX,Y)

*

*

*

The numbers a and a l a r e r e g r e s s i o n c o e f f i c i e n t s of 0 Y on X , and t h e o p t i m a l l i n e a r p r e d i c t i o n i s of t h e form 'predicted

*

= a.

t

*

alX.

After determining t h e optimal I-inear prediction, t h e r e a r i s e s t h e problem o f e v a l u a t i n g t h e q u a l i t y o f t h i s p r e d i c t i o n . The b a s i c q u a n t i t y which c h a r a c t e r i z e s t h e q u a l i t y of p r e d i c t i o n i s t h e c o r r e l a t i o n between t h e c r i t e r i o n and t h e o p t i m a l l i n e a r p r e d i c t i o n ( i . e . t h a t l i n e a r combination of p r e d i c t o r s which y i e l d s minimal mean s q u a r e e r r o r ) . T h i s c o e f f i c i e n t is c a l l e d t h e mult i p l e correlation coefficient. I n c a s e when t h e p r e d i c t o r s a r e t e s t s , t h e m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t i s t a k e n as a measure o f t h e v a l i d i t y of t e s t s f o r t h e c r i t e r i o n i n q u e s t i o n . To sum up, t h e l i n e a r r e g r e s s i o n , as opposed t o t r u e r e g r e s s i o n , does n o t have t o b e t h e b e s t among a l l p r e d i c t o r s ; i n o t h e r words, t h e l i n e a r and t r u e r e g r e s s i o n d o n o t need t o c o i n c i d e . The m o t i v a t i o n f o r u s e o f t h e l i n e a r r e g r e s s i o n l i e s , however, i n t h e f a c t ( b e s i d e s t h e a l r e a d y mentioned s i m p l i c i t y ) t h a t i n t h e p r a c t i -


53

c a l l y most i m p o r t a n t c a s e when t h e j o i n t d i s t r i b u t i o n o f t h e p r e d i c t o r s and t h e c r i t e r i o n i s normal, t h e lin e a r and t r u e r e g r e s s i o n c o i n c i d e . I n o t h e r words, i f t h e j o i n t d i s t r i b u t i o n i s normal, t h e t r u e r e g r e s s i o n is linear. A t t h e e n d , i t i s w o r t h t o s t r e s s t h a t t h e use of t h e m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t as a measure o f t h e q u a l i t y of p r e d i c t i o n i s s p e c i f i c a l l y r e l a t e d t o t h e use o f mean s q u a r e e r r o r as t h e l o s s f u n c t i o n . I n t h e o r y , one c a n i n t r o d u c e o t h e r loss f u n c t i o n s as measures o f e r r o r ; i n s u c h a c a s e , t h e c o e f f i c i e n t of v a l i d i t y would b e some number, s p e c i f i c a l l y r e l a t e d t o loss function, other than m u l t i p l e correlation coefficient.

P r e d i c t i n g a c r i t e r i o n i s o n l y one of t h e p o s s i b l e t a s k s of a p s y c h o l o g i s t u s i n g t e s t s o r a b a t t e r y o f t e s t s as a s o u r c e of i n f o r m a t i o n a b o u t t h e s u b j e c t . These t a s k s may b e v a r i e d : i t may b e , f o r i n s t a n c e , a c l i n i c a l d i a g n o s i s , an o p i n i o n about t h e i n t e l l i g e n c e l e v e l , a d v i c e on c h o o s i n g a p r o f e s s i o n , p r e d i c t i o n of f u t u r e succ e s s e s or f a i l u r e s , and s o on. I n e a c h c a s e , we have a s e t o f p o s s i b l e d e c i s i o n s , and t h e p s y c h o l o g i s t must make one of them. For i n s t a n c e , i n c a s e of c l i n i c a l d i a g n o s i s , t h e d e c i s i o n s a r e simply t h e c a t e g o r i e s ; i n c a s e of d e t e r m i n n g t h e l e v e l of i n t e l l i g e n c e , t h e d e c i s i o n s may be s i m p l y t h e v a l u e s of a c e r t a i n i n d e x , s u c h a s I&. I n o t h e r c a s e s , t h e d e c i s i o n s e t may c o n t a i n j u s t two e l e m e n t s , e . g . "send t o a s p e c i a l s c h o o l " a g a i n s t " n o t t o send t o a s p e c i a l school". The p s y c h o l o g i s t ' s o b j e c t i v e i s t o choose t h e d e c i s i o n

which i s o p t i m a l w i t h r e s p e c t t o t h e g i v e n s u b j e c t .

54

CHAPTER I

O p t i m a l i t y depends h e r e , of c o u r s e , on t h e a c c e p t e d c r i t e r i o n . It may g e n e r a l l y be assumed t h a t t h i s c r i t e r i o n has t h e form of a c e r t a i n v a r i a b l e , whose v a l u e i s unknown a t t h e time of d e c i s i o n making. A d e c i s i o n r u l e i s a r u l e , which c o n n e c t s d e c i s i o n s w i t h t h e observed v a l u e s of v a r i a b l e s . A r u l e l e a d s t o

e x p e c t e d l o s s connected w i t h u s i n g i t , s i n c e any r u l e may sometimes l e a d t o wrong d e c i s i o n s . An o p t i m a l de-

c i s i o n r u l e i s such which minimizes t h e e x p e c t e d loss, and t h e l a t t e r may s e r v e as a measure of v a l i d i t y o f t h e v a r i a b l e s which s e r v e f o r making d e c i s i o n ( p r e d i c t -

ors) w i t h r e s p e c t to t h e g i v e n c r i t e r i o n .

9.2. C o n s t r u c t i o n o f b a t t e r y o f p r e d i c t o r s by s c r e e n i n g

Thus f a r , we d i s c u s s e d t h e problem of d e t e r m i n i n g t h e v a l i d i t y of a t e s t o r a b a t t e r y of t e s t s w i t h r e s p e c t

to a g i v e n c r i t e r i o n . L e t us c o n s i d e r now, for a g i v e n c r i t e r i o n , t h e problem o f c h o i c e o f a b a t t e r y o f t e s t s ( f r o m a g i v e n p o o l ) which would be most v a l i d for t h i s c r i t e r i o n . More p r e c i s e l y : t h e p o o l of t e s t s and t h e c r i t e r i o n are f i x e d , and t h e problem c o n s i s t s e i t h e r on f i n d i n g a b a t t e r y w i t h t h e g i v e n s i z e and h i g h e s t v a l i d i t y , or a b a t t e r y w i t h g i v e n v a l i d i t y and s m a l l e s t possible size. Thus f a r , t h e s e problems have n o t been s o l v e d t h e o r e t i c a l l y i n a s a t i s f a c t o r y way, i . e . i n a way which would a l l o w p r a c t i c a l i m p l e m e n t a t i o n . The o b v i o u s sug g e s t i o n of i n s p e c t i n g a l l p o s s i b l e s e t s of a g i v e n s i z e and f i n d i n g t h e o p t i m a l one i s i m p r a c t i c a l , even


55

w i t h t h e use of computers, s i n c e t h e number o f s e t s t o be i n s p e c t e d i s e x c e e d i n g l y l a r g e .

I n p r a c t i c e , two approaches have been s u g g e s t e d ; none o f t h e m , however, g u a r a n t e e s t h e o p t i m a l c h o i c e . The f i r s t a p p r o a c h t e l l s t o choose a c c o r d i n g t o t h e following principles. Include i n the battery: ( a ) t h o s e t e s t s which c o r r e l a t e h i g h l y w i t h t h e c r i t e r i o n , and have p o s s i b l y l o w e s t i n t e r c o r r e l a t i o n s ; ( b ) t h o s e t e s t s which c o r r e l a t e low w i t h t h e c r i t e r i o n ,

but c o r r e l a t e h i g h l y w i t h t h e t e s t s o f t h e b a t t e r y . The i n t u i t i v e j u s t i f i c a t i o n of t h e s e p r i n c i p l e s i s as f o l l o w s . 8 ~r e g a r d s ( a ) , t h e recommendation t h a t t h e t e s t s s h o u l d c o r r e l a t e h i g h l y w i t h t h e c r i t e r i o n requi r e s no comments. On t h e o t h e r hand, t h e recommendation t h a t t h e t e s t s s h o u l d have low i n t e r c o r r e l a t i o n s i s connected w i t h t h e i n t e n t i o n of h a v i n g t e s t s , e a c h o f which s u p p l i e s new i n f o r m a t i o n about t h e c r i t e r i o n . I n d e e d , i f t h e s e t e s t s had h i g h i n t e r c o r r e l a t i o n s , t h e i n f o r m a t i o n s u p p l i e d by them would be o v e r l a p p i n g . Recommendation ( b ) i s connected w i t h t h e p o s s i b i l i t y o f t h e f o l l o w i n g phenomenon. The i n t e r c o r r e l a t i o n s o f t h e v a r i a b l e s which have h i g h c o r r e l a t i o n w i t h t h e c r i t e r i o n may be d u e , a t l e a s t i n p a r t , t o t h e f a c t t h a t t h e y measure n o t only t h e c r i t e r i o n , b u t a l s o som e o t h e r v a r i a b l e , u n r e l a t e d t o i t . The t e s t s chosen a c c o r d i n g t o recommendation ( b ) , c a l l e d s u p p r e s s o r var i a b l e s , are aimed a t e l i m i n a t i n g (by s u b t r a c t i o n ) o f t h i s u n r e l a t e d v a r i a b l e , t h u s enhancing the v a l i d i t y

CHAPTER 1

56

o f t h e b a t t e r y as a whole. It i s w o r t h s t r e s s i n g t h a t i n p r a c t i c e , good s u p p r e s s o r v a r i a b l e s are h a r d t o f i n d . Moreover, some of t h e v a r i a b l e s which were t h o u g h t t o be s u p p r e s s o r s , l o s t t h i s q u a l i t y u n d e r i n c r e a s e of t h e sample. The second a p p r o a c h t o t h e c o n s t r u c t i o n of a b a t t e r y of t e s t s by s c r e e n i n g , i s based on t h e n o t i o n of i n c r e mental v a l i d i t y . T h i s v a l i d i t y concerns a s i n g l e t e s t w i t h r e s p e c t t o a g i v e n c r i t e r i o n and b a t t e r y and i s d e f i n e d as t h e d e g r e e t o which i n c l u s i o n of t h i s t e s t

t o t h e b a t t e r y would i n c r e a s e t h e v a l i d i t y o f t h e b a t te r y

.

The programs of c h o i c e of t e s t s t o t h e b a t t e r y a r e b u i l t

on t h e b a s i s o f t h i s n o t i o n . These programs e i t h e r i n clude a

t e s t which would yie1.d h i g h e s t i n c r e m e n t a l va-

l i d i t y of a b a t t e r y w h i c h h a s alrieady been s e l e c t e d , o r p r o c e e d "backwards", t h r o u g h a n a l y s i s of t h e l o s s i n v a l i d i t y which would r e s u l t . from removing a g i v e n t e s t from t h e b a t t e r y . A s a l r e a d y mentioned, n e i t h e r o f t h e s e programs g u a r a n t e e s t h e o p t i m a l c h o i c e . The r e a s o n s a r e c o n n e c t e d w i t h t h e f o l l o w i n g phenomenon. It may namely happen t h a t i f we t a k e t h e o p t i m a l b a t t e r y of s i z e k , and add t o i t t h e t e s t w i t h h i g h e s t i n cremental v a l i d i t y , tnen t h e r e s u l t i n g b a t t e r y of s i z e ktl need n o t be o p t i m a l . Already i n t h e s i m p l e s t c a s e , i f we t a k e t h e o p t i m a l t e s t arid add t o i t t h e t e s t w i t h h i g h e s t i n c r e m e n t a l v a l i d i t y , we may o b t a i n a p a i r which i s n o t o p t i m a l . The f i n a l number of t e s t s which a r e n e c e s s a r y f o r a p r e d i c t i o n w i t h s p e c i f i e d l e v e l of v a l i d i t y cannot be


57

d e t e r m i n e d a p r i o r i , and depends on t h e c o m p l e x i t y o f the t e s t s andthe criterion. I n g e n e r a l , t h e t o p i c s of o p t i m a l c h o i c e of b a t t e r y of t e s t s c o n t a i n s t i l l many open problems.

9 . 3 . The e x p e c t e d and a c t u a l q u a l i t y o f p r e d i c t i o n Both t h e o p t i m a l p r e d i c t i o n of t h e c r i t e r i o n f o r a f i x ed t e s t (or b a t t e r y of t e s t s ) , and t h e c h o i c e o f a bat t e r y from a p o o l of t e s t s , are based on t h e knowledge of means, v a r i a n c e s and c o v a r i a n c e s between t e s t s , and between t e s t s and c r i t e r i o n . I n p r a c t i c e , t h e s e p a r a m e t e r s a r e seldom known e x a c t l y , and have t o be e s t i mated from t h e sample. S i n c e t h e a c t u a l c h o i c e o f t e s t s t o t h e b a t t e r y , and d e t e r m i n a t i o n of t h e r e g r e s s i o n coe f f i c i e n t s i s based on e s t i m a t e d v a l u e s , t h e r e a l goodn e s s of p r e d i c t i o n ( t h e v a l u e of t h e m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t ) d i f f e r s from t h e e s t i m a t e d o n e , and t h e l a t t e r a r e always h i g h e r t h a n t h e f o r m e r . Empirica l l y , t h i s means t h a t when t h e o b t a i n e d p r e d i c t i o n equa t i o n i s a p p l i e d t o a new sample, i t t u r n s o u t t h a t i t s goodness i s always lower t h a n e x p e c t e d . T h i s phenomenon i s c a l l e d s h r i n k a g e , and may o c c a s i o n a l l y be s o s e v e r e t h a t t h e p r e d i c t i o n t u r n s o u t t o be w o r t h l e s s . A famous example o f s u c h s h r i n k a g e a r e t h e r e s u l t s of

M o s t e l l e r and Wallace (1964), where t h e v a l i d i t y of p r e d i c t i o n for t h e sample was 0 . 7 , and when t h e same p r e d i c t o r s were a p p l i e d t o a d i f f e r e n t sample, t h e v a l i d i t y dropped down t o about 0 . 0 4 . The cause o f t h i s phenomenon i s t h e f a c t t h a t b o t h t h e

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c h o i c e of t e s t s t o t h e b a t t e r y and d e t e r m i n a t i o n o f r e g r e s s i o n c o e f f i c i e n t s ( w e i g h t s of p r e d i c t o r s ) a r e bas e d n o t on t h e t r u e v a l u e s of means, v a r i a n c e s and cov a r i a n c e s , but on t h e v a l u e s o b s e r v e d i n t h e sample. F i r s t l y , t h e c h o i c e of t e s t s t o t h e b a t t e r y ( c h o i c e o f p r e d i c t o r s ) uses t y p i c a l l y t h e h i g h e s t observed c o r r e l a t i o n s between t e s t s and t h e c r i t e r i o n . However, bec a u s e o f random f l u c t u a t i o n s from sample t o sample, t h e v a l u e s which were o b s e r v e d as h i g h e s t i n one p a r t i c u l a r sample need n o t be h i g h e s t i n r e a l i t y . Thus, t h e s e l e c t e d b a t t e r y need n o t be o p t i m a l , w i t h t h e d a n g e r o f c h o o s i n g a s u b o p t i m a l b a t t e r y bee-Qming h i g h e r w i t h i n c r e a s e of t h e s i z e of t h e p o o l o f t e s t s , d e c r e a s e o f t h e s i z e o f t h e b a t t e r y and d e c r e a s e o f t h e sample s i z e . The c h o i c e o f t h e b a t t e r y based on t h e o b s e r v e d , i n s t e a d of t r u e , v a l u e s o f c o r r e l a t i o n c o e f f i c i e n t s i s b i a s e d by what Lord and Novick c a l l c a p i t a l i z a t i o n on chance. The second cause of s h r i n k a g e i s connected w i t h t h e f a c t t h a t t h e c a l c u l a t i o n of r e g r e s s i o n c o e f f i c i e n t s i s a l s o based on t h e o b s e r v e d v a l u e s o f c o r r e l a t i o n s , means and v a r i a n c e s , i n s t e a d of t h e t r u e v a l u e s (which a r e n o t known). A s a r e s u l t , t h e o b t a i n e d p r e d i c t i o n , b e i n g o p t i m a l for t h e v a l u e s f o r which i t was o b t a i n e d , i s suboptimal f o r o t h e r v a l u e s . This implies t h a t t h e a c t u a l q u a l i t y o f p r e d i c t i o n w i l l a l w a y s be lower t h a n t h e expected q u a l i t y . I n p r a c t i c e , t o d e c r e a s e t h e phenomenon o f s h r i n k a g e , t h e f o l l o w i n g r o u t i n e p r o c e d u r e i s a c c e p t e d . The f i r s t s t e p i s s c r e e n i n g , or t h e c h o i c e o f t e s t s from t h e p o o l t o t h e b a t t e r y , based on t h e r e s u l t s o b t a i n e d i n t h e


59

s c r e e n i n g sample. Next, on t h e b a s i s o f a n o t h e r sample ( t h e s o - c a l l e d c a l i b r a t i o n s a m p l e ) , one c a l c u l a t e s t h e r e g r e s s i o n c o e f f i c i e n t s . F i n a l l y , t h e v a l i d i t y of p r e d i c t i o n i s t e s t e d on s t i l l a n o t h e r sample, c a l l e d t h e v a l i d a t i o n sample

.

I t i s t o be s t r e s s e d t h a t s u c h a p r o c e d u r e i s v e r y i m p o r t a n t as a p r o c e d u r e f o r c h o o s i n g p r e d i c t o r s ; i n f a c t , b e c a u s e of l a c k of s a t i s f a c t o r y t h e o r e t i c a l s o l u t i o n s , i t i s e s s e n t i a l f o r a t t a i n i n g meaningful p r e d i c t i o n s . I n c a s e of b r e a k i n g of t h e s e r u l e s , t h e p r e d i c t i o n obt a i n e d on t h e b a s i s o f one sample o n l y has only i l l u sory v a l i d i t y .

1 0 . SOME PROBLEMS OF THE CONSTRUCTION OF TESTS

I n most c a s e s o c c u r r i n g i n p r a c t i c e , t e s t s a r e b u i l t o u t of items ( t e s t p r o b l e m s , q u e s t i o n s , e t c . ) and t h e t e s t s c o r e i s t h e sum of t h e s c o r e s i n p a r t i c u l a r i t e m s . There a r i s e s t h e r e f o r e t h e problem of f i n d i n g r e l a t i o n s between a p p r o p r i a t e p a r a m e t e r s of i t e m s and p a r a m e t e r s of t h e t e s t as a whole. The p r a c t i c a l i m p o r t a n c e o f t h i s problem ' l i e s i n t h e f a c t t h a t one u s u a l l y s t a r t s w i t h a p o o l of i t e m s which a r e s u b j e c t t o some s t a t i s t i c a l o b s e r v a t i o n s , and t h e n one chooses some o f t h e se items t o t h e t e s t , It i s t h e r e f o r e e s s e n t i a l t o c o n s i d e r only t h o s e p a r a m e t e r s o f i t e m s which a r e i n v a r i a n t w i t h r e s p e c t t o

t h e c h o i c e o f t h e sample o f s u b j e c t s . I n o t h e r words, t h e p o i n t i s n o t t o c o n s i d e r p a r a m e t e r s which would be s p e c i f i c f o r a g i v e n group of s u b j e c t s o n l y .

CHAPTER I

60

are t h e o b s e r v e d s c o r e s f o r t h e i t e m s which e n t e r t h e t e s t , t h e n

If Y l a ,

Y 2 a a * - * , Yna

(10.1)

i s t h e o b s e r v e d s c o r e of p e r s o n a . Taking e x p e c t a t i o n s

w i t h r e s p e c t t o a ( i . e . w i t h r e s p e c t t o t h e sampling d i s t r i b u t i o n ) , we o b t a i n

C o n s e q u e n t l y , t h e e x p e c t e d val.ue f o r a randomly s e l e c t ed p e r s o n e q u a l s n

= E(Y ) i s t h e e x p e c t e d s c o r e o f t h e randomg g* l y chosen p e r s o n i n i t e m g . The p a r a m e t e r v i s c a l l e d g sometimes t h e d i f f i c u l t y o f i t e m g .

where

v

If yga d e n o t e s t h e o b s e r v e d s c o r e ( i n a n n element sam-

p l e ) of t h e p e r s o n a i n i t e m g , t h e n t h e u n b i a s e d e s t i mate o f t h e d i f f i c u l t y v i s g

0 ) , or i f t h e r e e x i s t s a n i n t e r a c t i o n between t e s t s and p e r s o n s ( i . e .

The example below, t a k e n from Lord and Novick (1968, pp.

179-180) i l l u s t r a t e s t h e s i g n i f i c a n c e of t h e g e n e r i c e r r o r v a r i a n c e . Imagine t h a t i n a c c e p t i n g c a n d i t a t e s f o r a c e r t a i n p o s i t i o n , one a p p l i e s a t e s t which has a number o f nominally p a r a l l e l v e r s i o n s . The v e r s i o n s are sampled a t random, and i t i s known t h a t t h e r e s u l t s have a p p r o x i m a t e l y normal d i s t r i b u t i o n w i t h s t a n d a r d d e v i a t i o n 3 . 0 0 . T o become a c c e p t e d , t h e c a n d i d a t e must s c o r e more t h a n 70.

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L e t u s imagine t h a t M r . Smith s c o r e d 6 5 , hence was n o t a c c e p t e d . Suppose t h a t he d e c i d e s t o t r y a g a i n , a f t e r two weeks ( s a y ) ; we assume t h a t ; d u r i n g t h e s e two weeks h i s t r a i t d o e s n o t change. What; i s t h e p r o b a b i l i t y t h a t he w i l l be a c c e p t e d on t h e second a t t e m p t ? To answer t h i s q u e s t i o n , l e t u s o b s e r v e t h a t t h e d i f f e r -

e n c e between two s c o r e s (assuming t h a t t h e y were chosen a t random from t h e p o p u l a t i o n of s c o r e s w i t h unknown mean and s t a n d a r d d e v i a t i o n 3 . 0 0 , and t h a t t h e r e s u l t s a r e i n d e p e n d e n t ) i s a random v r a r i a b l e w i t h mean 0 and variance 3

2

t

32

=

18, hence w i t h s t a n d a r d d e v i a t i o n

equal 3 6 . Consequently , t h e p r o b a b i l i t y t h a t t h i s d i f f e r e n c e w i l l exceed 5/(3'@) = 1.18 i s a b o u t 0 . 2 4 . Thus, w i t h p r o b a b i l i t y 0 . 1 2 M r . Smith w i l l s c o r e o v e r 7 0 on t h e second t r i a l , and w i t h t h e same p r o b a b i l i t y 0 . 1 2 he w i l l g e t a s c o r e below 6 0 . The o u t l i n e d here t h e o r y of g e n e r i c t r u e s c o r e s (which i s a s p e c i a l c a s e o f t h e t h e o r y o f g e n e r a l i z a b i l i t y ) desp i t e a n a l o g i e s w i t h t h e c l a s s i c a l theory of t e s t s , d i f f e r s i n many e s s e n t i a l a s p e c t s from t h e l a t t e r . The most i m p o r t a n t among t h e s e d i f f e r e n c e s c o n s i s t s o f t h e f a c t t h a t now t h e g e n e r i c e r r o r s f o r d i f f e r e n t p e r s o n s a r e correlated. I n s t e a d of d e r i v i n g t h e a p p r o p r i a t e f o r m u l a s , i t w i l l be i n t u i t i v e l y e x p l a i n e d why t h i s phenomenon o c c u r s . Suppose t h a t we c o i l s i d e r a s e t o f n o m i n a l l y p a r a l l e l v e r s i o n s of t h e t e s t , c o n s i s t i n g of ( s a y ) t h r e e v e r s i o n s , and t h a t one of them i s chosen a t random f o r a d m i n i s t r a t i o n . Suppose t h a t f o r p e r s o n a t h e t r u e s c o r e s i n v e r s i o n s A ,


75

= 87 ( s o t h a t , aA = 80, TaB = 85 and ‘taC a t l e a s t f o r p e r s o n a , v e r s i o n A i s t h e most d i f f i c u l t , and v e r s i o n C i s t h e e a s i e s t ) . The g e n e r i c t r u e s c o r e , b e i n g t h e a v e r a g e of t r u e s c o r e s , e q u a l s h e r e ( 8 0 + 85 + 8 7 ) / 3 = 84.

B and C a r e 7

C o n s e q u e n t l y , one may e x p e c t t h a t i f t h e v e r s i o n C i s c h o s e n , t h e g e n e r i c e r r o r s ( d e v i a t i o n s of observed s c o r e from t h e g e n e r i c t r u e s c o r e 84) w i l l t e n d t o be p o s i t i v e , while f o r v e r s i o n A they w i l l t e n d t o be n e g a t i v e . Suppose t h a t a n a l o g o u s s i t u a t i o n h o l d s f o r p e r s o n b ; l e t t h e t r u e s c o r e s be e q u a l h e r e ZbA = 30, ‘rbB = 38 and ‘Cbc = 43 (which g i v e s t h e g e n e r i c t r u e s c o r e 3 7 ) . Here a g a i n v e r s i o n A i s t h e most d i f f i c u l t , and v e r s i o n C i s the e a s i e s t ) . It i s c l e a r t h a t i f v e r s i o n C i s c h o s e n , t h e g e n e r i c

e r r o r s w i l l be p o s i t i v e b o t h for p e r s o n a and b y w h i l e under c h o i c e o f v e r s i o n A , b o t h e r r o r s w i l l t e n d t o be n e g a t i v e . G e n e r a l l y , one may prove t h a t f o r a f i x e d v e r s i o n o f t h e t e s t , t h e g e n e r i c e r r o r s of two p e r s o n s i n t h i s v e r s i o n w i l l t e n d t o be p o s i t i v e l y c o r r e l a t e d ( t h e r e w i l l be a tendency f o r t h e e r r o r s c o r e s t o b e o f t h e same s i g n ) . The d i f f e r e n c e s , as compared w i t h t h e c l a s s i c a l t e s t t h e o r y , c o n c e r n i n g a s i n g l e t e s t , and n o t a s e t o f nominally p a r a l l e l t e s t s , are as follows:

For a f i x e d t e s t g: 1) t h e g e n e r i c e r r o r s

# 0).

C

ga

may be b i a s e d ( i . e . E( Cga)

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2 ) t h e g e n e r i c e r r o r s c o r e s may be c o r r e l a t e d w i t h t h e

generic true scores;

3 ) The g e n e r i c e r r o r s c o r e s o f one v e r s i o n o f t h e t e s t may be c o r r e l a t e d w i t h g e n e r i c e r r o r s c o r e s i n a n o t h e r version;

4) t h e g e n e r i c e r r o r s c o r e s of a p e r s o n , under r e p e t i t i o n of a t e s t ( u s i n g a n o t h e r nominally p a r a l l e l v e r s i o n ) may be l i n e a r l y e x p e r i m e n t a l l y d e p e n d e n t . These f a c t s , n a t u r a l l y , c o m p l i c a t e t h e t h e o r y c o n s i d e r a b l y . It i s w o r t h t o m e n t i o n , however, t h a t t h e s e com-

p l i c a t i o n s make t h e t o those situations ricians: situations a r e b i a s e d , and n o t under r e p e t i t i o n .

theory of generic scores applicable which a r e t r o u b l i n g t h e psychometi n which t h e e r r o r s o f measurements l i n e a r l y e x p e r i m e n t a l l y independent

11.3. Schemes of e x p e r i m e n t s The c o n s i d e r a t i o n s t h u s f a r concerned o n l y t h e i n f l u -

e n c e o f v a r i o u s v e r s i o n s o f t h e t e s t on t h e s c o r e and on t h e v a r i a n c e of e r r o r s c o r e . To i l l u s t r a t e t h e r i c h n e s s o f t h e schemes of a n a l y s i s of v a r i a n c e , i t i s worth w h i l e t o s y s t e m a t i c a l l y l i s t a l l p o s s i b i l i t i e s which a r i s e i n t a k i n g i n t o a c c o u n t two f a c t o r s . Thus, imagine t h a t w e c o n s i d e r a group o f p e r s o n s , d e n o t e d by 1 , 2 , N , and we a n a l y s e t h e i n f l u e n c e o f two f a c t o r s , w i t h f a c t o r I o p e r a t i n g on l e v e l s A , B , C , and In special f a c t o r I1 o p e r a t i n g on l e v e l s ~ , b , c , may d e n o t e v e r s i o n s o f t h e t e s t , s i t u a t i o n s , A,B, while a , b , c , may d e n o t e e x p e r i m e n t a l c o n d i t i o n s ( e . g .

...,

...

.. .

... .

...


77

t h e y may s t a n d for a r b i t e r s , d u r a t i o n s o f t e s t i n g , lighting conditions, e t c . ) . G e n e r a l l y , one may d i s t i n g u i s h two c a t e g o r i e s of t h e w a y s i n which t h e p e r s o n s and l e v e l s of f a c t o r s a r e combined: c r o s s e d d e s i g n and n e s t e d d e s i g n . I f e a c h p e r s o n i s t e s t e d on e a c h l e v e l of a g i v e n f a c -

t o r , we s p e a k of a c r o s s e d d e s i g n of p e r s o n s w i t h t h i s f a c t o r ; s i m i l a r l y , i f e a c h l e v e l o f one f a c t o r i s comb i n e d w i t h e a c h l e v e l of a n o t h e r f a c t o r , we s a y t h a t these factors are crossed. If e a c h o f t h e p e r s o n s i s t e s t e d under some l e v e l s of

a f a c t o r ( n o t a l l ) , or i f t h e l e v e l s o f one f a c t o r a r e combined w i t h some l e v e l s of t h e o t h e r f a c t o r o n l y , w e speak of a n e s t e d d e s i g n . For one f a c t o r , t h e s e p o s s i b i l i t i e s a r e b e s t i l l u s t r a t ed on t h e a p p r o p r i a t e diagrams ( s e e F i g . 11.1 and 1 1 . 2 ) . Factor A

B

C

D

Persons

F i g . 11.1.

Crossed d e s i g n

E

F

78

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Factor A

B

C

D

E

F

G

H

I

F i g . 1 1 . 2 . Nested d e s i g n

The g r a p h i c a l r e p r e s e n t a t i o n of' s e v e r a l p o s s i b l e exper i m e n t a l schemes i n c a s e of two f a c t o r s i s g i v e n on

Factor I A

B

C

D

E

F

Factor

Persons

Fig.

11.3. F u l l c r o s s e d d e s i g n for four p e r s o n s and two f a c t o r s


Factor I

B

A

F a c t o r I1

a

c

b

d

19

C

e

f

g

h

i

I

I

1

I

Persons

4

Fig. 11.4.

N e s t i n g o f F a c t o r , I1 i n F a c t o r I , c r o s s ed w i t h p e r s o n s .

Factor I

A

F a c t o r I1

a

B

b

c

d

I

1

Persons

Fig.

-t

1 1 . 5 . A mixed d e s i g n : c r o s s i n g o f f a c t o r I1 a n d persons, nested with F a c t o r I.

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A c o n v e n i e n t s y s t e m a t i z a t i o n of' schemes o f e x p e r i m e n t s

may be o b t a i n e d when one i n t r o d u c e s t h e f o l l o w i n g n o t a t i o n ( s e e Cronbach e t a l . , 1 9 7 2 ) : t h e s e t of p e r s o n s , as w e l l as s e t s of l e v e l s of f a c t o r s , a r e r e p r e s e n t e d by c e r t a i n a r e a s , w i t h t h e c r o s s e d d e s i g n d e n o t e d as

on F i g . 1 1 . 6 , and a n e s t e d d e s i g n - as on F i g . 1 1 . 7 .

F i g . 1 1 . 6 . Symbolic representation of

F i g . 1 1 . 7 . Symbolic r e p r e s e n t a t i o n of

a crossed design

a nested design

D e n o t i n g by c o n t i n u o u s l i n e t h e s e t of p e r s o n s , and by dashed and d o t t e d l i n e s f a c t o r s I and 11, v a r i o u s schemes may be r e p r e s e n t e d by c o n f i g u r a t i o n s o f domains on t h e p l a n e .

For i n s t a n c e , t h e f i r s t scheme, when a l l f a c t o r s a r e c r o s s e d , i s shown on F i g . 11. 8. The second scheme i s p r e s e n t e d on F i g . 1 1 . 9 . The convenience h e r e c o n s i s t s p r i m a r i l y of t h e f a c t t h a t one can r e a d immediately which components of v a r i a n c e o f t h e o b s e r v e d s c o r e can be e s t i m a t e d from d a t a a r i s i n g from a g i v e n e x p e r i m e n t a l scheme.

81


/---

/

-

I

/

I

/

/ \

i

\ I

F i g . 1 1 . 8 . Schematic r e p r e s e n t a t i o n of s y s t e m from F i g . 11.3

F i g . 1 1 . 9 . Schematic representation of s y s t e m from F i g . 1 1 . 4

I n g e n e r a l , under s u c h g r a p h i c a l r e p r e s e n t a t i o n , a n a r e a which b e l o n g s t o one domain o n l y r e p r e s e n t s t h e p o s s i b i l i t y of e s t i m a t i n g t h e e f f e c t of t h e f a c t o r which i s r e p r e s e n t e d by t h i s domain. Also, a n a r e a which bel o n g s t o two domains, r e p r e s e n t s t h e e f f e c t of i n t e r a c t i o n of t h e c o r r e s p o n d i n g f a c t o r s , w h i l e t h e i n t e r s e c t i o n o f a l l t h r e e domains r e p r e s e n t s t h e i n t e r a c t i o n of a l l t h r e e f a c t o r s p l u s e r r o r . D e n o t i n g by S t h e i n f l u e n c e of t h e s u b j e c t , by I and I1 t h e e f f e c t s of p a r t i c u l a r f a c t o r s , by e t h e e r r o r of measurement, and by p a i r s o f t h e form ( S , I ) , (1,II) e t c . t h e c o r r e s p o n d i n g i n t e r a c t i o n s , one may e s t i m a t e a l l 7 components i n t h e complete c r o s s e d d e s i g n ( s e e Fig. 11.10). I n o t h e r words, a complete c r o s s e d d e s i g n a l l o w s t o estimate t h e e f f e c t s from p e r s o n s (S), o f f a c t o r s I and 11, i n t e r a c t i o n of p e r s o n s and f a c t o r I , e t c . The t r i p -

82

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l e i n t e r a c t i o n ( S , I , I I ) may be s e p a r a t e d from t h e e r r o r

e only i n c a s e o f r e p e a t e d measurements. If f o r e v e r y p e r s o n and e v e r y combination of l e v e l s of f a c t o r s I and I1 one makes o n l y one measurement, t h e t r i p l e i n t e r a c t i o n cannot be s e p a r a t e d from e r r o r of measurement. I n t h e c a s e o f t h e second scheme under c o n s i d e r a t i o n , t h e s i t u a t i o n i s as on F i g . 1.11.

/--

'\

S

1 I

_ -

\

--. \

,,

.

/

IJ

F i g . 1 1 . 1 0 . Components o f v a r i a n c e i n a complete c r o s s e d d e s i g n I n t h i s scheme one may s e p a r a t e t h e e f f e c t s of p e r s o n s S , f a c t o r I , and i n t e r a c t i o n between S and I , but one


83

.-

,/'

, .,

,

.*

.

1

,

I'

\ \

/

i \

i

(S,

F i g . 11.11. Components o f v a r i a n c e i n nested design

c a n n o t e s t i m a t e s e p a r a t e l y t h e e f f e c t s of f a c t o r I1 and

i n t e r a c t i o n between 1 and I1 ( t h e s e a r e n o t s e p a -

r a b l e ) . Thus , t h i s scheme o f e x p e r i m e n t s a l l o w s t h e p a r t i t i o n o f t h e v a r i a n c e of o b s e r v e d s c o r e i n t o 5 components. N a t u r a l l y , for e a c h e x p e r i m e n t a l scheme t h e r e e x i s t comp u t a t i o n a l f o r m u l a s f o r e s t i m a t i o n o f d i f f e r e n t compon e n t s o f v a r i a n c e ; t h e s e f o r m u l a s w i l l be p r e s e n t e d h e r e o n l y f o r t h e s i m p l e s t c a s e o f one f a c t o r and f u l l crossed design. To simplify t h e formulations,

l e t u s assume t h a t t h e l e v e l s of t h e f a c t o r a r e v e r s i o n s of t h e t e s t , g = 1,

...,

2, n , and t h a t t h e r e a r e N s u b j e c t s ( a =,l,...,N). We s h a l l c o n s i d e r two c a s e s : e x p e r i m e n t w i t h o u t r e p e t i t i o n s , and e x p e r i m e n t w i t h r e p e t i t i o n s (r t i m e s , where r > 1). I n o t h e r words, each person i s connected w i t h

84

CHAPTER 1

e a c h Level of t h e f a c t o r ( v e r s i o n o f t h e t e s t ) , w i t h e i t h e r one o r r measurements. I n t h e f i r s t c a s e , of scheme w i t h o u t r e p e t i t i o n s , we d e a l t h e r e f o r e w i t h a scheme p r e s e n t e d on F i g . 1 1 . 1 2 . Versions of t h e t e s t

1

2

...

3

...

...

n-1

n

... ...

F i g . 1 1 . 1 2 . Scheme of e x p e r i m e n t s w i t h o u t repetitions I n c a s e of r e p e t i t i o n s , t h e y may be t r e a t e d as l e v e l s of some f a c t o r , and t h e scheme o f e x p e r i m e n t s looks as on F i g . 1 1 . 1 3 . Thus, i n t h i s case each person responds r times t o each

v e r s i o n o f t h e t e s t ( i n g e n e r a l , r = 2 , as a l r e a d y mentioned).

a) ga

Scheme w i t h o u t r e p e t i t i o n s . N o t a t i o n s :

-

t h e s c o r e of p e r s o n a i n t e s t g;

1 N yg+ = R a=1 Yga

z

-

mean s c o r e i n t e s t g;


1

n

85

N

- t h e g e n e r a l mean s c o r e . g=l a=l

F i n a l l y , l e t G d e n o t e t h e number of t e s t s used i n t h e e x p e r i m e n t , of which n a r e s e l e c t e d a t random ( i n p a r t i c u l a r c a s e we may have G = n , when a l l v e r s i o n s a r e a n a l y s e d , of G = , which c o r r e s p o n d s t o t h e s i t u a t i o n when t h e r e a r e i n f i n i t e l y many v e r s i o n s ) . XJ

The e s t i m a t o r s o f components of v a r i a n c e a r e t h e n r e p r e s e n t e d i n T a b l e 11.1. The f o u r t h column of t h i s t a b l e c o n t a i n s q u a n t i t i e s , whose e s t i m a t o r s a p p e a r i n t h e 2 second column. Here B z i s t h e v a r i a n c e o f t h e g e n e r i c 2 t r u e s c o r e , 6 (I**) i s t h e v a r i a n c e of t h e g e n e r i c e r i s t h e v a r i a n c e due t o i n t e r a c t i o n , and : 0 is r o r , c2 $54 t h e v a r i a n c e connected w i t h d i f f e r e n c e s o f d i f f i c u l t y of v a r i o u s v e r s i o n s . A s may be s e e n , scheme w i t h o u t r e p e t i t i o n does n o t a l 2 2

low t o e s t i m a t e s e p a r a t e l y

G

(I**) and

r-

.

b ) Scheme w i t h r e p e t i t i o n s . N o t a t i o n s

'gak

-

t h e s c o r e of p e r s o n a i n v e r s i o n g of t h e t e s t ,

a t k - t h r e p e t i t i o n ( k = 1,. 'gar

- 1' - - kz' Ygak -- mean s c o r e of p e r s o n a i n v e r s i o n g; =l N

Yg**

.., r ) ;

-k

rN

r

2 2 a=l

k=l

Ygak

--

mean s c o r e i n t e s t g;

86

CHAPTER 1

.

n

N

r

The r e m a i n i n g n o t a t i o n s a r e as b e f o r e . A s may be s e e n from T a b l e 1 1 . 2 , i n t h i s c a s e i t i s pos s i b l e to s e p a r a t e t h e e f f e c t o f i n t e r a c t i o n from t h e v a r i a n c e of e r r o r s c o r e . Repetitions

n Ver s i ons of t e s t

1

2

/

r

,

... +

! . ,'1. . /. . . /.ii

i

2

,

,/ .

I

1

...

~

.

,

... ...

1

Persons

1

.

/

r

i

/ / I

i '

11

' '

I

,i

i

* ,,

,,

.. .. ... I

N

...

F i g . 1 1 . 1 3 . The scheme of e x p e r i m e n t w i t h

r repetitions

v n

+ v

H

n

+ **

c \

H

**

v

+ **

I

I

ri

G

I

ri

n

z

V

n

N

(d

** +h * h I * M M

(d

h I

I

h

**

n

N

z r:

v

ri

n

b

cu

H

b

N

r-! ri

I

r:

I

z

cu \3

V


c

0 .ri

P V

d

P

a,

a x

cii

k 0

~~

cu n

N n

I

h

* *

rn

I

h

**

k 5

a,

0

(d

v)

0

k

5

v)

87

T a b l e 1 1 . 2 . A n a l y s i s of v a r i a n c e i n c r o s s e d d e s i g n with r e p e t i t l o n s

(from L o r d and Novick, 1 9 6 8 )

'

S o u r c e of variability

Sum o f s q u a r e s

1

DegEFes freedom

I

' %

Between subjects

c N ( Y * ~ +- y***

rn

T-

a=1 n

I 1 I

Between

tests

I

Repetitions

r

1

N-1

!

i n-1 I

N 2 L ( Y ~ ~ * - Y ~ * ~ - Y * ~ , + Y , * *) (N-l)(n-l) g=l a = l

i'

n N -

2-

I-'

r i (ygak

g=l a=l k=l I

- Y*** ) 2

1

n

Interaction

I

z (Yg** g= 1

rN

I

i

)2

-

yga* ) *

nN(r-1)

Expect a t i o n


89

For n e s t e d d e s i g n s , t h e f o r m u l a s f o r t h e c o r r e s p o n d i n g e s t i m a t o r s a r e s i m i l a r , and w i l l n o t b e r e p r o d u c e d h e r e . A t t h e end i t i s w o r t h w h i l e t o p o i n t o u t t h e f a c t t h a t

w h i l e t h e complete c r o s s e d d e s i g n a l l o w s f o r e s t i m a t i n g a l l components o f v a r i a n c e , i n c l u d i n g a l l i n t e r a c t i o n s , t h i s i s a t t a i n e d a t t h e c o s t of performing a d i f f i c u l t e x p e r i m e n t : e a c h p e r s o n must be t e s t e d a t e v e r y combination of t h e l e v e l s o f f a c t o r s . Nested d e s i g n s are much e a s i e r t o implement i n p r a c t i c e , b u t t h e y p r o v i d e l e s s i n f o r m a t i o n , by n o t a l l o w i n g

t o s e p a r a t e t h e e f f e c t s o f c e r t a i n f a c t o r s from some interactions. I n g e n e r a l , one c o n s i d e r s two t y p e s of f a c t o r s . Thus, w e speak of f i x e d e f f e c t s , i f t h e l e v e l s of f a c t o r a r e d e t e r m i n e d b y t h e e x p e r i m e n t e r i n some s y s t e m a t i c w a y , and e x h a u s t a l l p o s s i b i l i t i e s . S e c o n d l y , w e s p e a k o f random e f f e c t s , if t h e l e v e l s are s e l e c t e d a t random from a c e r t a i n u n i v e r s e . T h i s may b e s t be s e e n from t h e f o r m u l a s p r e s e n t e d i n T a b l e s 11.1 and 1 1 . 2 . If t h e e x p e r i m e n t e r h a s G v e r s i o n s o f t h e t e s t a t h i s

d i s p o s a l , and p r o c e e d s t o a n a l y s e a l l o f them s y s t e m a t i c a l l y ( n = G ) , t h e n v e r s i o n of t h e t e s t i s a f a c t o r w i t h c o n s t a n t e f f e c t s . Such a p r o c e d u r e i s p o s s i b l e o n l y f o r r e l a t i v e l y small G . If t h e s e t o f t e s t s Is numerous, s o t h a t i t i s impossib l e t o a n a l y s e a l l of them, one samples n t e s t s f o r t h e s t u d y . We d e a l i n s u c h a c a s e w i t h random e f f e c t s . I n p a r t i c u l a r , one may assume G = aand t r e a t t h e n t e s t s as s e l e c t e d a t random from a c e r t a i n i n i f i n i t e "uni-

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CHAPTER I

v e r s e " of t e s t s p o s s i b l e i n a given s i t u a t i o n . G e n e r a l l y s p e a k i n g , i n t e s t i n g problems, one d e a l s most o f t e n w i t h random e f f e c t s . It i s n o t p o s s i b l e to g i v e t h e b e s t e x p e r i m e n t a l schem e : t h e c h o i c e h e r e depends on t h e g o a l of s t u d y , and

a l s o on t h e s u b j e c t i v e c o n v i c t i o n o f t h e e x p e r i m e n t e r t h a t s u c h and s u c h i n t e r a c t i o n s a r e z e r o . I f he t h i n k s t h a t c e r t a i n i n t e r a c t i o n s v a n i s h , he may choose t h e corresponding nested design, thus simplifying t h e s t u dy. F i n a l l y , i t i s w o r t h to m e n t i o n t h a t t h e t h e o r y of gen e r a l i z a b i l i t y , while leading i n general t o a decrease o f e r r o r o f measurement (by s e p a r a t i n g v a r i o u s compon e n t s of v a r i a n c e ) , p r o v i d e s o f t e n u n s a t i s f a c t o r y e s t i mation of t h e t r u e s c o r e . To i l l u s t r a t e t h i s phenomenon, and a l s o i n o r d e r to s k e t c h new a p p r o a c h e s to e s t i m a t i o n o f t h e t r u e s c o r e , i t i s b e s t t o use a n example. I n g e n e r a l , under t h e a s s u m p t i o n t h a t t h e d i s t r i b u t i o n of e r r o r s c o r e s , i.e. d e v i a t i o n s between t h e t r u e and observed s c o r e i s normal, one c o n s t r u c t s c o n f i d e n c e i n t e r v a l s f o r t h e unknown t r u e s c o r e .

For i n s t a n c e , i n c a s e o f c o n f i d e n c e l e v e l 0 . 9 5 , t h e c o n f i d e n c e i n t e r v a l (X - 1 . 9 6 6 , X t 1.966) w i l l c o v e r t h e unknown t r u e s c o r e w i t h p r o b a b i l i t y 0 . 9 5 . C o n s t r u c t i o n of confidence i n t e r v a l s i s of g r e a t pract i c a l i m p o r t a n c e i n t h e p h y s i c a l s c i e n c e s , where t y p i c a l l y , t h e e r r o r o f measurement i s small as compared

91


w i t h t h e w i d t h o f " u n c e r t a i n t y band", i n which

i n g the t h e experimenter which he m e a s u r e s .

--

--

accordl i e s t h e unknown t r u e v a l u e

For example, i n t h e p h y s i c a l s c i e n c e s i t i s s e n s i b l e t o a s s e r t t h a t t h e t r u e weight of a g i v e n o b j e c t e q u a l s 15.03 ? 0 . 0 2 grams, where t h e v a l u e 0 . 0 2 was o b t a i n e d from p r e v i o u s s t u d i e s o f t h e mean s q u a r e e r r o r of t h e measurement t o o l (one u s u a l l y o m i t s h e r e t h e c o n f i d e n c e l e v e l ) . Such a n a s s e r t i o n c o n t a i n s s i g n i f i c a n t informa t i o n : b e f o r e measurement was t a k e n , from an i n s p e c t i o n of t h e o b j e c t , knowledge, e x p e r i e n c e , and s o f o r t h , one c o u l d a s s e r t o n l y ( s a y ) t h a t i t s weight l i e s w i t h i n t h e l i m i t s 1 2 and 18 grams. The p o i n t i s t h a t i n most c a s e s , t h e w i d t h of " u n c e r t a i n t y band" ( 1 2 - 18 grams) i s c o n s i d e r a b l y l a r g e r t h a n t h e w i d t h o f t h i s band a f t e r t h e measurmement was t a k e n . I n p s y c h o l o g y , t h e s i t u a t i o n i s , a s a r u l e , q u i t e oppos i t e : t h e w i d t h of " u n c e r t a i n t y band" a p r i o r i ( b e f o r e t h e measurement was t a k e n ) i s n o t much l a r g e r t h a n t h e w i d t h o f t h e band a f t e r measurement.

For i n s t a n c e , imagine t h a t we measure I&of a psychol o g y j u n i o r . Without any measurement, we may r e a s o n a b l y e x p e c t h i s I& t o be somewhere between 1 1 0 and 130: t h e r e w i l l be o n l y few s t u d e n t s w i t h I&lower t h a n 1 1 0 , and a l s o few w i t h I& e x c e e d i n g 130. Suppose t h a t t h e measuremer.t gave t h e r e s u l t 118, and w e a l s o know t h a t t h e s t a n d a r d d e v i a t i o n i s 5 . Then t h e c o n f i d e n c e i n t e r v a l , w i t h c o n f i d e n c e l e v e l 0 . 9 5 , w i l l be 118 ? 1.96.5 which i s about ( 1 0 8 , 1 2 8 ) . The p r o f i t from measurement i s h e r e r a t h e r mediocre: a f t e r measurement w e know, i n e s s e n c e , as much as we knew b e f o r e anyway.

92

CHAPTER 1

Such a s i t u a t i o n , o f r a t h e r c o n s i d e r a b l e p r i o r knowledge and l a r g e e r r o r of measurement, i s t y p i c a l i n measurement i n psychology, and t h e s o c i a l s c i e n c e s i n gen e r a l . It c a s t s some d o u b t s on t h e s e n s e o f u s i n g confidence intervals. A p o s s i b l e a p p r o a c h h e r e i s t o use B a y e s i a n e s t i m a t o r s

( s e e e . g . Novick 1969, 1971; Novick and G r i z z l e 1965; Novick and J a c k s o n 1970, o r Novick, Jackson and T h a y e r 1 9 7 1 ) . These e s t i m a t o r s u s e t h e a p r i o r i knowledge, r e p r e s e n t e d i n form o f an a p p r o p r i a t e p r i o r d i s t r i b u t i o n , and c o n s i s t o f t a k i n g t h e mode of t h e p o s t e r i o r d i s t r i b u t i o n ( g i v e n t h e observed s c o r e ) as a n e s t i m a t e of t h e t r u e s c o r e . F o r m a l l y , l e t f ( T ) be t h e p r i o r d e n s i t y of t h e t r u e s c o r e , and l e t g ( X [ T ) be t h e d e n s i t y o f t h e o b s e r v e d s c o r e X f o r g i v e n t r u e s c o r e T . Then

i s t h e c o n d i t i o n a l d e n s i t y of t h e t r u e s c o r e T g i v e n

t h e observed s c o r e X . A s B a y e s i a n e s t i m a t o r , one u s u a l l y t a k e s t h e mode o f the d i s t r i b u t i o n h ( X / T ) , t h a t i s t h e v a l u e of T (de-

p e n d i n g on X ) which m a x i m i z e s h ( T / X ) . The problem, n a t u r a l l y , l i e s i n f i n d i n g t h e p r i o r d i s -

t r i b u t i o n which would b e i n t u i t i v e l y a c c e p t a b l e as a f a i t h f u l r e p r e s e n t a t i o n o f t h e c o n v i c t i o n s and knowledge o f t h e e x p e r i m e n t e r , and a t t h e same t i m e which would lead t o r e l a t i v e l y s i m p l e c o m p u t a t i o n a l f o r m u l a s f o r estimators.


93

1 2 . B I A S I N SELECTION PROCEDURES

The use o f t e s t s i s o f t e n s e v e r e l y c r i t i c i z e d because of t h e p o s s i b i l i t y o f d i s c r i m i n a t i o n i n s o c i a l p o l i c i e s . We b e g i n w i t h p r e s e n t i n g t h e f o r m u l a s f o r t h e b e s t lin e a r e s t i m a t o r o f t h e t r u e s c o r e of t h e s u b j e c t , given h i s observed s c o r e X .

Y = Yp i s t h e mean s c o r e i n t h e p o p u l a t i o n s u b j e c t , and w = wp i s t h e r e l i a b i l i t y o f t h e p o p u l a t i o n P , t h e n t h e r e g r e s s i o n estimate o f s c o r e o f t h e s u b j e c t who s c o r e d X on t h e t e s t If

z

=

wx t ( 1 - w ) Y

P of the test in the true is (12.1)

( s e e Lord and Novick 1 9 6 8 ) . Formula ( 1 2 . 1 ) i m p l i e s t h a t i f t h e t e s t i s h i g h l y r e l i a b l e ( w i s c l o s e t o l), t h e n t h e o b s e r v e d s c o r e X i s g i v e n much weight i n t h e assessment o f t h e t r u e s c o r e . I f t h e t e s t i s n o t h i g h l y r e l i a b l e , l e s s weight i s g i v e n t o t h e o b s e r v e d s c o r e X , and more t o t h e g e n e r a l i n f o r m a t i o n about t h e s u b j e c t , namely t h a t he i s a member o f a p o p u l a t i o n whose e x p e c t e d t r u e score i s Y . T h i s appears r e a s o n a b l e and c o n v i n c i n g ; y e t , i t may l e a d

t o somewhat p a r a d o x i c a l r e s u l t s . Suppose, f o r i n s t a n c e , t h a t t h e t e s t i s used t o s e l e c t b e s t c a n d i d a t e s from a number o f groups ( e . g . s c h o o l s , or c l a s s e s w i t h i n t h e same s c h o o l , s o c i a l g r o u p s , e t c . ) . To f i x i d e a s , s u p p o s e t h a t a f o u r t h - g r a d e r z1 from some s c h o o l s c o r e d 4 5 on t h e t e s t i n q u e s t i o n , and t h e same s c o r e ( 4 5 ) was o b t a i n e d by a fif'th-grader z 2 . Suppose t h a t t h e r e l i a b i l i t i e s o f t h e t e s t i n t h e p o p u l a t i o n s of f o u r t h - g r a d -

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e r s and i n t h e p o p u l a t i o n o f f i f t h - g r a d e r s a r e t h e same and e q u a l 0 . 8 0 . F i n a l l y , suppose t h a t t h e mean s c o r e i n t h e p o p u l a t i o n of f o u r t h g r a d e r s i s Y = 4 0 , and i n t h e population of f i f t h graders it i s Y = 50. I n t h i s case, the estimator (12.1) yields f o r subject

z1 ( a f o u r t h - g r a d e r ) t h e e s t i m a t e d t r u e s c o r e Z = 0 . 8 . 4 5 + 0.2-40 = 44. S i m i l a r l y , for t h e f i f t h - g r a d e r z2 w e get Z = 0.8.45

+

t r u e s c o r e of z

1

0.2.50

=

46. We s e e t h a t t h e e s t i m a t e d

i s lower t h a n t h a t of z2, d e s p i t e t h e

f a c t t h a t z1 s c o r e d above a v e r a g e i n h i s r e f e r e n c e g r o u p , w h i l e z2 s c o r e d below a v e r a g e i n h i s r e f e r e n c e g r o u p . I n a sense, z

1 i s b e i n g "punished" f o r membership i n a group w i t h low a v e r a g e , w h i l e z2 i s b e i n g "rewarded" for membership i n a group w i t h h i g h a v e r a g e . I n a w i d e r c o n t e x t , t h e problems i l l u s t r a t e d h e r e conc e r n b i a s e s i n t e s t i n g and s e l e c t i o n p r o c e d u r e s . F o r a v e r y t h o r o u g h and e x h a u s t i v e d i s c u s s i o n of t h i s c r u c i a l t o p i c , and a d e t a i l e d a n a l y s i s o f t h e e m p i r i c a l d a t a collected thus far, see Jensen (1980). F i r s t l y , t h e s e l e c t i o n i s based u s u a l l y on p r e d i c t e d val u e s of some c r i t e r i o n U ( e . g . f u t u r e p e r f o r m a n c e , e t c . ) T y p i c a l l y , U i s r e l a t e d t o t h e t r u e s c o r e X on t h e t e s t i n q u e s t i o n ; t h u s , t e s t s c o r e X i s f i r s t used t o o b t a i n

a p r e d i c t i o n U = U(X) o f t h e c r i t e r i o n v a r i a b l e U, and t h e n t h e o b t a i n e d v a l u e s U a r e used f o r s e l e c t i o n procedure: t h e c a n d i d a t e s a r e o r d e r e d a c c o r d i n g t o t h e est i m a t e d v a l u e s of U , and t h e n t h e s e l e c t i o n p r o c e e d s from t h e t o p , u n t i l e i t h e r a r e q u i r e d number o f candidat e s i s f o u n d , o r u n t i l t h e v a l u e s o f U become lower t h a n some p r e - a s s i g n e d t h r e s h o l d .

The example above shows

t h a t ( i n t h e p a r t i c u l a r c a s e when t h e e s t i m a t e d t r u e

95


s c o r e i s t h e v a r i a b l e used as a c r i t e r i o n ) , s u c h a proc e d u r e would d e c r e a s e c h a n c e s o f s e l e c t i o n f o r c a n d i d a t e s from g r o u p s w i t h l o w e r means ( i n c a s e of t e s t s w i t h r e l i a b i l i t y less than one). The p r e d i c t i o n o f U on t h e b a s i s o f X h a s t h e form o f regression equation U = a t b X

(12.2)

where t h e c o e f f i c i e n t s a and b ( t h e i n t e r c e p t and t h e s l o p e ) , as w e l l a s t h e e r r o r o f t h e e s t i m a t e , may depend on t h e r e f e r e n c e g r o u p . Suppose now t h a t we have two g r o u p s o f s u b j e c t s , s a y A and B , and t h e s e l e c t i o n i s t o be a p p l i e d t o b o t h

g r o u p s j o i n t l y . The c o n c e p t s r e l e v a n t h e r e a r e t h o s e o f b i a s o f t h e t e s t , and o f f a i r n e s s of i t s u s e .

for d e f i n i n g t h e n o t i o n o f b i a s o f a t e s t ( w i t h

Firstly,

r e s p e c t t o t h e two g r o u p s i n q u e s t i o n ) , l e t

U

=

a 1 t blX,

U = a2

+

b2X

(12.3)

be t h e r e g r e s s i o n e q u a t i o n s f o r p r e d i c t i n g U on t h e b a s i s

o f X i n g r o u p s A and B r e s p e c t i v e l y . The s t a n d a r d e r r o r o f p r e d i c t i o n e q u a l s

(12.4) where r Y 2 i s t h e v a r i a n c e of t h e c r i t e r i o n U , and rXU U i s t h e c o e f f i c i e n t o f c o r r e l a t i o n between t e s t and c r i t e r i o n . N a t u r a l l y , we have h e r e two v a l u e s , s and s2, 1

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CHAPTER 1

connected w i t h t h e f a c t t h a t t h e c o r r e l a t i o n c o e f f i c i e n t between X and U , and v a r i a n c e o f U , may d i f f e r i n groups A and B . We may now s a y t h a t t e s t X i s u n b i a s e d f o r U , w i t h r e s p e c t t o groups A and B , i f

a l = a 2,

bl

= b2,

s

1

= s

2'

(12.5)

N a t u r a l l y , t h e above c o e f f i c i e n t s a r e t h e o r e t i c a l param e t e r s , which must be e s t i m a t e d from t h e sample. Consequently, i n practice the t e s t i s biased, i f there i s 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 d i f f e r e n c e between a t l e a s t one o f t h e p a i r s o f v a l u e s i n ( 1 2 . 5 ) . Also, one s h o u l d t a k e i n t o a c c o u n t some a p p r o p r i a t e c o r r e c t i o n s f o r r e l i a b i l i t y o f t e x t X ( i . e . requirement ( 1 2 . 5 ) a p p l i e s t o t h e i d e a l i z e d c a s e of p e r f e c t l y r e l i a b l e t e s t s ) . Observe t h a t t h i s d e f i n i t i o n o f u n b i a s e d n e s s i s e n t i r e l y u n r e l a t e d t o f a i r n e s s , t h e l a t t e r being a concept pert a i n i n g t o u s e o f t h e t e s t . B i a s and u n b i a s e d n e s s a r e purely "neutral" s t a t i s t i c a l notions. Observe a l s o t h a t t h e d e f i n i t i o n o f u n b i a s e d n e s s d o e s n o t r e q u i r e t h a t groups A and B have t h e same means o r v a r i a n c e s i n t e s t X . Note, t o o , t h a t t h e f a c t t h a t t h e t e s t X i s b i a s e d does n o t , by i t s e l f , d i s q u a l i f y i t from u s e . An example g i v e n by J e n s e n ( 1 9 8 0 ) might c l a r i f y t h i s point. C o n s i d e r t h e groups A and B d e t e r m i n e d b y s e x , and l e t U be t h e p r e d i c t e d h e i g h t a t mature a g e . A s a p r e d i c t o r X we t a k e t h e mean h e i g h t o f p a r e n t s , i . e . t h e m i d p o i n t between t h e h e i g h t o f f a t h e r and m o t h e r . I n t h i s


97

c a s e , t h e r e g r e s s i o n o f U on X i s d i f f e r e n t f o r men and f o r women, simply because t h e f o r m e r t e n d t o b e , on t h e a v e r a g e , t a l l e r t h a n women. Thus, t h e a v e r a g e h e i g h t of p a r e n t s i s a b i a s e d p r e d i c t o r o f t h e h e i g h t a t mature a g e ; t h i s f a c t , however, d o e s n o t d i s q u a l i f y t h e use o f X . To a p p r e c i a t e t h e i n t u i t i v e j u s t i f i c a t i o n o f r e q u i r e m e n t (12.5) f o r unbiasedness, it i s best t o consider the consequences o f v i o l a t i n g t h e e q u a l i t i e s i n ( 1 2 . 5 ) . I f a1 # a2 a n d / o r b1

#

b 2 , t h e n f o r t h e same o b s e r v e d

s c o r e X , t h e p r e d i c t e d v a l u e s would, i n g e n e r a l , be d i f f e r e n t f o r groups A and B . C o n s e q u e n t l y , t h e e s t i m a t e of t h e c r i t e r i o n i n one group i s h i g h e r t h a n i n t h e other: t h e c r i t e r i o n i s underestimated o r overestimated (or both). If a l = a*, b1

--

b2, but

s1 # s2, t h e b i a s l i e s i n t h e

f a c t t h a t w h i l e t h e same s c o r e X y i e l d s t h e same p r e d i c t ed c r i t e r i o n , t h e p r o b a b i l i t y o f p r e d i c t i o n b e i n g i n e r r o r by a g i v e n amount i s d i f f e r e n t i n group A and i n group B . Now, as r e g a r d s t h e problem o f s e l e c t i o n ( w h e t h e r w i t h a b i a s e d o r u n b i a s e d t e s t ) , one would l i k e i t t o meet t h e c o n d i t i o n s of f a i r n e s s . The l a t t e r n o t i o n may be e x p l i c a t e d i n v a r i o u s w a y s . Hun-

t e r and Schmidt ( 1 9 7 6 ) c l a s s i f y a l l methods of s e l e c t i o n s u g g e s t e d t h u s f a r i n t o t h r e e broad c a t e g o r i e s , c a l l e d u n q u a l i f i e d i n d i v i d u a l i s m , q u a l i f i e d i n d i v i d u a l i s m , and q u o t a s y s t e m . These t h r e e methods w i l l be o u t l i n e d below, f o r t h e c a s e o f t w o g r o u p s , A and B.

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According t o t h e p r i n c i p l e o f u n q u a l i f i e d i n d i v i d u a l i s m , t h e c a n d i d a t e s s h o u l d be chosen a c c o r d i n g t o t h e b e s t a v a i l a b l e p r e d i c t o r o f t h e i r f u t u r e performance U . It i s not r e q u i r e d t h a t t h e same p r e d i c t o r s be a p p l i e d t o any p e r s o n , as l o n g one u s e s t h e p r e d i c t o r which i n a g i v e n i n s t a n c e has t h e h i g h e s t p o s s i b l e v a l i d i t y . The v a r i a b l e s which e n t e r t h e p r e d i c t o r may t h e r e f o r e comprise t h e s c o r e s on v a r i o u s t e s t s , as w e l l as group membership (e.g. sex, race, e t c . ) . This principle guarantees

t h e h i g h e s t a v e r a g e perform-

ance l e v e l among t h e chosen c a n d i d a t e s , and a l s o minimiz e s t h e e x p e c t e d number o f e r r o r s ( a c c e p t i n g a c a n d i d a t e who l a t e r f a i l s t o meet t h e r e q u i r e d performance l e v e l , and r e j e c t i n g a c a n d i d a t e who would meet t h i s l e v e l i f a c c e p t e d ) . The c h o i c e i s f a i r , i n t h e s e n s e t h a t no p e r s o n i s s e l e c t e d o r r e j e c t e d because o f h i s group membership. The main o b j e c t i o n a g a i n s t t h e u s e o f t h i s p r i n c i p l e l i e s i n t h e f a c t t h a t t h e e x i s t i n g p r e d i c t o r s may have d i f f e r e n t v a l i d i t i e s f o r d i f f e r e n t g r o u p s . I n such a cas e , p e r s o n s from t h e group w i t h p r e d i c t o r s o f a lower v a l i d i t y may have lower chances o f b e i n g s e l e c t e d . The second p r i n c i p l e , of q u a l i f i e d i n d i v i d u a l i s m , d i f f e r s

from t h e p r e c e d i n g one i n one r e s p e c t , namely t h a t one i s n o t a l l o w e d t o use t h e group membership as a v a r i a b l e entering t h e predictors. In t h i s case therefore, t h e same v a r i a b l e s s h o u l d b e used i n b o t h g r o u p s , and t h e same common e q u a t i o n f o r performance p r e d i c t i o n U has t o be u s e d .

I n t h i s c a s e , t h e a v e r a g e l e v e l o f performance among t h e


99

s e l e c t e d p e r s o n s w i l l , i n g e n e r a l , be l o w e r t h a n i n t h e f i r s t case, i f t h e predictor

has d i f f e r e n t v a l i d i t y i n

c a s e of d i f f e r e n t g r o u p s . T h i s i s compensated by t h e f a c t t h a t no one can c l a i m t h a t he was t r e a t e d i n any s p e c i a l ( f a v o u r i n g o r d i s c r i m i n a t i n g ) way b e c a u s e of h i s group membership. However, i n a n a t t e m p t t o i n c r e a s e t h e v a l i d i t y of t h e p r e d i c t o r s u s e d , one o f t e n u s e s v a r i a b l e which c o r r e l a t e h i g h l y b o t h w i t h t h e performance l e v e l and w i t h t h e group membership. Thus, group membership e n t e r s i n a d i s g u i s e d way ( e . g . t h r o u g h s o c i o e c o n o m i c a l v a r i a b l e s ) , and i t i s n o t e a s y t o d e t e r m i n e which v a r i a b l e s may and which may n o t be used i n p r e d i c t i o n . F i n a l l y , i n t h e q u o t a system,

one r e s i g n s from t h e

p r i n c i p l e of m i n i m i z i n g t h e e r r o r , or maximizing t h e l e v e l of f u t u r e p e r f o r m a n c e , and a l l o c a t e s some q u o t a s t o g r o u p s A and B. Thus, w i t h i n t h e groups t h e c a n d i d a t e s a r e s e l e c t e d by t h e i r p r e d i c t e d l e v e l o f performanc e , b u t it may happen t h a t a lower s c o r i n g c a n d i d a t e w i l l be s e l e c t e d and a h i g h e r s c o r i n g c a n d i d a t e w i l l be

r e j e c t e d i f t h e y a r e from v a r i o u s g r o u p s . I n more g e n e r a l s e t u p , by q u o t a s y s t e m one means any method o f s e l e c t i o n under which a g i v e n c a n d i d a t e w i t h lower p r e d i c t e d performance may be c h o s e n o v e r a c a n d i d a t e w i t h h i g h e r p r e d i c t e d performance. Quota s y s t e m s are t h o u g h t o f as means of improving t h e e x i s t i n g s o c i a l s t r u c t u r e s and undoing some p r e v i o u s i n j u s t i c e . The d e c i s i o n as t o t h e c h o i c e of q u o t a s i s a m a t t e r of s o c i a l p o l i c y . Once t h i s i s d e c i d e d , however, one may s u g g e s t numerous s e l e c t i o n schemes s o as t o o p t i m i z e

CHAPTER I

100

c e r t a i n c r i t e r i a , meeting t h e q u o t a c o n s t r a i n t s . A t t h e e n d , w e s h a l l p r e s e n t some of t h e s u g g e s t e d s o l -

utions i n the quota system. C o n s i d e r one g r o u p , and assume t h a t t h e j o i n t d i s t r i b u t i o n o f t h e c r i t e r i o n U and t e s t X i s normal. Note t h a t U i s n o t o b s e r v a b l e a t t h e time of d e c i s i o n making.

Suppose ( s e e F i g . 1 2 . 1 ) t h a t U

*

i s the required l e v e l n of performance (for a g i v e n g r o u p ) , and t h a t X i s t h e * c u t - o f f p o i n t f o r t h i s group (so t h a t X i s t h e c o n t r o l

X* F i g . 1 2 . 1 . Choice a r e a s i n q u o t a s y s t e m v a r i a b l e ) . The a r e a s A , B , C

and D on F i g . 1 2 . 1 r e p r e s e n t

respectively: A

--

c a n d i d a t e s who a r e a c c e p t e d and whose performance exceeds t h e r e q u i r e d l e v e l ( c o r r e c t acceptance)


101

B

--

c a n d i d a t e s who are r e j e c t e d , b u t whose l e v e l o f performance would meet t h e r e q u i r e d l e v e l (wrong rejection);

C

--

c a n d i d a t e s who a r e r e j e c t e d , and whose l e v e l o f performance d o e s n o t meet t h e s t a n d a r d ( c o r r e c t rejections) ;

D

--

c a n d i d a t e s who a r e a c c e p t e d , b u t whose l e v e l o f performance d o e s n o t meet t h e r e q u i r e d s t a n d a r d (wrong a c c e p t a n c e ) .

To g i v e j u s t few examples o f t h e s u g g e s t e d s e l e c t i o n p r o c e d u r e s : one may r e q u i r e t h a t i n e a c h group, t h e rat i o o f t h o s e a c c e p t e d t o t h o s e r e j e c t e d s h o u l d be t h e same. T h i s s e l e c t i o n p r o c e d u r e r e q u i r e s t h e r e f o r e t h a t * t h e c u t o f f p o i n t X (and hence a l s o t h e r e q u i r e d p e r * formance l e v e l U ) a r e s e t s o t h a t t h e r a t i o ( A + D ) / ( B - k C s h o u l d be t h e same i n e a c h group.

Thorndike ( 1 9 7 1 ) proposed a n o t h e r model o f s e l e c t i o n , a c c o r d i n g t o which t h e r a t i o o f a c c e p t a n c e t o s u c c e s s s h o u l d be t h e same i n e a c h group, i . e . he p o s t u l a t e d t h a t ( A t D ) / ( A t B ) s h o u l d be t h e same i n e a c h group. I n s t i l l a n o t h e r p r o c e d u r e , Cole

(1973) s u g g e s t e d t h a t

p r o b a b i l i t y o f s e l e c t i o n among t h o s e who succeed ( a r e * above U ) s h o u l d be t h e same i n b o t h g r o u p s . T h i s r e q u i r e s e q u a l i z i n g t h e r a t i o A / ( A + B) i n b o t h g r o u p s . F i n a l l y , P e t e r s e n and Novick (19762 a r g u e t h a t a l l acc e p t e d a p p l i c a n t s s h o u l d have t h e same c h a n c e s of b e i n g s u c c e s s f u l ; t h i s , i n t u r n , means t h a t t h e r a t i o s A / ( A + B should be e q u a l i n b o t h groups.

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13. M A I N DIRECTIONS OF RESEARCH I N TEST THEORY

It a p p e a r s t h a t t h e development o f t e s t t h e o r y i n t h e

l a s t t e n or s o y e a r s p r o c e e d e d i n t h r e e main d i r e c t i o n s : d e v e 1o pme n t o f

I'

c 1as s i c a 1 t e s t t he o r y , g e n e r a 1i z a b i 1i t y

t h e o r y , and t h e o r y o f l a t e n t t r a i t s . A s r e g a r d s t h e development of' c l a s s i c a l t e s t t h e o r y , t h e most i m p o r t a n t c o n t r i b u t i o n s a p p e a r t o i n v o l v e t h e a p p l i c a t i o n of t h e t h e o r y o f s t o c h a s t i c p r o c e s s e s . To mention j u s t few e x a m p l e s , t h e t h e o r e m s on r e l a t i o n s between r e l i a b i l i t y and l e n g t h of t h e t e s t may be r e i n t e r p r e t e d a s s t a t e m e n t s about t h e p r o p e r t i e s of s t o c h a s t i c processes occurring i n continuous time, representi n g t h e f r a c t i o n o f i t e m s answered. Such a n a p p r o a c h h a s been s u g g e s t e d by Novick 1 9 7 6 . A n o t h e r p o s s i b i l i t y i s c o n n e c t e d w i t h t h e problem o f d e t e r m i n i n g t h e b e s t i n t e r v a l o f time between t e s t i n g and r e - t e s t i n g ,

so as t o o b t a i n t h e b e s t e s t i m a t e of

r e l i a b i l i t y by t e s t - r e t e s t c o r r e l a t i o n ( s e e M o r r i s o n

1 9 8 1 ) . Here t h e main i s s u e i s t h e dilemma: i f t h e second t e s t i n g comes t o o e a r l y , t h e memory e f f e c t s may be a p p r e c i a b l e . On t h e o t h e r hand, i n c a s e o f t e s t i n g t o o f a r a p a r t i n t i m e , t h e t r u e s c o r e may undergo a p p r e c i a b l e c h a n g e s . Thus, M o r r i s o n s u g g e s t s a model o f b o t h memory f a d i n g and c h a n g e s of t r u e s c o r e , and d e r i v e s t h e f o r mulas for t h e o p t i m a l t i m e s e p a r a t i n g t e s t i n g and r e t e s t ing. A s regards generalizability theory, several directions

of r e s e a r c h a r e w o r t h n o t i n g . F i r s t l y , i t o u g h t t o be o b s e r v e d t h a t t h e a p p l i e d methods o f e s t i m a t i o n of t h e


103

components o f v a r i a n c e are based on f o r m u l a s which

ex-

p r e s s t h e s e components as sums o f o t h e r components ( s e e T a b l e s i n S e c t i o n 11). C o n s e q u e n t l y , a n e s t i m a t e o f a v a r i a n c e component i s , t y p i c a l l y , a l i n e a r combination of sums o f s q u a r e s o f o b s e r v a t i o n s , w i t h some c o e f f i c i e n t s b e i n g n e g a t i v e . S i n c e t h e e s t i m a t e s a r e s u b j e c t t o sampl i n g e r r o r s , i t may o c c u r t h a t t h e e s t i m a t e s y i e l d negat ive values. Moreover, t y p i c a l l y t h e more i m p o r t a n t and i n t e r e s t i n g v a r i a n c e components w i l l have h i g h e r s a m p l i n g e r r o r s , which a f f e c t s p r e c i s i o n of e s t i m a t i o n and a l s o makes them more prone t o assume n e g a t i v e v a l u e s . One of t h e a p p r o a c h e s i s t o r e p l a c e n e g a t i v e e s t i m a t e s by z e r o , and p r o c e e d w i t h f u r t h e r e s t i m a t e s u s i n g t h e o b t a i n e d v a l u e s . However, as p o i n t e d o u t by ScheffC ( 1 9 5 9 ) , t h e s a m p l i n g d i s t r i b u t i o n of e s t i m a t o r s which i n v o l v e s u b s t i t u t i o n of e s t i m a t o r s o b t a i n e d by s u c h r e p l a c e m e n t s o f n e g a t i v e v a l u e s by z e r o , i s q u i t e complicated. A p r o m i s i n g s o l u t i o n h e r e i s t h e use of Bayesian methods,

imposing a n o n n e g a t i v i t y c o n s t r a i n t ( s e e e . g . Novick and J a c k s o n 1 9 7 4 o r Box and T i a o 1 9 7 3 ) . Numerous s i m u l a t i o n s t u d i e s ( s e e e . g . Smith 1978) seem t o i n d i c a t e t h a t sampling e r r o r s i n g e n e r a l i z a b i l i t y s t u d i e s a r e t y p i c a l l y q u i t e l a r g e , and t h a t i n o r d e r t o g e t s t a b l e e s t i m a t e s one needs u n i v e r s e s w i t h l a r g e numbers o f f a c e t s . The s i t u a t i o n becomes much more c o m p l i c a t e d when t h e d e s i g n o f e x p e r i m e n t i n unbalanced ( i . e . t h e numbers o f o b s e r v a t i o n s i n c e l l s do n o t meet p r o p o r t i o n a l i t y

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conditions). A s another l i n e of r e s e a r c h i n g e n e r a l i z a b i l i t y t h e o r y

one s h o u l d mention h e r e t h e a t t e m p t s t o d e v e l o p a m u l t i d i m e n s i o n a l v a r i a n t of t h e t h e o r y ( s e e e . g . J o e and Woodward 1 9 7 6 ) . For a thorough review of g e n e r a l i z a b i l i t y t h e o r y , i t s a p p l i c a t i o n , and a n e x t e n s i v e b i b l i o g r a p h y of t h e subj e c t , s e e S h a v e l s o n and Webb ( 1 9 8 1 ) . F i n a l l y , t h e t h i r d l i n e o f r e s e a r c h concerns t h e theory o f l a t e n t t r a i t s , s u g g e s t e d o r i g i n a l l y be L a z a r s f e l d (1959, 1 9 6 0 ) . T h i s t h e o r y d e a l s w i t h s i m u l t a n e o u s measurement by a

number o f t o o l s , s u c h as i t e m s ( i n c a s e o f a t e s t ) . The c e n t r a l i d e a i s t o use t h e n o t i o n o f l o c a l c o n d i t i o n a l independence ( c a l l e d a l s o symmetric dependence, o r e x c h a n g e a b i l i t y ; see F e l l e r 1966) : a l a t e n t s t r u c t u r e i s c o m p l e t e , if i n any group o f p e r s o n s i d e n t i c a l w i t h respect t o a l l l a t e n t t r a i t s , t h e observable v a r i a b l e s ( i t e m r e s p o n s e s , f o r i n s t a n c e ) a r e independent. It f o l l o w s i n p a r t i c u l a r , t h a t t h e o b s e r v e d v a r i a b l e s

a r e i n d e p e n d e n t f o r e a c h p e r s o n s e p a r a t e l y , s i m p l y bec a u s e a s i n g l e p e r s o n i s a s e p e c i a l c a s e o f a group of p e r s o n s w i t h i d e n t i c a l l a t e n t t r a i t s . The a s s u m p t i o n o f c o n d i t i o n a l independence o f r e s p o n s e s i s not v e r i f i a b l e , because of t h e d i f f i c u l t y i n g e t t i n g i n d e p e n d e n t r e p e t i t i o n s o f measurements o f t h e same p e r s o n . I n t h i s s e n s e , t h i s a s s u m p t i o n p l a y s t h e same r o l e as a s s u m p t i o n o f l i n e a r e x p e r i m e n t a l independence

CONTJZMPORARY THEORY OF PSYCHOLOGICAL TESTS

105

i n classical t e s t theory. The a s s u m p t i o n a b o u t independence of o b s e r v a t i o n s i n

a group o f p e r s o n s w i t h t h e same v a l u e s of l a t e n t t r a i t s s e r v e s as a b a s i s f o r i d e n t i f i c a t i o n of s u c h g r o u p s . To d e t e r m i n e t h e l a t e n t s t r u c t u r e of a p o p u l a t i o n , i t s u f f i c e s t o p a r t i t i o n t h i s p o p u l a t i o n i n t o groups, such t h a t measurements a r e i n d e p e n d e n t w i t h i n g r o u p s . These groups , by d e f i n i t i o n , a r e g r o u p s w i t h i d e n t i c a l v a l u e s of l a t e n t t r a i t s . I n o t h e r words, t h e p o i n t i s t o p a r t i t i o n t h e p o p u l a t i o n i n t o g r o u p s , s a y G1, G 2 , with t h e following property. L e t Xs, Y s , be t h e random v a r i a b l e s d e s c r i b i n g t h e measurements of p e r s o n s w i t h v a r i o u s t o o l s ( e . g . t h e s e random v a r i a b l e s may be r e s p o n s e s t o v a r i o u s i t e m s ) . Then, g i v e n s c G t h e random v a r i a b l e s X s , Y s , . . . are i’ independent, s o t h a t

...

...

Consequently, g i v e n t h e i n f o r m a t i o n t h a t s i s i n Gi ’ t h e knowledge of h i s r e s u l t s i n some t e s t s d o e s n o t g i v e any i n f o r m a t i o n a b o u t h i s v a l u e s on o t h e r t e s t s . I n s t i l l o t h e r words, t h e u n d e r l y i n g i n t u i t i v e i n t e r p r e t a t i o n i s t h a t a l l p s y c h o l o g i c a l i n f e r e n c e a b o u t t h e subj e c t i s t o b e based only on t h e i n f o r m a t i o n about h i s group membership; once t h i s i s known, f u r t h e r i n f o r m a t i o n a b o u t t h e v a l u e s of v a r i o u s measurements i s psychologically irrelevant. N a t u r a l l y , when a p a r t i t i o n w i t h t h i s p r o p e r t y i s obt a i n e d , t h e r e i s s t i l l a t a s k of i t s p s y c h o l o g i c a l i n terpretation.

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D e s p i t e g r e a t g e n e r a l i t y and c o n c e p t u a l s i m p l i c i t y , t h e theory of l a t e n t t r a i t s encounters i n p r a c t i c a l applica t i o n s c o n s i d e r a b l e t h e o r e t i c a l and t e c h n i c a l d i f f i c u l t i e s . A s a t i s f a c t o r y development i s r e s t r i c t e d t o oned i m e n s i o n a l t r a i t s , which means ‘chat p s y c h o l o g i c a l a p p l i c a b i l i t y r e d u c e s m o s t l y t o a b i l i t i e s and a t t i t u d e s , which o f t e n may be t a k e n as one-dimensional. However, even i n t h e one-dimensional c a s e t h e l a t e n t t r a i t t h e o r y enounters c e r t a i n d i f f i c u l t i e s . To e x p l a i n t h e s i t u a t i o n , l e t u s o u t l i n e t h e s i m p l e s t c a s e . Imagine t h a t we have r b i n a r y i t e m s , which may be answered c o r r e c t l y o r i n c o r r e c t l y . L e t u be t h e l a t e n t t r a i t . I f X . i s t h e random v a r i a b l e which assumes t h e v a l u e 1 1

i t e m i s c o r r e c t and v a l u e 0 i f t h e answer i s i n c o r r e c t , t h e n t h e f u n c t i o n i f t h e answer t o i - t h

P(Xi

=

llu) =

fib)

i s c a l l e d t h e item c h a r a c t e r i s t i c curve ( I C C ) of i - t h i t e m . Given t h e assumption of l o c a l c o n d i t i o n a l i n d e pendence, t h e p r o b a b i l i t y of r e s p o n s e = (xl, ...,x r ) on a l l r items b y a p e r s o n w i t h v a l u e u o f t h e t r a i t i s r p ( x l u ) = l-7 P ( X i = X i J U ) i=l

x

and e a c h t e r m o f t h i s p r o d u c t i s e i t h e r f i ( u ) o r l - f i ( u ) . T o e n a b l e t h e s t a t i s t i c a l , and hence a l s o p s y c h o l o g i c a l ,

i n f e r e n c e about t h e s u b j e c t , one i n t r o d u c e s a p a r a m e t r i c form o f f i ( u ) . The b e s t known model h e r e i s t h a t o f l o g i s t i c i t e m c h a r a c t e r i s t i c c u r v e (Rasch 1960; s e e a l s o Birnbaum 1 9 6 8 ) . I n t h i s model i t i s assumed t h a t


utk Ce

fib) = 1

+

107

i

u+ki Ce

where ki i s a c o n s t a n t connected w i t h i - t h i t e m , C i s a s c a l i n g c o n s t a n t , and u i s a one-dimensional t r a i t r e p r e s e n t i n g a b i l i t y ( s a y ) . When u i n c r e a s e s , t h e prob a b i l i t y f i ( u ) o f c o r r e c t answer t e n d s t o 1, w h i l e w i t h a decrease of u, t h i s probability decreases t o zero. Also, f o r a g i v e n u, t h e p r o b a b i l i t y o f c o r r e c t r e s p o n s e i s l a r g e r f o r l a r g e r k i , s o t h a t - k i i s a measure o f i t e m difficulty. The g e n e r a l problem i s now t o e s t i m a t e t h e v a l u e s ul, * ,un f o r n i n d i v i d u a l s and v a l u e s k l , . . . , k r for r i t e m s , g i v e n t h e d a t a on r e s p o n s e s o f t h e s e i n d i v i d u a l s . F o r t h e e s t i m a t i o n p r o c e d u r e s , goodness o f f i t o f t h e model, e t c . s e e f o r i n s t a n c e Anderson 1973, Cohen 1979, G o l d s t e i n 1979, G u s t a f s s o n 1980 or Sanathanan and Blument ha1 197 8 .

..

The d i f f i c u l t i e s connected w i t h Rasch model, and a l s o w i t h some o t h e r l a t e n t t r a i t models, were poJnted o u t by G o l d s t e i n ( 1 9 8 0 ) . These d i f f i c u l t i e s a r e r e l a t e d t o t h e f a c t t h a t t h e model p r e s u p p o s e s t h a t u i s measured on some s c a l e w i t h known p r o p e r t i e s , w h i l e i n f a c t t h e a s s u m p t i o n s about t h e s c a l e a r e n o t t e s t a b l e . The p r a c t i c a l consequences of t h i s i s t h a t t h e a b i l i t i e s are d i s c r i m i n a t e d d i f f e r e n t l y on t h e ends of t h e s c a l e t h a n i n t h e m i d d l e , which l e a d s t o some s c a l e v a l u e s a p p e a r i n g c o n t r a r y t o i n t u i tion. F o r a t h o r o u g h r e v i e w o f r e c e n t development o f l a t e n t

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t r a i t t h e o r y s e e Hambleton e t a l . 1978. I n a d d i t i o n , one s h o u l d mention a l s o t h e newest r e s u l t s o f Holland

(1981), who i n t r o d u c e s n o t i o n s somewhat weaker t h a n t h a t of l o c a l c o n d i t i o n a l i n d e p e n d e n c e , and p r o v i d e s i n t h i s way c e r t a i n means o f t e s t i n g e m p i r i c a l l y for t h i s i n d e pendence. Comments. The p s y c h o l o g i c a l t h e o r y o f t r a i t s , b e i n g t h e

f o u n d a t i o n o f t e s t t h e o r y , only i n d i r e c t l y takes i n t o a c c o u n t s i t u a t i o n a l f a c t o r s . I n most r e c e n t development o f t e s t theory ( i n p a r t i c u l a r g e n e r a l i z a b i l i t y ) , t h e r e s e a r c h goes t o w a r d s r e f i n e d r e s p o n s e a n a l y s i s , s o as t o be a b l e t o d i s t i n g u i s h t h e f a c t o r s c o n n e c t e d w i t h t h e s e t of t r a i t s , as w e l l as t h e s e t o f s i t u a t i o n a l c h a r a c t e r i s t i c s . The s i t u a t i o n , however, i s u n d e r s t o o d i n a r a t h e r narrow s e n s e , and w i t h o u t a w i d e r t h e o r y ( u s i n g mainly c o r r e l a t i o n a l m e t h o d s ) . To deepen t h e p s y c h o l o g i c a l p i c t u r e s u p p l i e d by t e s t s , i t would b e f r u i t f u l t o t a k e i n t o account t h e o r y o f s i t u a t i o n s , u n d e r s t o o d as i t s s t r u c t u r e and knowledge a b o u t i t . It seems t h a t t e s t t h e o r y would p r o f i t from r e - i n t e r p r e t a t i o n o f i t s b a s i c n o t i o n s , s u c h as t r a i t . P e r h a p s some r e f r e s h i n g o f t e s t t h e o r y , and more u s e f u l n e s s i n p r e s e n t l y d e v e l o p i n g o t h e r domains ( s u c h as e . g . c o g n i t i v e s c i e n c e s , a r t i f i c i a l i n t e l l i g e n c e ) , c o u l d be o b t a i n e d by some change o f i n t e r p r e t a t i o n of t h e c o n c e p t o f t r a i t , t r e a t i n g i t as a c e r t a i n c o n f i g u r a t i o n of d i s p o s i t i o n s , g e n e t i c a l l y d e t e r m i n e d and e n v i r o n m e n t a l l y m o d i f i e d , t o (1) some t y p e s o f p e r c e p t i o n o f s i t u a t i o n and s e n s i t i v i t y t o some e l e m e n t s o f t h e i r s t r u c t u r e , ( 2 ) a b i l i t y t o a c q u i r i n g some t y p e s o f knowledge s t r u c t u r e and ( 3 ) c h o i c e of a c t i o n s imposed by g i v e n t y p e of p e r c e p t i o n and knowledge. I n o t h e r words, more s t r e s s would b e p u t on knowledge s t r u c t u r e i n some c l a s s e s o f s i t u a t -


109

i o n s , which would i n v o l v e g r e a t e r c o n t r i b u t i o n o f l i n g u i s t i c s and t e x t t h e o r y on t h e one hand, and c o g n i t i v e p r o c e s s e s ( i n p a r t i c u l a r p e r c e p t i o n models) on t h e o t h e r . Subsequent c h a p t e r s w i l l show some p o s s i b l e e x t e n s i o n s o f t h e s e ideas.

110

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BIBLIOGRAPHY TO CHAPTER 1

. .

ANDERSEN , E B

1 9 7 3 A Goodness of F i t T e s t for t h e R a s c h Model. P s y c h o m e t r i k a . 38; 123-140. BIRNBAUM, A .

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N o v i c k , S t a t i s t i c a l T h e o r i e s of M e n t a l T e s t S c o r e s . R e a d i n g s , Mass. Addison-Wesley

BOX, G . E . P . ,

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TIAO,

G.C.

B a y e s i a n I n f e r e n c e i n S t a t i s t i c a l A n a l y s i s . Rea d i n g s , Mass. Addison-Wesley

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Approximate E x p r e s s i o n s for Parameter E s t i m a t e s i n t h e R a s c h Model. B r i t i s h J o u r n a l of Mathe-

matical and S t a t i s t i c a l Psychology.

32. 113-120.

COLE, N.S.

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B i a s i n S e l e c t i o n . J o u r n a l of E d u c a t i o n a l Meas u r e m e n t . 1 0 ; 237-255.

CRONBACH, L . J .

1951 C o e f f i c i e n t a l p h a a n d t h e I n t e r n a l S t r u c t u r e of T e s t s . P s y c h o m e t r i k a . 1 6 ; 297-334. CRONBACH, L . J . ,

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GLESER, G . C . ,

NANDA, H . ,

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The D e p e n d a b i l i t y of B e h a v i o r a l M e a s u r e m e n t s . New York. Wiley.

FELLER, W .

1966

An I n t r o d u c t i o n t o P r o b a b i l i t y Theory a n d I t s A p p l i c a t i o n s . V o l . 2 . N e w York. W i l e y .

GOLDSTEIN , H .

1979

C o n s e q u e n c e s o f U s i n g t h e R a s c h Model for Edu-

111


c a t i o n a l A s s e s s m e n t . B r i t i s h E d u c a t i o n a l Resea r c h J o u r n a l . 5; 211-220. 1980

D i m e n s i o n a l i t y , B i a s , I n d e p e n d e n c e and Measurement S c a l e P r o b l e m s i n L a t e n t T r a i t T e s t S c o r e Models. B r i t i s h J o u r n a l of M a t h e m a t i c a l a n d S t a -

t i s t i c a l P s y c h o l o g y . 33; 2 3 4 - 2 4 6 . GUSTAFSSON, J . E . 1980

Testing

a n d O b t a i n i n g F i t of .;Data t o t h e R a s c h

Model. B r i t i s h J o u r n a l of M a t h e m a t i c a l a n d S t a t i s t i c a l P s y c h o l o g y . 33; 2 0 5 - 2 3 3 . GULLIKSEN, H . T h e o r y of M e n t a l T e s t s . N e w York. W i l e y .

1950 GUTTMAN,

1945

L. A Basis for A n a l y s i n g T e s t - R e t e s t

Reliability.

P s y c h o m e t r i k a . 1 0 ; 255-282. HAMBLETON, R . K . ,

D.R.,

SWAMINATHAN, H . ,

COOK, L . L . ,

EIGNOR,

GIFFORD, J . A .

1978

Developments i n L a t e n t T r a i t T h e o r y : M o d e l s , T e c h n i c a l I s s u e s a n d A p p l i c a t i o n s . Review of E d u c a t i o n a l R e s e a r c h . 48;

HOLLAND,

467-510.

.

P W.

1 9 8 1 When Are Item R e s p o n s e Models C o n s i s t e n t w i t h O b s e r v e d Data?. P s y c h o m e t r i k a . 4 6 ; 79-92. HUNTER, J . E . ,

1976

SCHMIDT, F . L .

A C r i t i c a l A n a l y s i s o f t h e S t a t i s t i c a l and E t h i c a l I m p l i c a t i o n s o f V a r i o u s D e f i n i t i o n s of " T e s t

B i a s " . P s y c h o l o g i c a l B u l l e t i n . 83; 1053-1071. JENSEN, A . R . 1980

Bias i n M e n t a l T e s t i n g . The F r e e P r e s s . N e w York.

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CHAPTER I

JOE, G . N . ,

WOODWARD, J . A .

1976 Some Developments i n M u l t i v a r i a t e G e n e r a l i z a b i l i t y . Psychometrika. RICHARDSON, M . W .

KUDER, G.F.,

1937

4 1 ; 205-217.

The Theory of E s t i m a t i o n o f T e s t R e l i a b i l i t y . Psychometrika.

2;

151-160.

LAZARSFELD, P.F.

1959 L a t e n t S t r u c t u r e A n a l y s i s , I n S . Koch ( e d . ) P s y c h o l o g y . A Study o f S c i e n c e . V o l . 3 . 4 7 6 - 4 5 2 . New York. McGraw-Hill.

1960

L a t e n t S t r u c t u r e A n a l y s i s and T e s t T h e o r y . I n H : G u l l i k s e n , S . Messick ( e d s . ) P s y c h o l o g i c a l S c a l i n g . Theory and A p p l i c a t i o n s .

York. W i l e y

83-95. N e w

.

LOEVINGER, J .

1957

O b j e c t i v e Tests a s I n s t r u m e n t s of P s y c h o l o g i c a l Theory. P s y c h o l o g i c a l R e p o r t s . 3; 635-694.

LORD. F.M.,

1968

NOVICK, M.R.

S t a t i s t i c a l T h e o r i e s of Mental T e s t S c o r e s . R e a d i n g s , Mass. Addison-Wesley.

MORRISON, D . G .

1 9 8 1 A S t o c h a s t i c Model for T e s t - R e t e s t C o r r e l a t i o n . P s y c h o m e t r i k a . 46; 143-151. MOSTELLER, F . , WALLACE, D .

1964

I n f e r e n c e and D i s p u t e d A u t h o r s h i p : The F e d e r a l -

ist.

R e a d i n g s , Mass. Addison-Wesley

.

N O V I C K , M.R.

1966

The Axioms and P r i n c i p a l R e s u l t s of C l a s s i c a l T e s t T h e o r y . J o u r n a l of M a t h e m a t i c a l P s y c h o l o g y .

3; 1-

la.

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196 9

Multiparameter Bayesian I n f e r e n c e Procedures J . Royal S t a t i s t i c a l Socie-

(with Discussion).

Q.

S e r . B . 31; 2 9 - 6 4 .

197 1 B a y e s i a n C o n s i d e r a t i o n s i n E d u c a t i o n a l I n f o r m a t i o n Systems. I n E o c . o f t h e I n t e r n a t i o n a l Testi n g C o n f e r e n c e . 77-88.

P r i n c e t o n , Educ. T e s t i n g

Service. NOVICK, M.R.,

1965

GRIZZLE, J . E .

A Bayesian I n f e r e n c e Procedure. J . Amer.

ASSOC. NOVICK, M.R.,

1970

6 0 ; 1104-1117. JACKSON, P . H .

B a y e s i a n Guidance T e c h n o l o g y . Rev. Educ. Rese-

arch. 1974

St&.

40; 459-494.

S t a t i s t i c a l Methods f o r E d u c a t i o n a l a n d Psychol o g i c a l R e s e a r c h . N e w York. McGraw-Hill.

N O V I C K , M.R.

JACKSON, P . H . ,

THAYER, D.S.

197 1 B a y e s i a n I n f e r e n c e and t h e C l a s s i c a l T e s t Theo r y Model. R e l i a b i l i t y a n d T e s t S c o r e s . Psy chometrika. 36; 261-288. PETERSEN, N.S.,

1976

NOVICK, M.R.

An E v a l u a t i o n o f Some Models f o r C u l t u r e - F a i r S e l e c t i o n . J o u r n a l o f E d u c a t i o n a l Measurement.

13; 3-29. RASCH, G . 1960

P r o b a b i l i s t i c Models f o r Some I n t e l l i g e n c e a n d Attainment T e s t s .

Copenhagen: D a n i s h I n s t i t u t e

f o r Educational Research. SANATHANAN, L , BLUMENTHAL, S .

1978

The L o g i s t i c Model and E s t i m a t i o n of L a t e n t S t r u c t u r e s . J o u r n a l o f t h e American S t a t i s t i c a l Association.

73; 794-799.

114

CHAPTER I

SCHEFFE, H.

1959

The A n a l y s i s of V a r i a n c e . N e w York. W i l e y .

SHAVELSON, R . J . ,

WEBB, N . M .

1 9 8 1 G e n e r a l i z a b i l i t y T h e o r y . 1973-1980. B r i t i s h J o u r n a l of Mathematical and S t a t i s t i c a l Psychol o g y . 34; 133-166. SMITH, P . L .

1978

S a m p l i n g Errors of V a r i a n c e Components i n S m a l l Sample M u l t i f a c e t G e n e r a l i z a b i l i t y S t u d i e s . J o u r n a l of E d u c a t i o n a l S t a t i s t i c s . 3 ; 3 1 9 - 3 4 6 .

THORNDIKE, 1964

R.L.

R e l i a b i l i t y . I n P r o c . o f t h e 1963 C o n f e r e n c e on T e s t i n g Problems.

1971

Princeton, N.5.

Education-

a l Testing Service. C o n c e p t s of C u l t u r e F a i r n e s s . J o u r n a l o f Educ a t i o n a l M e a s u r e m e n t . 8; 63-70.

115

CHAPTER 2 FACTOR A N A L Y S I S : ARBITRARY D E C I S I O N S W I T H I N M A T H E M A T I C A L MODEL

The aim of t h e p r e s e n t c h a p t e r i s a d i s c u s s i o n o f t h o se a s p e c t s of f a c t o r a n a l y s i s which may l e a d t o method o l o g i c a l r e s e r v a t i o n s . The l a t t e r a r e c o n n e c t e d , f i r s t l y , w i t h making m u l t i p l e a r b i t r a r y d e c i s i o n s i n apply-

ing factor chological cessity of t e r of t h e

a n a l y s i s , and s e c o n d l y ( e s p e c i a l l y i n p s y and s o c i o l o g i c a l a p p l i c a t i o n s ) w i t h t h e nemaking s t r o n g a s s u m p t i o n s about t h e c h a r a c t s c a l e s on which one measures t h e u n d e r l y i n g

variables. S e c t i o n 1 d e a l s w i t h a p r e s e n t a t i o n and d i s c u s s i o n of t h e model o f f a c t o r a n a l y s i s and o f t h e a l t e r n a t i v e model o f m u l t i d i m e n s i o n a l s c a l i n g . Section 2 presents a discussion of an a p p l i c a t i o n of f a c t o r a n a l y s i s t o t h e c o n s t r u c t i o n of C a t t e l l ' s persona l i t y theory.

1. SOME M U L T I V A R I A T E T E C H N I Q U E S I N PSYCHOLOGY

1.1. F a c t o r a n a l y s i s : t h e m a t h e m a t i c a l model

L e t us c o n s i d e r t h e s i t u a t i o n i n which for e a c h o f t h e

116

CHAPTER 2

e l e m e n t s under s t u d y ( i t i s assumed t h a t t h e s e e l e m e n t s form a sample from some p o p u l a t i o n ) one makes a c e r t a i n number n of measurements. I n t y p i c a l p s y c h o l o g i c a l a p p l i c a t i o n s t h e e l e m e n t s o f t h e sample a r e s u b j e c t s , s e l e c t e d a t random from some p o p u l a t i o n ; t h e r e s u l t s o f measurement would t y p i c a l l y be s c o r e s on some t e s t s , coded answers t o v a r i o u s q u e s t i o n s , e v a l u a t i o n s of beh a v i o u r o f s u b j e c t s , a s g i v e n by j u d g e s ( f r o m o u t s i d e t h e s a m p l e ) , o r even c e r t a i n a n t h r o p o l o g i c a l measurements, age of t h e s u b j e c t s , e t c . I n a p p l i c a t i o n s from o u t s i d e p s y c h o l o g y , t h e o b j e c t s s t u d i e d may be samples o f s o i l a n a l y s e d from t h e p o i n t o f view o f a number of c h a r a c t e r i s t i c s , m u n i c i p a l counc i l s a n a l y s e d from t h e p o i n t of view of v a r i o u s c r i t e r i a , and s o f o r t h .

G e n e r a l l y , l e t x j s ) d e n o t e t h e r e s u l t o f measurement o f i - t h v a r i a b l e f o r t h e s - t h element o f t h e sample; h e r e t h e lower i n d e x r e f e r s t o t h e measurement o f t h e c h a r a c t e r i s t i c under s t u d y , and t h i s i n d e x may assume as many v a l u e s as t h e r e a r e c h a r a c t e r i s t i c s . The upper i n d e x r e f e r s t o t h e element i n t h e sample, and may assume as many v a l u e s as t h e r e a r e e l e m e n t s . Thus, t h e raw d a t a may be r e p r e s e n t e d i n form o f a m a t r i x

.................

(1.1)

where n i s t h e number of c h a r a c t e r i s t i c measured, and

FACTOR ANALYSIS

117

k i s t h e s i z e o f t h e sample. Thus, e v e r y row c o r r e s p o n d s

t o a s e t of measurements o f a n element o f t h e sample, and e a c h column c o r r e s p o n d s t o measurements o f one char a c t e r i s t i c i n t h e sample. The a s s u m p t i o n t h a t t h e sample r e p r e s e n t s t h e p o p u l a t i o n under s t u d y may b e e x p r e s s e d as s a y i n g t h a t e a c h row of t a b l e (1.1) i s a r e a l i z a t i o n o f a c e r t a i n n-dim e n s i o n a l random v a r i a b l e

x'

= (xl,

...,

Xn).

Here d e n o t e s t r a n s p o s i t i o n , i . e . change o f rows t o columns. I n t h i s c h a p t e r , v e c t o r s ( d e n o t e d b y a n underl i n e d l e t t e r , w i t h o u t t h e s i g n * ) w i l l be r e g a r d e d as columns. For g r a p h i c a l c o n v e n i e n c e , t h e y a r e w r i t t e n as t r a n s p o s i t i o n s , i . e . row v e c t o r s . The a i m o f f a c t o r a n a l y s i s i s t o decompose t h e random

v a r i a b l e 2 i n t o some o t h e r random v a r i a b l e s . It i s assumed t h a t 5 has f i n i t e moments o f t h e second o r d e r ( h e n c e v a r i a n c e s and c o v a r i a n c e s e x i s t ) . L e t

...,

m n ) . Thus, mi i s t h e e x p e c t e d s c o r e and m* = ( m l , on i - t h measurement i n t h e p o p u l a t i o n under s t u d y . Next (1.3) Thus, x;

5

i s t h e c o v a r i a n c e m a t r i x of t h e random v e c t o r

on t h e i n t e r s e c t i o n o f i - t h row and j - t h column of - m i ) ( x j - m . ) = cov ( x i , x j ) .

t h i s m a t r i x we have E ( x i

118

CHAPTER 2

In particular, on the diagonal we have the elements of the form E(xi - mi) c , hence variances of the components of the vector 5 . r

Both the vector m and the covariance matrix may be estimated from the sample (1.1). In particular, the estimator of m i is

while the estimator of the covariance Cov (xi,x.) is J

-1

k

(xit) - Ci)(x (t)

t=l

j

-

m.). J

(1.5)

In particular,

the estimator of the variance of the random variable x is i

-1 E. (Xi(t)

- * mi) 2 .

(1.6)

t=l

h

h

In the sequel, g and will denote the estimates of the expectation fi and covariance matrix El. The model of factor analysis may be formulated as follows: for the vector 2 of the observed random variables we l o o k for a representation in the form x =

g + Af

+

g,

(1.7)

where f ' = (fl,..., f ) is a certain r-dimensional ranr dom variable, whose components are called common factors, & is a matrix of dimension n x r (with n rows and r columns), called the matrix of factor loadings, and = (ul, un ) is an n-dimensional random variable, whose components are called specific factors.

...,

119

FACTOR ANALYSIS

The a s s u m p t i o n s c o n c e r n i n g t h e r e p r e s e n t a t i o n ( 1 . 7 ) a r e as f o l l o w s :

which means t h a t t h e number of common f a c t o r s i s l e s s t h a n t h e number o f v a r i a b l e s ; (b)

E(2lf)

=

9

f o r a l l v a l u e s of

1,

t h i s assumption i m p l i e s t h a t t h e r e g r e s s i o n of 5 on

t h e f a c t o r f i s l i n e a r . To prove i t , i t s u f f i c e s t o w r i t e ( 1 . 7 ) i n t h e form

and t a k i n g e x p e c t a t i o n s c o n d i t i o n a l on f one o b t a i n s

or

From a s s u m p t i o n ( b ) i t f o l l o w s a l s o t h a t E ( 2 ) = 0, s i n - = E [ E ( u i f ) ] = E ( 2 ) = 2, and t h a t t h e random v a r ce E(u) i a b l e s ui ( i = l,.. ,n) and f. (j = l,.. . , r ) are unJ correlated (since E(u.f.) = E[E(u.f.lf)l = E[fjE((uilf)l

.

1 J

= 0).

The n e x t assumption has t h e form

1 3

120

CHAPTER 2

D

(c)

=

F(=')

i s a diagonal m a t r i x .

T h i s assumption means t h a t t h e s p e c i f i c f a c t o r s u

are m u t u a l l y u n c o r r e l a t e d ( t h e y a r e a l s o , as shown, not c o r r e l a t e d w i t h common f a c t o r s ) . j

We assume a l s o t h a t ; (d)

H = E (ff?)

d

i s not s i n g u l a r .

T h i s a s s u m p t i o n means t h a t common f a c t o r s fl,...,f

1"

are

l i n e a r l y i n d e p e n d e n t , and c o n s e q u e n t l y , t h e r e e x i s t s no r e p r e s e n t a t i o n ( 1 . 7 ) w i t h a smaller number of f a c tors. F i n a l l y , t h e l a s t assumption a s s e r t s t h a t (e)

h2 = 1 f o r e a c h j, where h2 i s t h e d i a g o n a l jj jj

element of m a t r i x

3.

T h i s a s s u m p t i o n , l e a s t i m p o r t a n t among t h e a s s u m p t i o n s ,

s t a t e s t h a t v a r i a n c e s of common f a c t o r s are 1, i . e . t h a t common f a c t o r s are s t a n d a r d i z e d . T h i s may a l w a y s be a c h i e v e d by a change of f a c t o r l o a d i n g s i n m a t r i x

A.

From t h e above a s s u m p t i o n s one may d e r i v e t h e f o l l o w i n g important r e s u l t concerning t h e decomposition of t h e of vector 5 : covariance matrix

( s e e Lord and Novick 1968, p . 5 3 3 ) . I n d e e d , from t h e d e f i n i t i o n we have 5 = E ( q - E ) ? , hence i n view of x - m = Af + u ( s e e f o r m u l a ( 1 . 7 ) ) one g e t s

m)(x

FACTOR ANALYSIS

= E[A ff'A' = AE(E')A'

t

Afu

t Uf'A'

-

t E (u u ' ) = AHA'

121

&'I t D.

I n p a r t i c u l a r , i f t h e common f a c t o r s a r e u n c o r r e l a t e d , then = 2, where 2 i s t h e i d e n t i t y m a t r i x of s i z e ~ x r , hence

R

=

t

D.

(1.10)

Such a d e c o m p o s i t i o n o f m a t r i x zation.

i s called its factori-

1.2. Discussion

1.2.1.

Arbitrary decisions i n applications of factor

a n a l y s i s . To sum up t h e c o n t e n t o f t h e p r e c e d i n g s e c t i o n , one may s a y t h a t t h e model of f a c t o r a n a l y s i s cons i s t s of e x p l a i n i n g t h e whole r e l a t i o n s h i p between t h e o b s e r v e d v a r i a b l e s 5 by t h e a p p e a r e n c e of common f a c t ors: a f t e r s u b t r a c t i n g t h e a p p r o p r i a t e weighted a v e r a g e s o f t h e s e f a c t o r s (and s u b t r a c t i o n o f t h e common - m - Af which are mean) t h e r e remain r e s i d u a l s 1 = mutually uncorrelated.

x

The b a s i c theorems ( f o r m u l a s ( 1 . 9 ) and ( 1 . 1 0 ) ) a s s e r t

t h a t under a s s u m p t i o n s ( a )

rix

-

( e ) , the covariance m a t may b e decomposed i n t o t h e m a t r i x of c o v a r i a n c e s

122

CHAPTER 2

H , m a t r i x of f a c t o r l o a d i n g s A , and t h e d i a g o n a l m a t -

r i x D of v a r i a n c e s of s p e c i f i c f a c t o r s . The f o l l o w i n g q u e s t i o n s , b a s i c f o r t h e model o f f a c t o r a n a l y s i s , may be a s k e d . 1) Does a d e c o m p o s i t i o n g i v e n by ( 1 . 9 ) a l w a y s e x i s t ?

I n o t h e r words, can e v e r y c o v a r i a n c e m a t r i x be decomposed? The answer t o t h i s q u e s t i o n i s p o s i t i v e : f o r e v e r y covariance matrix that the matrix

E

t h e r e e x i s t m a t r i c e s g and A s u c h - A H A ' i s d i a g o n a l . What i s more,

one may prove ( s e e Lord and Novick 1968, p . 5 3 3 ) t h a t t h e rank of matrix i s u n i q u e l y d e t e r m i n e d . Thus, t h e covariance matrix common f a c t o r s .

d e t e r m i n e s u n i q u e l y t h e number o f

T h i s f a c t h a s , however, o n l y a l i m i t e d p r a c t i c a l con-

sequence, s i n c e t h e covariance m a t r i x known; one knows o n l y i t s e s t i m a t e

n

i s u s u a l l y un-

E.

The f a c t t h a t i n a p p l i c a t i o n s o f f a c t o r a n a l y s i s one d e a l s w i t h t h e e s t i m a t e s o f c o v a r i a n c e m a t r i x has a l s o more s e r i o u s consequences t h a n i n c o m p l e t e n e s s o f knowl e d g e of p a r a m e t e r s i n seemingly s i m i l a r s t a t i s t i c a l problems. The p o i n t i s t h a t i n o r d e r t o a p p l y t h e s t a n d a r d e s t i m a t i o n t e c h n i q u e s , s u c h a s maximum l i k e l i h o o d ,

i t i s n e c e s s a r y t o have t h e c o r r e s p o n d i n g e q u a t i o n s , i n which t h e p a r a m e t e r s a p p e a r as unknowns. I n c a s e o f f a c t o r a n a l y s i s , w r i t i n g such equations i s impossible ( u n l e s s o f assumes a p r i o r i t h e number o f f a c t o r s , which i s an unknown). There i s no need t o p r e s e n t h e r e t h e t e c h n i c a l l y com-

123

FACTOR ANALYSIS

p l i c a t e d f o r m u l a s . G e n e r a l l y , t h e s o l u t i o n s c o n s i s t of a p p l y i n g t h e known t e c h n i q u e s o f f a c t o r e x t r a c t i o n u n t i l the matrix (1.11) w i l l be " s u f f i c i e n t l y c l o s e " t o a d i a g o n a l m a t r i x , i n

t h e s e n s e o f a c r i t e r i o n , e x p r e s s e d t h r o u g h a n appropr i a t e f u n c t i o n o f non-diagonal e l e m e n t s of (1.11). One may a l s o a p p l y t h e s c r e e t e s t s u g g e s t e d by C a t t e l l for t h e number of f a c t o r s ( s e e C a t t e l l 1 9 6 6 a ) : f o r t e s t i n g t h e number o f f a c t o r s i n t h e m a t r i x 5 C a t t e l l s u g g e s t s u s i n g t h e i n t u i t i v e e x p e c t a t i o n t h a t i n s u c c e s s i v e matrices - =' for i n c r e a s i n g number of f a c t o r s , t h e r e s h o u l d be a r a p i d i n c r e a s e of t h e q u a l i t y of a p p r o x i m a t i o n t o a d i a g o n a l m a t r i x when t h e t r u e number of f a c t o r s i s exceeded. For o t h e r t e c h n i q u e s o f d e t e r m i n i n g t h e number of f a c t o r s , s e e e . g . C a t t e l l 1958 o r Henrysson 1957. I t s h o u l d , however, be mentioned t h a t t h e s c r e e t e s t was c r i t i c i z e d , e . g . by G u i l f o r d ( s e e G u i l f o r d and Hoepfner 1971 ) .

Another methods a r e based on assuming a g i v e n number of f a c t o r s ( s a y , t w o ) . Then one may d e v i s e t e c h n i q u e s , based e . g . on maximum l i k e l i h o o d p r i n c i p l e , t o e s t i m a t e t h e m a t r i x o f f a c t o r l o a d i n g s ( s e e e . g . Harman 1976) I t seems t h e r e f o r e t h a t t h e d e c i s i o n c o n c e r n i n g t h e number o f f a c t o r s i s t h e f i r s t i n a s e r i e s o f d e c i s i o n s

which a r e made i n a more or l e s s a r b i t r a r y way, hence d e c i s i o n s which b i a s t h e r e s u l t s o f f a c t o r a n a l y s i s by s u b j e c t i v i t y . The n e x t s u c h d e c i s i o n s concern r o t a t i o n

-

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and i n t e r p r e t a t i o n . I n d e e d , one has t o d e c i d e e i t h e r t o

use a program which e x t r a c t s a f i x e d number of f a c t o r s , o r a program which has a " b u i l t i n " s t o p p i n g r u l e , which t e r m i n a t e s t h e e x t r a c t i o n when a n a p p r o p r i a t e c r i t e r i o n i s m e t . N a t u r a l l y , a f t e r s u c h a program i s s e l e c t e d , t h e r e s u l t s of f a c t o r a n a l y s i s are o b j e c t i v e , i n t h e s e n s e t h a t t h e y a r e d e t e r m i n e d by t h e d a t a . The p o i n t i s , however, t h a t t h e c h o i c e o f t h e program i s l e f t t o the experimenter's discretion. The n e x t q u e s t i o n connected w i t h t h e model of f a c t o r analysis i s t h e following: 2 ) I s t h e f a c t o r i z a t i o n o f t h e c o v a r i a n c e m a t r i x uni-

quely determined? It t u r n s o u t t h a t h e r e t h e answer i s n e g a t i v e : t h e r e a l w a y s e x i s t i n f i n i t e l y many f a c t o r i z a t i o n s o f m a t r i x R , b o t h o f t h e form (1.9), as w e l l as of t h e form ( 1 . 1 0 ) . The c h o i c e o f one p a r t i c u l a r d e c o m p o s i t i o n i s c a l l e d t h e problem of r o t a t i o n ( o f t h e e x t r a c t e d s t r u c t u r e ) . To show t h a t f a c t o r i z a t i o n i s n e v e r u n i q u e , s u p p o s e that = m t Af t g, s o t h a t R = AHA' t D i s a f a c t o r i n t o t h e m a t r i x of i z a t i o n of t h e c o v a r i a n c e m a t r i x covariances between common f a c t o r s , m a t r i x 4 o f f a c t o r l o a d i n g s , and m a t r i x g o f v a r i a n c e s of s p e c i f i c f a c t o r s . Suppose t h a t 5 i s a m a t r i x of dimension r x r , and l e t 2 d e n o t e a n a r b i t r a r y n o n - s i n g u l a r m a t r i x of t h e s i z e r x r . L e t us d e f i n e t h e new v e c t o r of common -1 f a c t o r s g b y t h e f o r m u l a g = Tf, i . e . t = T. g. Then x = m t g - ' g t g, and t h e c o v a r i a n c e m a t r i x I3 w i l l be * -1 * R = -AT _ H- ( A T - l ) ' t ll, where = E(gg') = E ( T f ) ( T f ) ' - 1- w e = E(T ff " T ' ) = EE(ff')T' = E'.D e n o t i n g = AT

x

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obtain another factorization

E

*

= BH B' t

g.

I n a s i m i l a r way one may show t h a t f a c t o r i z a t i o n i s n o t u n i q u e , even i f one r e s t r i c t s t h e c o n s i d e r a t i o n t o o r t h o gonal s t r u c t u r e s only. i . e . t o f a c t o r i z a t i o n s of t h e form ( 1 . 1 0 ) . I n d e e d , i f 5 = g t Af + 2 where E (ff') = I , i . e . E = g't g, t h e h one may t a k e g = T f , where T i s an a r b i t r a r y orthogonal matrix, i . e . such t h a t -1 * = T - I . Then 2 = m t 2 g t 2, as b e f o r e , and g = E(=') = E E ( f f * ) T ' = TIT' = I . Consequently, = BB' + D, with = AT-l-is a f a c t o r i z a t i o n of m a t r i x g o f t h e form ( 1 . 1 0 ) .

r'

I n p r a c t i c e , t h e problem o f r o t a t i o n i s u s u a l l y s o l v e d i n s e v e r a l c o n s e c u t i v e s t e p s , e a c h of them r e q u i r i n g an arbitrary choice. Naturally, i n these choices t h e e x p e r i m e n t e r i s guided by h i s knowledge, i n t u i t i o n , p r i o r a s s u m p t i o n s about t h e phenomenon s t u d i e d , common s e n s e , and f i n a l l y - tendency t o parsimony of t h e obtained structure. Thus, t h e f i r s t d e c i s i o n c o n c e r n s t h e c h o i c e between a n o r t h o g o n a l s t r u c t u r e , and one o u t o f i n f i n i t e l y many o b l i q u e s t r u c t u r e s ( f o r m a l l y s p e a k i n g , t h i s c h o i c e does not concern r o t a t i o n , s i n c e t h e e x t r a c t e d s t r u c t u r e , o r t h o g o n a l or n o t , must s t i l l be r o t a t e d ) . Here t h e r e s e a r c h e r must t a k e i n t o a c c o u n t t h e f a c t t h a t , on t h e one hand, o r t h o g o n a l s t r u c t u r e s are e a s i e r t o h a n d l e by s t a t i s t i c a l t e c h n i q u e s , and a r e i n some sense more p a r s i m o n i o u s t h a n o b l i q u e s t r u c t u r e s . On t h e o t h e r hand, t h e r e e x i s t s i t u a t i o n s i n which t h e phenomenon a n a l y s e d i s a l r e a d y d e s c r i b e d , a t l e a s t p a r t i a l ly, i n terms of some f a c t o r s w i t h f i x e d and w e l l e s tablished t r a d i t i o n . If these factors are correlated,

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i t i s u s u a l l y b e t t e r t o use a n o b l i q u e s t r u c t u r e , w i t h correlated factors. An i l l u m i n a t i n g example h e r e i s g i v e n by T h u r s t o n e . It i s narhely known t h a t weight and h e i g h t a r e c o r r e l a t e d . One can c e r t a i n l y f i n d two l i n e a r c o m b i n a t i o n s o f we--i g h t and h e i g h t which would b e u n c o r r e l a t e d , b u t i t d o e s n o t seem r e a s o n a b l e t o u s e s u c h l i n e a r c o m b i n a t i o n s i n practice. N e x t , as r e g a r d s a r o t a t i o n of t h e e x t r a c t e d s t r u c t u r e , a l l known t e c h n i q u e s seem t o be b a s e d on t h e p r i n c i p l e o f t h e s o - c a l l e d s i m p l e s t r u c t u r e , s u g g e s t e d by Thurstone (1947). T h i s p r i n c i p l e i s based on a s u p p o s i t i o n t h a t i n t h e

phenomenon under s t u d y e a c h of t h e v a r i a b l e s i s l e s s complex f a c t o r i a l l y t h a n t h e whole phenomenon; if s o , one should be a b l e t o o b t a i n a m a t r i x o f f a c t o r l o a d i n g s s u c h t h a t t h e v a r i a b l e s would c o r r e l a t e h i g h l y w i t h few f a c t o r s and have low c o r r e l a t i o n s w i t h t h e r e m a i n i n g f a c t o r s . I n s u c h a c a s e , i n e a c h row o f t h e m a t r i x A of f a c t o r l o a d i n g s t h e r e s h o u l d be p o s s i b l y many z e r o s . Moreover, t o f a c i l i t a t e i n t e r p r e t a t i o n , one imposes t h e a d d i t i o n a l c o n d i t i o n , t h a t p o s s i b l y many z e r o s s h o u l d a l s o a p p e a r i n e a c h column ( s o t h a t e a c h f a c t o r would be r e p r e s e n t e d by few v a r i a b l e s o n l y ) ; f o r e a c h two columns t h e r e s h o u l d e x i s t v a r i a b l e s which have h i g h c o r r e l a t i o n s i n one column and low c o r r e l a t i o n s i n a n o t h e r ( s o as t o e n s u r e o r t h o g o n a l i t y o f f a c t o r s ) ; moreover, t h e r e s h o u l d b e many v a r i a b l e s w i t h z e r o ( l o w ) c o r r e l a t i o n s i n b o t h columns, and only few w i t h h i g h c o r r e l a t i o n s i n b o t h columns. These q u a l i t a t i v e conditions characterize simple s t r u c t u r e s .

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121

It i s w o r t h w h i l e t o p o i n t o u t h e r e t h a t t h e e x i s t e n -

ce of a s i m p l e s t r u c t u r e i s a c e r t a i n h y p o t h e s i s ; i t may p e r h a p s be r e a s o n a b l y e x p e c t e d t o h o l d i n most cases, but it i s not automatically s a t i s f i e d . T h i s p r i n c i p l e i s v e r y g e n e r a l , and i n p a r t i c u l a r i n -

s t a n c e s i t must b e a p p r o p r i a t e l y s p e c i f i e d , s i n c e t h e r e may e x i s t many r o t a t i o n s which l e a d t o m a t r i x o f f a c t or l o a d i n g s s a t i s f y i n g t h e r e q u i r e m e n t s of s i m p l e s t r u c t u r e . Thus, e a c h program f o r a computer which e x t r a c t s a s i m p l e s t r u c t u r e i s based on a c e r t a i n c r i t e r i o n , which r e q u i r e s m a x i m i z a t i o n of m i n i m i z a t i o n o f a c e r t a i n f u n c t i o n ( e . g . programs V a r i m a x , Q u a r t i m a x , Q u a r t imin, e t c ) . There e x i s t s a r a t h e r wide s p r e a d c o n v i c t i o n t h a t t h e results of factor analysis are objective (at least with r e s p e c t t o t h e r o t a t i o n ) , s i n c e t h e y a r e o b t a i n e d as a r e s u l t o f a c e r t a i n o b j e c t i v e p r o c e d u r e . It i s c l e a r , t h a t when t h e program i s s e l e c t e d , t h e r e s u l t s a r e dat a - d e t e r m i n e d ; t h e p o i n t i s , however, t h a t t h e c h o i c e o f t h e program i s t o a l a r g e e x t e n t s u b j e c t i v e : one c h o o s e s , as a r u l e , s u c h a program which l e a d s t o a s t r u c t u r e which i s e a s i e s t t o i n t e r p r e t .

The problems o f i n v a r i a n c e . I n o r d e r t o be a b l e t o a s s i g n meaningful i n t e r p r e t a t i o n t o f a c t o r s , t h e l a t t e r cannot be a c c i d e n t a l , i . e . s p e c i f i c f o r t h e g i v e n c o v a r i a n c e m a t r i x . The same f a c t o r s s h o u l d have a r e a s o n a b l e chance o f a p p e a r i n g i n o t h e r s e t s o f d a t a . T h i s r e q u i r e m e n t , o f t h e e x t r a c t e d and r o t a t e d c o n f i g u r a t i o n t o be r e p e t i t i v e , i s u s u a l l y c a l l e d t h e r e q u i r e 1.2.2.

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ment of i n v a r i a n c e . I t would p r e s e n t l i t t l e s t a t i s t i c a l d i f f i c u l t i e s i f i t were r e s t r i c t e d t o r e p e t i t i o n s o f c o n f i g u r a t i o n s of f a c t o r s from s e t s o f d a t a o b t a i n e d by u s i n g a n a l o g o u s measurement t o o l s f o r o t h e r samp l e s from t h e same p o p u l a t i o n . I n d e e d , a f a c t o r may be r e g a r d e d a s a f u n c t i o n d e f i n e d on t h e s e t o f v a r i a b l e s , namely as a v e c t o r of f a c t o r l o a d i n g s . F o r t h e same domain, i . e . f o r t h e same s e t o f v a r i a b l e s , two f a c t o r s are i d e n t i c a l , i f t h e c o r r e s p o n d i n g f u n c t i o n s c o i n c i d e , i . e . i f t h e y have t h e

same v e c t o r s o f f a c t o r l o a d i n g s . I n t h i s c a s e one may r e l a t i v e l y easily construct a c r i t e r i o n f o r t e s t i n g t h e d i f f e r e n c e between two f a c t o r s . Such t e s t s may b e based on t h e l o c a t i o n o f maxima ( t h e s o - c a l l e d marker v a r i a b l e s ) , o r on t h e s u i t a b l y d e f i n e d d i s t a n c e between v e c t o r s of f a c t o r l o a d i n g s ( c o m p a r i s o n of p r o f i l e s ) . The problem, however, becomes more c o m p l i c a t e d i f one requires a higher order of generality of r e s u l t s , i . e . i f one r e q u i r e s i n v a r i a n c e a c r o s s p o p u l a t i o n s , o r

--

which i s even more d i f f i c u l t -- i n v a r i a n c e w i t h r e s p e c t t o a p a r t i a l o r complete change o f t o o l s of measurement. The l a t t e r problem was c o n s i d e r e d by C a t t e l l ( s e e Catt e l l 1957, 1 9 6 1 , 196613; C a t t e l l and Digman 1 9 6 4 ; C a t t e l l , Eber and T a t s u o k a 1970) i n c o n n e c t i o n w i t h h i s t h e o r y

of p e r s o n a l i t y . C a t t e l l d i s t i n g u i s h e d namely t h r e e s o r c e s o f i n f o r m a t i o n about p e r s o n a l i t y : s e l f - e v a l u a t i o n , c a l l e d Q d a t a (from: q u e s t i o n n a i r e d a t a ) , L d a t a , ( l i f e d a t a ) , b e i n g t h e e v a l u a t i o n s by o t h e r s , and T d a t a ( t e s t d a t a ) , from o b j e c t i v e t e s t s . C a t t e l l formula t e d a n i n t e r e s t i n g h y p o t h e s i s , which a s s e r t s t h a t t h e f a c t o r s t r u c t u r e e x t r a c t e d from e a c h of t h e s e s e t s of

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129

d a t a s h o u l d be t h e same. The i n t u i t i v e j u s t i f i c a t i o n

of t h i s h y p o t h e s i s i s t h a t e a c h o f t h e s e s e t s o f d a t a c o n t a i n s i n f o r m a t i o n about t h e same phenomenon, namel y p e r s o n a l i t y . If t h i s h y p o t h e s i s should t u r n o u t t o be t r u e , i t would mean t h a t one c o u l d p r e d i c t b e h a v i o u r of a s u b j e c t on t h e b a s i s of one t y p e o f d a t a o n l y . T h i s e x p l a i n s t h e name used b y C a t t e l l : h y p o t h e s i s o f i n d i f f e r e n c e o f media. I t i s w o r t h t o m e n t i o n , i n c i d e n t a l l y , t h a t t h e hypothe-

s i s o f i n d i f f e r e n c e o f media was n e v e r f u l l y confirme d . T h i s f a i l u r e o f C a t t e l l may be r e g a r d e d as one o f h i s most i n t e r e s t i n g r e s u l t s ( s e e Nowakowska 1 9 7 3 % and a l s o t h e subsequent s e c t i o n ) . The methodology f o r t e s t i n g i n v a r i a n c e , c r e a t e d mainly by C a t t e l l i n o r d e r t o t e s t h i s h y p o t h e s i s of i n d i f f e r n e c e of media, l i e s on t h e b o r d e r l i n e of s t a t i s t i c a l i n f e r e n c e and p s y c h o l o g i c a l a n a l y s i s of t h e c o n t e n t o f e x t r a c t e d and r o t a t e d f a c t o r s . From t h e s t a t i s t i c a l vie.wpoint, t h e methodology of t e s t i n g i n v a r i a n c e c o n s i s t s o f s i m u l t a n e o u s f a c t o r anal y s i s of t h e j o i n t covariance m a t r i x f o r various sets of d a t a ( s e e Tucker 1958; Browne 1979, 1 9 8 0 ) . Roughly, two f a c t o r s from two domains may be r e g a r d e d as i d e n t i c a l , i f t h e y c o r r e l a t e h i g h l y among t h e m s e l v e s , and a l s o have i d e n t i c a l c o r r e l a t i o n s w i t h o t h e r v a r i a b l e s , t a k e n as c r i t e r i a . T h i s r e q u i r e m e n t , however, i s onl y necessary: i t s f u l f i l l m e n t i s not s u f f i c i e n t f o r i n v a r i a n c e . To show t h a t two f a c t o r s a r e i d e n t i c a l n o t only s t a t i s t i c a l l y but a l s o p s y c h o l o g i c a l l y , i t i s necessary t o analyse t h e content of t h e f a c t o r s .

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1 . 2 . 3 . I n t e r p r e t a t i o n of r e s u l t s of f a c t o r a n a l y s i s . If a g i v e n c o n f i g u r a t i o n o f f a c t o r s s a t i s f i e s t h e cond i t i o n s of i n v a r i a n c e , one may a s s i g n meanings t o f a c t o r s . Roughly s p e a k i n g , two s t r a t e g i e s a r e c o n s i d e r e d h e r e . F i r s t l y , f a c t o r s may be r e g a r d e d merely as means of c l a s s i f i c a t i o n . I n t h i s c a s e , t h e i n t e r p r e t a t i o n c o n s i s t s s i m p l y of c h o o s i n g a n a p p r o p r i a t e name f o r t h e s e t o f v a r i a b l e s r e p r e s e n t i n g i t (i.e. s e t o f v a r i a b l e s with highest f a c t o r loadings), so as t o generalize t h e s e v a r i a b l e s ( s e e e . g . Horst 1 9 6 5 ) . Secondly ( s e e e . g . T h u r s t o n e 1947), f a c t o r s may be assi g n e d n o t o n l y names, b u t a l s o e x p l a n a t o r y h y p o t h e s e s , which a c c o u n t f o r t h e c o v a r i a n c e s between v a r i a b l e s . I n s u c h a c a s e , f a c t o r i s t r e a t e d as a d e t e r m i n a n t o f c o - v a r i a b i l i t y ; i n t h i s s e n s e , i t becomes a c o n s t r u c t , endowed w i t h s u r p l u s meaning. Thus, w h i l e b e f o r e i n t e r p r e t a t i o n f a c t o r c o n s t a i n s no more i n f o r m a t i o n t h a n t h e c o v a r i a n c e m a t r i x , t h e expl a n a t o r y h y p o t h e s e s , as i n v o l v i n g s u r p l u s meaning, p r o v i d e more i n f o r m a t i o n t h a n c o n t a i n e d i n t h e c o v a r i a n c e m a t r i x . Such an e x p l a n a t o r y h y p o t h e s i s i s , however, t o a l a r g e e x t e n t s u b j e c t i v e ; t h i s i s t h e p r i c e which one has t o pay f o r p a s s i n g from a r i g i d and p u r e l y d e s c r i p t i v e i n t e r p r e t a t i o n t o an explanator y i n t e r p r e t a t i o n , which a l l o w s t o l o c a t e t h e f a c t o r i n t h e network o f o t h e r t h e o r e t i c a l n o t i o n s . I n t h i s approach, t h e successive steps of f a c t o r analysis a r e i n f a c t steps i n t h e i n d u c t i o n p r o c e s s . The p r e l i m i n a r y h y p o t h e s i s , b e f o r e invariance i s established, yields an i n t e r p r e t a t i o n which may be confirmed or r e j e c t e d by f u r t h e r s t u d i e s concerning invariance.

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C o g n i t i v e l i m i t a t i o n s o f f a c t o r a n a l y s i s -- the s u b j e c t i v e b i a s . A s e x p l a i n e d above, t h e r e s u l t s o f f a c t o r a n a l y s i s a r e b i a s e d i n a t l e a s t two p l a c e s . F i r s t l y , t h e r e u s u a l l y e x i s t s a number o f v a r i o u s programs f o r e x t r a c t i o n and r o t a t i o n of f a c t o r s , e a c h l e a d i n g t o a s i m p l e s t r u c t u r e , which a l l o w s a c l e a r i n t e r p r e t ation. 1.2.4.

Secondly ( t h i s remark a p p l i e s when f a c t o r a n a l y s i s i s used n o t o n l y as a t a x o n o m i c a l method, b u t a l s o as a s o u r c e of e x p l a n a t o r y h y p o t h e s e s ) , s u b j e c t i v i t y i n t e r venes i n i n t e r p r e t a t i o n o f f a c t o r s , i . e . i n a s s i g n i n g t o them meanings c o n s i s t e n t w i t h known t h e o r i e s . Looking somewhat s u p e r f i c i a l l y one could s a y t h a t t h e same o b j e c t i o n s ( a t l e a s t t h e f i r s t o n e ) a p p l y a l s o t o o t h e r m a t h e m a t i c a l models. I n d e e d , one could c l a i m t h a t if someone a p p l i e s a model of P o i s s o n p r o c e s s t o t h e phenomenon o f r a d i o a c t i v e d e c a y , or t o t h e work o f a t e l e p h o n e exchange, he makes a s u b j e c t i v e c h o i c e o f one of many a l t e r n a t i v e models. T h i s , o f c o u r s e , i s t r u e , but t h e b a s i c d i f f e r e n c e l i e s i n t h e f a c t t h a t i n such a p p l i c a t i o n s t h e c o g n i t i v e r e s t r i c t i o n s a r e e i t h e r made precise a t t h e beginning, o r a r e studied empirically. No one would c l a i m ( a s i t was done by t h e p r o p o n e n t s of f a c t o r a n a l y s i s ) t h a t P o i s s o n p r o c e s s i s t h e d e s s r i p t i o n o f t h e " t r u e " and "unique" n a t u r e o f r a d i o a c t i v e p r o c e s s , or t h e work o f a t e l e p h o n e exchange. One c o u l d a l s o d e f e n d t h e o b j e c t i v l t y o f f a c t o r a n a l y s i s by u s i n g t h e argument o f v a l i d i t y o f p r e d i c t i o n s o f v a r i o u s e x t e r n a l c r i t e r i a , and c l a i m t h a t t h e same arguments a r e u s e d f o r t h e d e f e n s e o f P o i s s o n p r o c e s s as a model f o r r a d i o a c t i v e d e c a y . A moment o f r e f l e c t -

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i o n s u f f i c e s , however, t o r e a l i z e t h a t v a l i d i t y of p r e d i c t i o n shows o n l y t h e p r a g m a t i c v a l u e o f t h e model, but n o t i t s u n i q u e n e s s . I n f a c t , t h e p o s s i b i l i t y o f p r e d i c t i o n of one or more e x t e r n a l c r i t e r i a on t h e basis of a g i v e n s e t o f v a r i a b l e s depends o n l y on t h e j o i n t p r o b a b i l i t y d i s t r i b u t i o n of t h e s e v a r i a b l e s and c r i t e r i a . The o p t i m a l p r e d i c t i o n , g i v e n by t h e r e g r e s s i o n of t h e f i r s t k i n d ( s e e p r e c e d i n g c h a p t e r ) i s , in c a s e o f f a c t o r a n a l y s i s , irnpossible, since t h e data collected f o r f a c t o r a n a l y s i s do n o t u s u a l l y p r o v i d e t h e p o s s i b i l i t y of e s t i m a t i n g the j o i n t probability distribution. There remains t h e p o s s i b i l i t y o f u s i n g t h e methods of l i n e a r r e g r e s s i o n , where one t a k e s a s p r e d i c t o r s t h e a p p r o p r i a t e l i n e a r c o m b i n a t i o n s o f v a r i a b l e s ( t h e coe f f i c i e n t s b e i n g d e t e r m i n e d s o as t o minimize t h e mean s q u a r e e r r o r o f p r e d i c t i o n ) . Such a p r o c e d u r e r e q u i r e s o n l y t h e knowledge o f means, v a r i a n c e s and c o v a r i a n c e s of t h e v a r i a b l e s and t h e c r i t e r i o n , hence r e q u i r e s t h e d a t a which a r e u s u a l l y a c c e s s i b l e from t h e r e s u l t s of f a c t o r a n a l y s i s . Moreover, l i n e a r r e g r e s s i o n has t h e property t h a t it coincides with t r u e regression i n c a s e o f j o i n t normal d i s t r i b u t i o n o f v a r i a b l e s and c r i teria. D e t e r m i n i n g t h e l i n e a r combinations of v a r i a b l e s des t r o y s p a r t i a l l y t h e i n f o r m a t i o n c o n t a i n e d i n t h e values o f t h e s e v a r i a b l e s ( s i m p l y because t h e knowledge of t h e v a l u e o f l i n e a r combination does n o t s u f f i c e t o r e c o n s t r u c t t h e v a l u e s o f v a r i a b l e s ) . T h i s l o s s 01’ i n f o r m a t i o n may be more or l e s s s e v e r e , d e p e n d i n g on t h e joint probability distribution.

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I n c a s e of f a c t o r a n a l y s i s t h e s i t u a t i o n i s more comp l e x , s i n c e f a c t o r s are l i n e a r combinations of v a r i a b l e s . I f t h e problem c o u l d be r e d u c e d t o p r e d i c t i n g of one c r i t e r i o n o n l y , i t would p r o b a b l y b e b e t t e r t o b a s e t h e p r e d i c t i o n s on t h e v a r i a b l e s d i r e c t l y , i n s t e a d of u s i n g f a c t o r s which d e s t r o y p a r t i a l l y t h e i n f o r m a t i o n . The p o i n t i s , however, t h a t a s a r u l e one wants t o use t h e same f a c t o r s f o r p r e d i c t i n g v a r i o u s c r i t e r i a . I n s u c h c a s e s , i t seems r e a s o n a b l e t o f i n d a "good" i n i t i a l r e d u c t i o n of d a t a t o a s m a l l e r number o f v a r i a b l e s ( f a c t o r s ) -- b u t s o as n o t t o l o s e t o o much informa t i o n . T h i s i s t h e b a s i c i d e a of f a c t o r a n a l y s i s . However, even i f s u c h a r e d u c t i o n p r o v i d e s a s e t o f v a r i a b l e s which a r e u s e f u l i n p r e d i c t i n g v a r i o u s c r i t e r i a , a l l one may s a y i s t h a t t h e f a c t o r s a r e w e l l c h o s e n , u s i n g v a l i d i t y o f p r e d i c t i o n s as a n argument. There i s s t i l l no ground t o a s s e r t t h a t t h e c h o i c e leads t o t h e unique t r u e s t r u c t u r e o f t h e phenomenon, as some f a c t o r a n a l y s t s t e n d t o c l a i m , s i m p l y because there i s no r e a s o n why some o t h e r c h o i c e could n o t be b e t t e r . S e c o n d l y , even i f u n i q u e n e s s c o u l d be proved i n some way, i t would be u n i q u e n e s s w i t h r e s p e c t t o t h e s e l e c t ed s e t of v a r i a b l e s , and n o t " u n i v e r s a l " u n i q u e n e s s w i t h r e s p e c t t o t h e phenomenon. G e n e r a l l y , even i f t h e problems o f u n i q u e n e s s and i n v a r i a n c e c o u l d be s a t i s f a c t o r i l y r e s o l v e d , i t c o u l d a t b e s t s e r v e as a n argument t h a t f a c t o r a n a l y s i s a l l o w s u s t o e x t r a c t t h e t r u e n a t u r e o f t h e phenomenon as i t was d e f i n e d by t h e v a r i a b l e s chosen; i f some a s p e c t s a r e o m i t t e d i n t h e i n i t i a l s e t of v a r i a b l e s , t h e y w i l l not appear i n f a c t o r analysis.

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C o n s e q u e n t l y , i n p l a n n i n g a f a c t o r i a l e x p e r i m e n t , one s h o u l d choose t h e i n i t i a l s e t o f v a r i a b l e s i n an exhaus t i v e w a y , i n c l u d i n g a l l v a r i a b l e s which may c o n t r i b u t e t o the description.

I t is worth t o mention h e r e t h a t t h e r e e x i s t two " s t y l e s " of a p p l y i n g t h e model o f f a c t o r a n a l y s i s . F i r s t l y , one may p e r f o r m f a c t o r a n a l y s i s on a s e t o f d a t a w i t h o u t any p r i o r p r e - c o n c e p t i o n s

and h y p o t h e s e s a b o u t t h e

n a t u r e of t h e phenomenon under i n v e s t i g a t i o n . Simply , u s i n g s u f f i c i e n t l y many v a r i a b l e s which " c o v e r " t h e phenomenon, and p e r f o r m i n g f a c t o r a n a l y s i s , one o b t a i n s f a c t o r s r e g a r d e d as i n t r o d u c t o r y h y p o t h e s e s , which one needs t o f u r t h e r v e r i f y . S e c o n d l y , one may u s e f a c t o r a n a l y s i s as a t o o l f o r t e s t i n g i n t r o d u c t o r y hypotheses about t h e f a c t o r i a l s t r u c t u r e of t h e phenomenon ( f o r a d i s c u s s i o n o f t h e two s t r a t e g i e s , s e e e . g . Henrysson

1957).

These two s t y l e s , or s t r a t e g i e s , o f r e s e a r c h i n f a c t or a n a l y s i s a r e sometimes r e f e r r e d t o as e x p l o r a t o r y factor analysis

,

and c o n f i r m a t o r y f a c t o r a n a l y s i s . I t

ought t o be mentioned t h a t t h e f i r s t d o m i n a t e s t h e s e cond, b o t h as r e g a r d s t h e number o f t h e o r e t i c a l contr i b u t i o n s f o r e x p l a n a t o r y v s . c o n f i r m a t o r y f a c t o r anal y s i s , as w e l l as w i t h r e g a r d s t o t h e number o f empi-

r i c a l s t u d i e s . The same two s t r a t e g i e s c a r r y o v e r t o t h e t i o p i c s o f multidimensional s c a l i n g , a s e t of techn i q u e s w i t h aims l a r g e l y t h e same as f a c t o r a n a l y s i s , but based on weaker a s s u m p t i o n s a b o u t t h e d a t a . A s r e g a r d s f a c t o r a n a l y s i s , t h e b e s t known c o n f i r m a t o r y

t e c h n i q u e i s t h a t o f J d r e s k o g (1967), b a s e d on maximum

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l i k e l i h o o d p r i n c i p l e ( s e e a l s o Lee 1 9 8 1 ) . A t t h i s p l a c e i t i s w o r t h t o mention t h e work o f Davison (1980) who g i v e s a method o f t e s t i n g some h y p o t h e s e s a b o u t t h e o r d e r of items ( t e s t s , s t i m u l i , e t c . ) . Roughly s p e a k i n g , t h e s i t u a t i o n i s as f o l l o w s . Conside r t h e m a t r i x (1.1) o f r e s u l t s o f measurements, and suppose t h a t t h e i t e m s ( a s r e p r e s e n t e d b y columns) a r e dichotomous, s o t h a t x j j ) i s t h e r e s p o n s e ( s a y 1 o r 0 ) of j - t h p e r s o n t o i - t h i t e m . Suppose t h a t t h e items a r e o r d e r e d i n s u c h a way t h a t any p e r s o n who g i v e s answer " y e s " t o any i t e m s h o u l d a l s o g i v e answer " y e s " t o a l l i t e m s which follow i t i n t h e o r d e r i n g . T h i s a l lows us t o d i s t i n g u i s h t h e c l a s s of a d m i s s i b l e and t h e c l a s s o f i n a d m i s s i b l e r e s p o n s e p a t t e r n s . The problem l i e s i n t e s t i n g t h e h y p o t h e s i s about t h e o r d e r o f i t e m s i n t h e p r e s e n c e o f e r r o r s , i . e . assuming some d e g r e e of randomness i n r e s p o n s e s . Also, i t i s allowed t h a t some r e s p o n s e p a t t e r n s a r e e x c l u d e d by t h e c o n s t r a i n t s of t h e experiment.

1.2.5. The s t a t u s o f t h e v a r i a b l e s u n d e r l y i n g f a c t o r a n a l y s i s . T y p i c a l l y , i n p s y c h o l o g i c a l and s o c i o l o g i c a l a p p l i c a t i o n s o f f a c t o r a n a l y s i s , t h e v a r i a b l e s used a r e t h e r e s u l t s o f some t e s t s ( o r s i n g l e i t e m s ) f o r p a r t i c u l a r s u b j e c t s . To a p p l y t h e programs for e x t r a c t i n g f a c t o r s i t i s n e c e s s a r y t h a t t h e v a r i a b 3 e s a r e meas u r e d on a t l e a s t a n i n t e r v a l s c a l e , s o t h a t one may j u s t i f i a b l y c a l c h l a t e means., v a r i a n c e s and c o v a r i a n c e s . Moreover, f o r i n t e r p r e t a t i o n o f r e s u l t s , i t i s n e c e s s a r y t h a t one knows which t r a i t s a r e measured b y p a r t i cular tests (items).

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T h e o r e t i c a l l y , t o v e r i f y t h e s e two a s s u m p t i o n s , i t i s n e c e s s a r y f i r s t t o d e t e r m i n e t h e s t a t u s o f t h e measured t r a i t from t h e m e a s u r e m e n t - t h e o r e t i c a l p o i n t o f view, and t h e n check t h a t t h e i t e m i n q u e s t i o n p r o v i d e s a n a d e q u a t e measure of t h e t r a i t . To d e t e r m i n e t h e s t a t u s o f a g i v e n t r a i t from t h e p o i n t of view of measurement t h e o r y ( s e e e . g . Suppes and Z i nnes 1963; P f a n z a g l 1968, E l l i s 1966, Coombs, Dawes and Tversky 1970, Krantz e t a l . , 1970, o r R o b e r t s 1979), i t i s n e c e s s a r y t o a s s i g n t o t h i s t r a i t a c e r t a i n empir i c a l r e l a t i o n a l s y s t e m , and v e r i f y t h a t i t s a t i s f i e s t h e a p p r o p r i a t e axioms i m p l y i n g t h e e x i s t e n c e o f measurement on a s c a l e of a g i v e n t y p e ( s e e C h a p t e r 4 ) . I n c a s e o f i n t e r v a l s c a l e s , t h e r e must e x i s t a n u m e r i c a l r e p r e s e n t a t i o n of t h e t r a i t unique up t o a p o s i t i v e l i n e a r transformation. Another a p p r o a c h , a p p l i e d i n m e n t a l t e s t t h e o r y ( s e e Lord and Novick 1968) c o n s i s t s o f simply assuming t h a t t e s t r e s u l t s ( s c o r e s ) c o n s t i t u t e measurement on a t least i n t e r v a l t y p e scale. I n f a c t , one assumes even more, namely c o m p a r a b i l i t y of u n i t s o f measurement. The u s e of t h i s a s s u m p t i o n i s a p p a r e n t a l r e a d y on t h e l e v e l o f q u e s t i o n n a i r e i t e m s . I n d e e d , i f someone answers "yes" t o two q u e s t i o n s , s u c h as "DO you o f t e n f e e l t i r e d i n t h e morning?'' and "DO you t h i n k t h a t some numbers a r e u n l u c k y ? " , t h e n h i s s c o r e i n t h e " s u b t e s t " c o n s i s t i n g o f t h e above two i t e m s e q u a l s 2 . I f d i f f e r e n t w e i g h t s were a s s i g n e d to t h e s e items ( a s i t i s sometimes done i n c a s e o f t e s t s ) , i t would amount merely t o changing o f t h e u n i t s o f measurement.

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I n any c a s e , f o r t h e c o n d i t i o n s a l l o w i n g i n t e r p r e t a t i o n of t h e e x t r a c t e d and r o t a t e d f a c t o r s , i t i s e s s e n t i a l t h a n one knows what i s b e i n g measured by a g i v e n item (variable)

.

Theoretically, f o r t r a i t s defined through appropriate e m p i r i c a l systems i m p l y i n g i n t e r v a l t y p e s c a l e , s u c h a v e r i f i c a t i o n i s p o s s i b l e , i f o n l y t h e r e e x i s t s a way of m e a s u r i n g a g i v e n t r a i t w i t h o u t u s i n g t h e t e s t . Such an i d e a l s i t u a t i o n i f v e r y d i f f i c u l t , i f n o t i m possible t o a t t a i n i n practical situations, especially i n p s y c h o l o g i c a l r e s e a r c h , because t h e v a r i a b l e s a r e u s u a l l y on t o o h i g h l e v e l o f g e n e r a l i t y . I n p r a c t i c e , t o check t h a t a g i v e n p s y c h o l o g i c a l t o o l X ( t e s t , s a y ) measures a s e l e c t e d t r a i t Y, one has t o a p p l y t h e methods o u t l i n e d i n C h a p t e r 1: X s h o u l d c o r r e l a t e h i g h l y w i t h t h o s e t r a i t s which ( a c c o r d i n g t o p s y c h o l o g i c a l t h e o r y ) a r e r e l a t e d t o Y , and s h o u l d n o t c o r r e l a t e w i t h t h o s e t r a i t s which a r e n o t r e l a t e d t o Y. A t t h i s p l a c e i t i s worth t o mention t h a t a l a r g e

group o f p s y c h o m e t r i c i a n s ( e . g . Cowen 1963, Edwards 1957, Damarin and Messick 1965, Rorer 1965, E i s e n b e r g 1 9 4 1 , E i s e n b e r g and Wesnman 1941, Goldberg 1963) s p e n t a c o n s i d e r a b l e amount of e f f o r t on t e s t i n g t h e hypot h e s i s o f r e s p o n s e b i a s . G e n e r a l l y , i t was a r g u e d t h a t t e s t s measure n o t s o much t h e b a s i c d e t e r m i n a n t s of b e h a v i o u r ( t r a i t s ) , but s u c h v a r i a b l e s as r e s p o n s e s t y l e , e . g . p r e f e r e n c e towards some c a t e g o r i e s of a n s w e r s , o r t h e so-called s o c i a l d e s i r a b i l i t y .

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1 . 3 . Multidimensional s c a l i n g 1 . 3 . 1 . C o n c e p t u a l f o u n d a t i o n s . We s h a l l now mention b r i e f l y t h e group o f t e c h n i q u e s known under t h e name " m u l t i d i m e n s i o n a l s c a l i n g " , which a r e f r e e o f c e r t a i n o f t h e r e s t r i c t i v e assumptions o f f a c t o r a n a l y s i s . The b a s i c i d e a o f m u l t i d i m e n s i o n a l s c a l i n g may be p r e s e n t e d as f o l l o w s . We c o n s i d e r a c e r t a i n c l a s s o f obj e c t s ; t h e s e may be t e s t s , s u b j e c t s , or i n some c a s e s t h i s c l a s s may c o n s i s t o f b o t h s u b j e c t s and s t i m u l i ( e . g . o b j e c t s w i t h r e s p e c t to which one e x p r e s s e s p r e f e r e n ces). I n t h e s e t under c o n s i d e r a t i o n one o b s e r v e s c e r t a i n emp i r i c a l r e l a t i o n s . Depending on t h e c o n t e x t , t h e s e may be o r d e r i n g r e l a t i o n s o f " d i s t a n c e s " between t h e obj e c t s , t h e n o t i o n of " d i s t a n c e " o r "proximity" b e i n g a s s e s s e d s u b j e c t i v e l y , or measured by some means, e . g . c o r r e l a t i o n c o e f f i c i e n t ( i n c a s e o f t e s t s ) , and s o on. I n o t h e r s i t u a t i o n s , t h e o r d e r i n g s may be p r e f e r e n t i a l o r d e r i n g o f o b j e c t s e l i c i t e d from t h e s u b j e c t s . I n e a c h c a s e , t h e t e c h n i q u e of m u l t i d i m e n s i o n a l s c a l i n g c o n s i s t s of finding a geometrical representation of t h e s e t of o b j e c t s , i . e . l o c a t i n g these objects i n s p a c e w i t h p o s s i b l y l o w number of d i m e n s i o n s , i n s u c h a way t h a t m u t u a l l o c a t i o n s o f t h e p o i n t s r e p r e s e n t t h e r e l a t i o n s h i p s between o b j e c t s g i v e n i n t h e d a t a . To u s e a n example, two of t h e s i t u a t i o n s may be p r e s e n t -

ed i n an a b s t r a c t way as f o l l o w s . ( a ) Nonmetric a n a l y s i s o f s i m i l a r i t i e s . L e t us c o n s i -

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d e r t h e r e l a t i o n a l system o f t h e form

where A i s a c e r t a i n s e t , and S i s a r e l a t i o n which o r d e r s A X A . Let t h e symbol ( a i , a . ) S ( a k y a m ) mean t h a t J t h e p a i r ( a i , a . ) a p p e a r s i n t h e o r d e r i n g of S X S e a r l J i e r (not l a t e r ) than t h e p a i r (akyam)W . e assume t h a t from S we have f o r a l l ai yaj

* * *

The f i r s t t h r e e p o s t u l a t e s a s s e r t simply t h a t S i s a weak o r d e r i n g o f t h e s e t A x A , i . e . t h a t S i s r e f l e x i v e , t r a n s i t i v e and c o n n e c t e d . P o s t u l a t e ( d ) a s s e r t s t h a t p a i r s which d i f f e r o n l y by o r d e r of t h e i r e l e m e n t s do n o t p r e c e d e one a n o t h e r i n t h e o r d e r i n g . F i n a l l y , p o s t u l a t e ( e ) r e q u i r e s t h a t p a i r s w i t h i d e n t i c a l terms precede a l l o t h e r p a i r s . The above p o s t u l a t e s c o n c e r n f i n i t e s e t s A .

In case of i n f i n i t e s e t s , one imposes s t i l l more requirements ( s e e B e a l s , Krantz and Tversky 1968), c o n c e r n i n g continuity, etc.

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The problem l i e s i n c o n t r u c t i o n o f a s y s t e m

where Er i s t h e r - d i m e n s i o n a l s p a c e , d i s a c e r t a i n metric (not necessarily Euclidean) i n t h i s space, i . e .

a r e a l - v a l u e d f u n c t i o n d e f i n e d on E r x E' s u c h t h a t d(u,v,) 0 , w i t h d ( u , v ) = 0 i f f u = v , and a l s o s a t i s f y i n g d ( u , v ) = d ( v , u ) and t r i a n g l e i n e q u a l i t y d ( u , w ) ( d(u,v) t d(v,w).

>

-

Finally, f : A Er i s a n assignment of p o i n t s i n Er t o e l e m e n t s o f t h e s e t A , i . e . a mapping which t o ever y a 6 A a s s i g n s t h e p o i n t f ( a ) E Er s u c h t h a t

Moreover, i t i s r e q u i r e d t h a t r i s t h e s m a l l e s t p o s s i b l e number of' d i m e n s i o n s , i . e . t h a t t h e r e i s no system <En,d,f ) w i t h n

where Er and d have t h e same meaning as b e f o r e , w h i l e

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f: A

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B

--+

E’

i s a n a l l o c a t i o n o f b o t h o b j e c t s and

s u b j e c t s i n s p a c e Er , s u c h t h a t ( * * ) f o r e a c h b F- B and a i , a

J t

A the condition a R a

h o l d s i f and o n l y i f d [ f ( b ) , f ( a i ) ]

i b j

< d [ f ( b ) , f ( a5 ) I .

Moreover, as b e f o r e , one r e q u i r e s t h a t t h e d i m e n s i o n r be s m a l l e s t p o s s i b l e , s o t h a t t h e r e d o e s n o t e x i s t a system <En,d,f> s a t i s f y i n g ( * * ) w i t h n < r .

1 . 3 . 2 . The main d i r e c t i o n s o f d e v e l o p m e n t o f t h e t h e o r y o f m u l t i d i m e n s i o n a l s c a l i n g . The t h e o r e t i c a l r e s e a r c h i n m u l t i d i m e n s i o n a l s c a l i n g may be r o u g h l y d i v i d ed i n t o two g r o u p s . One o f them c o m p r i s e s t h e s o l u t i o n s , u s u a l l y accompanied w i t h c o m p u t a t i o n a l p r o g r a m s , which a l l o w us t o d e t e r m i n e , f o r g i v e n d a t a , t h e r e p r e s e n t a t i o n f . I n o t h e r words, here t h e problem l i e s i n f i n d i n g , f o r a given set of data, t h e r e p r e s e n t a t i o n f m e e t i n g u s u a l l y some a d d i t i o n a l c o n s t r a i n t s . F o r a r e v i e w and taxonomy o f v a r i o u s t y p e s o f p r o b l e m s h e r e , s e e Shepard 1972, 1974. F o r t h e main r e s u l t s i n t h e a r e a , see e . g . S h e p a r d 1 9 6 2 a , b . K r u s k a l 1 9 6 4 a , b ; Torg e r s o n 1952; L i n g o e s 1965; Guttman 1968; C a r r o l l and Chang 1970; B e n t l e r and Weeks 1978; Bloxom 1978; Weeks and B e n t l e r 1979, or Miyano a n d I n u k a i 1982. The s e c o n d d i r e c t i o n o f r e s e a r c h , much l e s s f r e q u e n t ,

c o n c e n t r a t e s on t h e p r o b l e m o f e x i s t e n c e o f t h e soluu t i o n s i n t h e two main g r o u p s ( a ) and ( b ) as d e s c r i b ed i n t h e p r e c e d i n g s e c t i o n . The main i s s u e i s t o f i n d empirically testable conditions f o r the existence

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of an a p p r o p r i a t e r e p r e s e n t a t i o n . I n c a s e o f problem ( a ) , of s c a l i n g s i m i l a r i t i e s , s u c h c r i t e r i a were formu l a t e d b y B e a l s , Krantz and Tversky (1968), who gave a n e m p i r i c a l l y t e s t a b l e axiom system f o r t h e e x i s t e n c e of g e n e r a l c l a s s o f m u l t i d i m e n s i o n a l r e p r e s e n t a t i o n s ( i n c a s e of i n f i n i t e s e t s of o b j e c t s ) . I n c a s e of t e c h n i q u e s of u n f o l d i n g s c a l e s ( b ) , t h e c r i t e r i a a r e based on t h e a n a l y s i s of a d m i s s i b l e permuta t i o n s o f o b j e c t s o f t h e s e t A . It a p p e a r s t h a t t h e number of a d m i s s i b l e p e r m u t a t i o n s i s c l o s e l y r e l a t e d t o t h e dimension of t h e s p a c e ( s e e e . g . H a y s and Benn e t t 1961). For i n s t a n c e , i n case of 4 o b j e c t s , there e x i s t s o n l y 7 p e r m u t a t i o n s f o r one-dimensional r e p r e s e n t a t i o n . I n consequence, i f t h e o b s e r v e d s e t of p e r m u t a t i o n s c o n t a i n s more t h a n s e v e n e l e m e n t s , l i n e a r r e p r e s e n t a t i o n ( r = 1) i s n o t p o s s i b l e . It i s a l s o w o r t h m e n t i o n i n g t h a t t h e e m p i r i c a l r e s u l t s

which s e r v e as a basis f o r m u l t i d i m e n s i o n a l s c a l i n g a r e o f t e n u n s t a b l e b e c a u s e of f l u c t u a t i o n o f a t t e n t i o n ( S h e p a r d 1 9 6 4 ) , and c o n t e x t u a l e f f e c t s ( T o r g e r s o n 1 9 6 5 ) . T h i s l a c k o f s t a b i l i t y i s n o t e a s i l y remedied w i t h i n t h e s t a t i c model of m u l t i d i m e n s i o n a l s c a l i n g . It seems t h a t i n o r d e r t o o b t a i n r e p r e s e n t a t i o n s which a r e p s y c h o l o g i c a l l y s e n s i b l e i t i s n e c e s s a r y t o impose c o n s t r a i n t s on t h e c l a s s of e x p e r i m e n t s . One s h o u l d a l s o mention some s i m u l a t i o n s t u d i e s ( e . g . Cohen and J o n e s 1973) f o r t e s t i n g . h y p o t h e s e s a b o u t s t a b i l i t y o f r e s u l t s under v a r i o u s a s s u m p t i o n s about f l u e t u a t i o n of a t t e n t i o n ( e . g . b a s i n g t h e e v a l u a t i o n s on some s e l e c t e d d i m e n s i o n s , which change from o c c a s i o n t o occasion).

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1 . 3 . 3 . M u l t i d i m e n s i o n a l s c a l i n g and f a c t o r a n a l y s i s . The a d v a n t a g e of t e c h n i q u e s of m u l t i d i m e n s i o n a l s c a l i n g o v e r f a c t o r a n a l y s i s c o n s i s t s n o t only on g r e a t e r a p p l i c a b i l i t y o f t h e f o r m e r . The main d i f f e r e n c e cons i s t s , f i r s t o f a l l , on t h e f a c t t h a t m u l t i d i m e n s i o n a l s c a l i n g i s based on weaker a s s u m p t i o n s , u s i n g o n l y t h e o r d e r i n f o r m a t i o n c o n t a i n e d i n t h e d a t a , and n o t t h e n u m e r i c a l v a l u e s (which are u s u a l l y s u b j e c t t o much h i g h e r e r r o r ) . Moreover, f a c t o r a n a l y s i s i s based on t h e a s s u m p t i o n t h a t t h e r e s u l t s f o r p a r t i c u l a r v a r i a b l e s may be r e p r e s e n t e d a d d i t i v e l y , as weighted sums o f f a c t o r s . T h i s a s s u m p t i o n h a s no c o n v i n c i n g s u p p o r t i n form of p s y c h o l o g i c a l h y p o t h e s e s . While i t i s r e l a t i v e l y e a s y to j u s t i f y , on t h e ground on some psychol o g i c a l t h e o r y , t h a t s u c h and s u c h f a c t o r s i n f l u e n c e the analysed reaction, the a s s e r t i o n t h a t t h i s influence i s e x p r e s s e d a d d i t i v e l y i s much h a r d e r t o j u s t i f y . A j u s t i f i c a t i o n would u s u a l l y r e q u i r e c h e c k i n g t h e axioms f o r c o n j o i n t measurement ( s e e C h a p t e r 4) , and such a procedure i s s u c c e s s f u l only i n rare i n s t a n c e s . A s opposed t o t h a t , t h e t e c h n i q u e s o f m u l t i d i m e n s i o n a l

s c a l i n g a r e based only on o r d e r r e l a t i o n s i n e m p i r i c a l data. A s r e g a r d s t h e i n t e r p r e t a t i o n of r e s u l t s , i . e . o f f a c -

tors i n c a s e of f a c t o r a n a l y s i s , or dimensions i n c a s e of m u l t i d i m e n s i o n a l s c a l i n g , t h e s i t u a t i o n i s s i m i l a r : i n both cases, it r e f e r s t o t h e psychological content of t h e o b j e c t s analysed ( t e s t s , items, e t c . ) . One c a n , however, h o l d t h e o p i n i o n t h a t t h e d i s a d v a n t a g e s of f a c t o r a n a l y s i s do n o t overweight t h e advantages: i f f a c t o r a n a l y s i s i s used as a s t a t i s t i c a l

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t e c h n i q u e w i t h a p p r o p r i a t e c h o i c e of d a t a , i t may s e r v e

as p o w e r f u l means of f o r m u l a t i n g h y p o t h e s e s and t h e o r e t i c a l l y f e r t i l e notions. Since t h i s i s , i n e f f e c t , one of t h e main o b j e c t i v e s o f any s c i e n t i f i c r e s e a r c h , any means of a t t a i n i n g i t s h o u l d b e r e g a r d e d as v a l u able.

I n f a c t , one c a n e n c o u n t e r two o p p o s i n g a t t i t u d e s t o wards f a c t o r a n a l y s i s among p s y c h o l o g i s t s : some are i n c l i n e d t o a t t a c h more s i g n i f i c a n c e t o c e r t a i n n o t i o n s b e c a u s e t h e y r e s u l t from a p p l c c a t i o n o f f a c t o r a n a l y sis, while others tend t o disregards these notions f o r p r e c i s e l y t h e same r e a s o n . Both a t t i t u d e s here seem somewhat e x a g g e r a t e d ; i t may be argued t h a t i f a n o t i o n s e r v e s w e l l t h e r o l e f o r which i t was c r e a t e d , i t matt e r s l i t t l e whether i t was o b t a i n e d from f a c t o r a n a l y sis o r not.

2 . AN EXAMPLE OF APPLICATION OF FACTOR ANALYSIS:

PERSONALITY THEORY OF CATTELL 2 . 1 . D e s c r i p t i v e approach t o p e r s o n a l i t y t h e o r y

The p e r s o n a l i t y t h e o r y p r e s e n t e d i n t h i s s e c t i o n i s p e r h a p s t h e b e s t example o f a n a p p l i c a t i o n o f f a c t o r a n a l y s i s , as i t a l l o w s t o t r a c e a l l c o g n i t i v e l i m i t a t i o n s o f t h i s method. The concept o f p e r s o n a l i t y i s o n e . o f t h e b a s i c c o n s t -

r u c t s of contemporary p s y c h o l o g y . I n most g e n e r a l t e r m s , i n a d e s c r i p t i v e a p p r o a c h , i t i s d e f i n e d t h r o u g h some o t h e r c o n s t r u c t s , c a l l e d p s y c h o l o g i c a l t r a i t s . The org a n i z a t i o n of t h e s e t o f t r a i t s , i . e . d e t e r m i n i n g t h e i r

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number, p s y c h o l o g i c a l c o n t e n t , and mutual r e l a t i o n s , l i e s a t t h e f o u n d a t i o n of e v e r y p e r s o n a l i t y t h e o r y which must a c c o u n t f o r t h e o r g a n i z a t i o n o f t h e s e t o f t r a i t s i n a s p e c i f i c w a y . Thus, a p s y c h o l o g i c a l t r a i t i s t r e a t e d as a h y p o t h e t i c a l component o f p e r s o n a l i t y , o b t a i n e d by a s s i g n i n g a name which g e n e r a l i z e s , and sometimes e x p l a i n s , t h e p s y c h o l o g i c a l c o n t e n t of a s e t of b e h a v i o u r s which have a tendency t o a p p e a r j o i n t l y . The f o l l o w i n g two premises l i e a t t h e f o u n d a t i o n o f

t h e d e s c r i p t i v e approach t o p e r s o n a l i t y : ( a ) o b s e r v a t i o n s i n d i c a t e a c e r t a i n s t a b i l i t y o f some e l e m e n t s of b e h a v i o u r i n s i m i l a r c l a s s e s of s i t u a t i o n s ; ( b ) r e g u l a r i t y and s i m i l a r i t y o f b e h a v i o u r s of d i f f e r -

e n t p e r s o n s a l l o w us t o assume t h a t t h e s e p e r s o n s have some common t r a i t s , and a l l o w u s t o d e s c r i b e t h e s e p e r s o n s i n t e r m s of t h e s e t r a i t s . I n o t h e r words, i n t h e b e h a v i o u r o f a s i n g l e p e r s o n , and a l s o i n t h e b e h a v i o u r o f a group o f p e r s o n s , one may d e t e c t r e g u l a r i t i e s o f s t a t i s t i c a l c h a r a c t e r , s u f f i c i e n t l y w e l l pronounced t o j u s t i f y , on t h e one hand, a s s i g n i n g a common name g e n e r a l i z i n g t h e c o n t e n t of t h e s e b e h a v i o u r s , and on t h e o t h e r hand -- s u f f i c i e n t t o allow f o r p r e d i c t i o n of f u t u r e behaviours. R e l a t i v e s t a b i l i t y of b e h a v i o u r s ( p r e m i s e ( a ) ) means h i g h p r o b a b i l i t y of a p p e a r i n g o f a g i v e n b e h a v i o u r or b e h a v i o u r s i n s i m i l a r s i t u a t i o n s , and i t s e r v e s as a basis f o r p r e d i c t i o n s o f f u t u r e b e h a v i o u r s , N a t u r a l l y , t h e s e p r e d i c t i o n s a r e n o t i n f a l l i b l e , and t h e s o u r c e of e r r o r s i s t w o f o l d . F i r s t l y , we d e a l here w i t h t h e

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r e g u l a r i t i e s of a s t a t i s t i c a l c h a r a c t e r , s o t h a t r a n domness i s i n h e r e n t t o t h e phenomenon. S e c o n d l y , t h e p r e d i c t i o n s a r e t y p i c a l l y based on o b s e r v a t i o n s o f a s e l e c t e d sample of b e h a v i o u r s , w h i l e t h e y concern o t h e r b e h a v i o u r s , n o t observed i n t h e sample. We d e a l h e r e , therefore, with a certain extrapolation. A s e x p l a i n e d i n C h a p t e r 1, t h e v a l u e o f a t r a i t o f a

g i v e n p e r s o n i s d e f i n e d as t h e e x p e c t a t i o n o f h i s prop e n s i t y d i s t r i b u t i o n , which c h a r a c t e r i z e s t h e s t a t i s t i c a l r e g u l a r i t i e s o f b e h a v i o u r s of t h i s p e r s o n w i t h r e s p e c t t o a g i v e n t e s t . A t any r a t e , t h e r e f o r e , t h e p r e d i c t i o n c a n be o n l y as good as t h e r e g r e s s i o n equat i o n s allow. The second p r e m i s e i s based on i n t u i t i o n s c o n c e r n i n g t h e n o t i o n o f a common t r a i t . I t i s namely assumed t h a t b o t h t h e s p e c i f i c and common t r a i t s d e v e l o p as a r e s u l t o f g e n e t i c and e n v i r o n m e n t a l f a c t o r s , w i t h common t r a i t s d e v e l o p i n g from c o n s t a n t e x p o s u r e t o s i m i l a r c o n f i g u r a t i o n s of e x t e r n a l s t i m u l i . These t r a i t s r e f l e c t s t a t i s t i c a l r e g u l a r i t i e s o f b e h a v i o u r i n groups o f s u b j e c t s , and t h e r e f o r e may be d e t e c t e d by s t u d y i n g groups i n s i milar s t i m u l u s s i t u a t i o n s . The p r e d i c t i o n s based on common t r a i t s a r e n e c e s s a r i l y l e s s p r e c i s e t h a n t h o s e based on b o t h s p e c i f i c and common t r a i t s . However, by r e s t r i c t i n g t h e c o n s i d e r a t i o n t o common t r a i t s o n l y one g e t s t h e p o s s i b i l i t y of comp a r i n g groups of s u b j e c t s . The b a s i c g o a l of a l l d e s c r i p t i v e t h e o r i e s o f p e r s o n a l i t y which r e l y on t h e n o t i o n o f common t r a i t i s : 1) t o d e t e r m i n e which, and how many t r a i t s a r e s u f f i c -

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i e n t t o describe personality; 2 ) t o c o n s t r u c t methods o f measurement of v a l u e s o f

t r a i t s f o r p a r t i c u l a r persons. The second c o n d i t i o n i s c o n n e c t e d w i t h t h e f a c t t h a t i n r e s t r i c t i n g t h e c o n s i d e r a t i o n s t o common t r a i t s o n l y , i n o r d e r t o maximize t h e v a l i d i t y o f p r e d i c t i o n s and a l s o t o be a b l e t o compare p e r s o n s , i t i s e s s e n t i a l t o have p o s s i b l y most e x a c t and s u b t l e q u a n t i t a t i v e methods of measurement.

2.2.

G e n e r a l c h a r a c t e r i s t i c s of C a t t e l l ' s a p p r o a c h

I n most g e n e r a l t e r m s , t h e a i m of C a t t e l l was t o make t h e d e s c r i p t i v e t h e o r y of p e r s o n a l i t y p r e c i s e and obj e c t i v e . P o s t u l a t i n g t h e e x i s t e n c e o f common t r a i t s , he d e c i d e d t o s e a r c h f o r a n e m p i r i c a l answer t o t h e q u e s t i o n about t h e number and c o n t e n t o f common t r a i t s . T h i s i s e q u i v a l e n t t o b u i l d i n g a taxonomy o f p e r s o n a l ity. It i s w o r t h t o s t r e s s h e r e t h a t t h e o r e t i c i a n s do n o t a g r e e as t o t h e number o f t r a i t s s u f f i c i e n t t o d e s c r i be p e r s o n a l i t y , n o r as t o t h e s t a b i l i t y and g e n e r a l i t y of t h e s e t r a i t s ( s e e e . g . F r e n c h 1953 o r C a t t e l l , Wagn e r and C a t t e l l 1970; t h e l a s t p a p e r c o n t a i n s a l s o C a t t e l l ' s remarks c o n c e r n i n g v a r i o u s p e r s o n a l i t y t h e o r i e s , e s p e c i a l l y t h a t o f Eysenck)

.

R e t u r n i n g t o C a t t e l l ' s o b j e c t i v e : i t was a c o n s t r u c t i o n o f a p r e c i s e , o b j e c t i v e , and i n d u c t i v e l y d e r i v e d from t h e d a t a taxonomy of p e r s o n a l i t y .

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D e s p i t e a p p a r e n t l y extreme e m p i r i c i s m , i . e . r e j e c t i o n o f any i n t r o d u c t o r y h y p o t h e s e s about t h e c o n t e n t and number of t r a i t s , i n o r d e r t o d e t e r m i n e t h e s e t of v a r i a b l e s t o b e measured, C a t t e l l had t o g i v e some v e r y g e n e r a l d e f i n i t i o n of p e r s o n a l i t y . He namely assumed t h a t p e r s o n a l i t y i s a l l , which a l l o w s t o p r e d i c t t h e b e h a v i o u r of a p e r s o n i n a g i v e n s i t u a t i o n ( C a t t e l l 1 9 5 0 ) . S i n c e any b e h a v i o u r may have p r o g n o s t i c v a l u e f o r some o t h e r b e h a v i o u r , t h e s e t of b e h a v i o u r s which d e t e r m i n e s p e r s o n a l i t y i s v e r y l a r g e -- it comprises s i m p l y a l l b e h a v i o u r s . To a p p l y some measurement proc e d u r e s , C a t t e l l had t o r e s t r i c t t h i s s e t . A s b r i e f l y mentioned i n t h e p r e c e d i n g s e c t i o n , i n t r y -

i n g t o c o v e r t h e s p h e r e of p e r s o n a l i t y i n a complete f a s h i o n , C a t t e l l s e l e c t e d t h r e e sources of information about t h e s u b j e c t s , which he c a l l e d t h e domains Q , L and T . The domain Q ( f r o m : Q u e s t i o n n a i r e d a t a ) was cons t r u c t e d on t h e assumption t h a t a l l e s s e n t i a l a s p e c t s of b e h a v i o u r a r e r e f l e c t e d i n t h e n a t u r a l l a n g u a g e . T h e Q domain, which s e r v e s as t h e main s o u r c e of i n f o r m a t i o n , was o b t a i n e d by c h o o s i n g t h e l i s t o f 1 7 1 adj e c t i v e s ( w i t h no synonyms) g i v i n g t h e complete s e t of terms r e l a t e d t o p e r s o n a l i t y . These a d j e c t i v e s were t h e n r e p r e s e n t e d as q u e s t i o n n a i r e items, l e a d i n g event u a l l y t o c o n s t r u c t i o n of C a t t e l l ' s p e r s o n a l i t y questionnaire. The domain L (from: L i f e d a t a ) c o n s i s t s of a s e r i e s o f e v a l u a t i o n s c a l e s , f o r a s s e s s i n g t h e behaviour of t h e s u b j e c t by j u d g e s . F i n a l l y , domain T ( f r o m : Test d a t a ) comprises a v e r y r i c h s e t of l a b o r a t o r y r t e e h n i q u e s , which c o u l d p o t e n t i a l l y s u p p l y i n f o r m a t i o n about t h e subject I s personality.

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For a d e s c r i p t i o n of t h e s e t e c h n i q u e s , s e e " O b j e c t i v e A n a l y t i c ( 0 - A ) P e r s o n a l i t y B a t t e r y " ( C a t t e l l 1957 , p p . 58-60, 225-275, and C a t t e l l and Warburton 1 9 6 7 ) . The o b j e c t i v e t e s t s a r e d e s i g n e d as s t a n d a r d i z e d s t i m u l u s s i t u a t i o n s . O b j e c t i v i t y of t h e s e t e s t s i s understood as o b j e c t i v i t y o f t h e d a t a , i . e . t h e i r independence f r D m t h e c o n s c i o u s or s u b c o n s c i o u s tendency t o t h e des i r e d r e s u l t s , and on t h e o t h e r hand -- as o b j e c t i v i t y of s c o r i n g t e c h n i q u e s , i . e . independence from t h e i n f l u e n c e of p e r s o n s who s c o r e , e v a l u a t e and a s s e s s t h e behaviour. Because o f a l a r g e number of v a r i a b l e s i n d a t a Q , L and T , i t was n e c e s s a r y t o r e s i g n from t h e c l a s s i c a l t w o - f a c t o r experiment aimed a t d e t e r m i n i n g t h e f u n c t i o n a l r e l a t i o n s between v a r i a b l e s , and u s e a n o t h e r model o f e x p e r i m e n t , namely m u l t i v a r i a t e d e s i g n , where one o b s e r v e s a l a r g e number of v a r i a b l e s f o r a l a r g e group of s u b j e c t s . Under t h i s d e s i g n one c a n n o t g e t informa t i o n about t h e f u n c t i o n a l r e l a t i o n s between v a r i a b l e s , and one g e t s o n l y t h e measures of l i n e a r r e l a t i o n s h i p s between v a r i a b l e s , a s e x p r e s s e d b y c o r r e l a t i o n coefficients. T h i s l o s s o f i n f o r m a t i o n may be q u i t e s e v e r e . F i r s t l y , i f t h e r e e x i s t s a c a u s a l r e l a t i o n between v a r i a b l e s ,

i t i s n o t r e c o v e r a b l e from t h e d a t a o f m u l t i v a r i a t e e x p e r i m e n t . A t b e s t , one may hope t o be a b l e t o deduce t h e e x i s t e n c e of s u c h a r e l a t i o n s h i p i n d i r e c t l y . Sec o n d l y , i f t h e r e l a t i o n s h i p s between t h e v a r i a b l e s a r e n o n l i n e a r , t h e y may be m i s r e p r e s e n t e d by c o r r e l a t i o n coefficients. Moreover, t h e use o f c o r r e l a t i o n c o e f f i c i e n t s as t h e

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b a s i s f o r i n f e r e n c e r e q u i r e s t h a t t h e y b e known e x a c t l y . T h i s , i n t u r n , i m p l i e s t h e u s e o f l a r g e and r e p r e -

s e n t a t i v e s a m p l e s , s e l e c t e d a t random from a g i v e n pop u l a t i o n . These d i s a d v a n t a g e s o f m u l t i v a r i a t e d e s i g n a r e compensated b y t h e p o s s i b i l i t y o f s i m u l t a n e o u s est i m a t i o n of a l a r g e number of p a r a m e t e r s . The r e s u l t s o f a m u l t i v a r i a t e a n a l y s i s a r e t y p i c a l l y p r e s e n t e d i n form o f a c o r r e l a t i o n m a t r i x . The e l e m e n t s of t h e m a t r i x may b e c o r r e l a t i o n s between v a r i a b l e s o b t a i n e d by u s l n g l a r g e number of t o o l s for a g i v e n group. T h i s i s t h e c o - c a l l e d t e c h n i q u e R , l e a d i n g t o common f a c t o r s i n t h e group. Another p o s s i b i l i t y i s t o t a k e c o r r e l a t i o n c o e f f i c i e n t s between measurements of t h e same p e r s o n a t d i f f e r e n t moments. T h i s i s t h e P technique, suggested by C a t t e l l , t o e x t r a c t t h e f a c t o r s s p e c i f i c f o r a given person.

One s h o u l d m e n t i o n , however, t h a t a p o s s i b l e o b j e c t i o n a g a i n s t h e use o f t e c h n i q u e P i s as f o l l o w s . I f t h i s t e c h n i q u e i s t o l e a d us t o e x t r a c t i o n of i n d i v i d u a l t r a i t s o f t h e s u b j e c t , i t i s n e c e s s a r y to assume t h a t t h e v a l u e s o f t h e t r a i t change f a s t enough, s o t h a t t h e c o v a r i a t i o n p a t t e r n s c o u l d a p p e a r as f a c t o r s . But i n s u c h a c a s e , a t l e a s t some o f t h e numbers p u b l i s h e d as r e l i a b i l i t i e s o f t h e t o o l s o f measurement o f t h e t r a i t s i n q u e s t i o n are n o t r e l i a b i l i t i e s , s i n c e what was t r e a t e d as e r r o r of measurement c o n t a i n s a compon e n t connected w i t h v a r i a b i l i t y of t h e t r a i t . A s a l r e a d y remarked, t h e main t e c h n i q u e used by C a t t e l l

i s f a c t o r a n a l y s i s : C a t t e l l t r e a t s t h e e x t r a c t e d fact o r s as a p p r o x i m a t i o n s t o p e r s o n a l i t y t r a i t s . A s a r e s u l t of f a c t o r a n a l y s i s , C a t t e l l d i s c o v e r e d a

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c e r t a i n number o f common f a c t o r s , which he c a l l e d s o u r c e t r a i t s . The 1 6 most u n i v e r s a l and most i m p o r t a n t among them were t h e n chosen, and a measurement t o o l w a s c o n s t r u c t e d f o r them (The S i x t e e n P e r s o n a l i t y F a c t o r Q u e s t i o n n a i r e , o r s h o r t l y 1 6 PF, p u b l i s h e d f i r s t i n

1 9 4 9 , and t h e n m o d i f i e d s e v e r a l times i n s u b s e q u e n t e d i t i o n s ) . According t o C a t t e l l , t h e s e 1 6 f a c t o r s cons t i t u t e t h e b a s i c taxonomical u n i t s of p e r s o n a l i t y . Thus, e a c h p e r s o n may be d e s c r i b e d and compared w i t h o t h e r s i n terms o f t h e 1 6 f a c t o r s , and t h e knowledge of t h e v a l u e s o f h i s f a c t o r s a l l o w s t h e p r e d i c t i o n and i n t e r p r e t a t i o n of h i s behaviour. I n c i d e n t a l l y , a f t e r 2 0 y e a r s of r e s e a r c h , t h e number of f a c t o r s was i n c r e a s e d t o 23, p l u s some p a t h o l o g i c a l d i m e n s i o n s ( s e e C a t t e l l , Eber and Tatsuoka 1 9 7 0 , pp. 258-259). P e r f o r m i n g f a c t o r a n a l y s i s on t h e m a t r i x o f i n t e r c o r r e l a t i o n s between t h e b a s i c 1 6 f a c t o r s , C a t t e l l d i s covered more g e n e r a l second-order f a c t o r s , t h e most i m p o r t a n t among them b e i n g t h e f a c t o r o f i n t r o v e r s i o n e x t r a v e r s i o n , and a n x i e t y - i n t e g r a t i o n . D e r i v a t i o n of f u n c t i o n a l u n i t y o f f a c t o r s of t h e second o r d e r , w i t h t h e p o s s i b i l i t y of s e p a r a t i n g them i n t o d e e p e r r e a c h i n g b a s i c f a c t o r s i s -- a c c o r d i n g t o C a t t e l l -- one o f h i s most i m p o r t a n t a c h i e v e m e n t s .

2 . 3 , The b a s i c h y p o t h e s i s o f C a t t e l l S t a r t i n g from t h e assumption t h a t f a c t o r a n a l y s i s l e a d s t o e x t r a c t i o n of t h e “ t r u e f a c t o r s t r u c t u r e ” of t h e phenomenon under s t u d y e x p l a i n i n g t h e o b s e r v e d r e l a t -

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i o n s h i p s between t h e v a r i a b l e s , C a t t e l l f o r m u l a t e d t h e f o l l o w i n g h y p o t h e s i s : s i n c e t h e v a r i a b l e s i n each of t h e t h r e e d i s t i n g u i s h e d domains ( Q , L and T ) c o n c e r n t h e same o b j e c t , namely p e r s o n a l i t y , t h e r e s u l t s o f f a c t o r a n a l y s i s performed on t h e s e t h r e e c l a s s e s o f v a r i a b l e s s h o u l d y i e l d t h e same f a c t o r s t r u c t u r e o f p e r s o n a l i t y . F o r i n s t a n c e ( s e e C a t t e l l 1957, p . 3 0 2 ) s c h i z o t h y m i a ought t o m a n i f e s t s i t s e l f i n o b s e r v a t i o n a l d a t a (L) as " k e e p i n g away from p e o p l e l ' a n d a c e r t a i n r i g i d i t y o f b e h a v i o u r ; i n s e l f - o b s e r v a t i o n d a t a (Q) -- as tendency t o l o n e l i n e s s and p r o f e s s i o n s which do n o t r e q u i r e c o n t a c t s w i t h p e o p l e ; f i n a l l y , i n o b j e c t i v e t e s t s (T)

--

as " p e r c e p t i o n of t h r e a t " , d i f f i c u l t y i n s e l f - d e s c r i p t i o n and l a c k o f s e l f - a s s u r a n c e i n new s i t u a t i o n s . I f t h e same f a c t o r s t r u c t u r e s c o u l d be o b t a i n e d from a l l t h r e e domains, i t would mean t h a t t h e assumed model o f p e r s o n a l i t y i s s u f f i c i e n t l y g e n e r a l and r e f l e c t s t h e " t r u e " n a t u r e o f t h e phenomenon. The l a t t e r -though viewed from d i f f e r e n t a n g l e s -- a p p e a r s i n v a r i a n t and l a r g e l y i n d e p e n d e n t from t h e i n i t i a l d a t a . I n t h i s s e n s e , o b s e r v a t i o n , s e l f - o b s e r v a t i o n and obj e c t i v e t e s t s would be simply t h r e e d i f f e r e n t w a y s o f c o l l e c t i n g i n f o r m a t i o n about t h e same p e r s o n a l i t y t r a i t s , u n d e r l y i n g t h e whole b e h a v i o u r . The v e r i f i c a t i o n o f t h i s h y p o t h e s i s was n o t s u c c e s s f u l ; i n t h e s e q u e l , t h e arguments w i l l b e p r e s e n t e d f o r t h e a s s e r t i o n t h a t t h i s p a r t i a l f a i l u r e of Cattell i s one o f t h e most i n t e r e s t i n g and v a l u a b l e c o n t r i b u t i o n s of h i s r e s e a r c h s t r a t e g y . 2.4. 2.4.1.

Some problems o f r e s e a r c h s t r a t e g y o f C a t t e l l P e r s o n a l i t y s t r u c t u r e from &-data. A s a l r e a d y

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mentioned, f a c t o r a n a l y s i s i s t h e main t o o l i n C a t t e l l ' s t h e o r y ; c o n s e q u e n t l y , a l l l i m i t a t i o n s and m e r i t s o f f a c t t o r analysis discussed i n t h e preceding sections proj e c t on t h e c o g n i t i v e c o n t e n t o f C a t t e l l ' s s y s t e m . A s a r e s u l t of f a c t o r a n a l y s e s o f e x p e r i m e n t a l d a t a ,

16 f a c t o r s were d i s c o v e r e d (as n o t e d , a t p r e s e n t t h e t h e o r y c o n t a i n s 2 3 f a c t o r s ; t h e d i s c u s s i o n below w i l l c o n c e r n , however, t h e c a s e o f 1 6 f a c t o r s measured by

16 PF). Most o f t h e f a c t o r s were d i s c o v e r e d i n 4 d a t a . According t o C a t t e l l , t h e f a c t o r s measured b y 1 6 PF sat i s f y t h e c r i t e r i o n of i n v a r i a n c e when t h e q u e s t i o n n a i -

r e i s a p p l i e d i n c r o s s - c u l t u r a l comparisons. Regarding v a l i d i t y , t h e 16 PF has been checked w i t h r e s p e c t t o c l i n i c a l , e d u c a t i o n a , i n d u s t r i a l and s o c i a l r e a l l i f e c r i t e r i a ( s e e e . g . C a t t e l l , Eber and T a t s u o k a 1 9 7 0 ) . T h i s s e t o f v a l i d a t i o n s s e r v e s n o t o n l y p r a c t i c a l purp o s e , b u t a l s o p r o v i d e s C a t t e l l w i t h an i n d i r e c t v e r i f i c a t i o n of h i s hypotheses r e g a r d i n g t h e i n t e r p r e t a t i o n o f f a c t o r s ( i t s h o u l d be remarked, t h a t C a t t e l l d i d n o t v e r i f y h y p o t h e s e s about f a c t o r s by c o n t r o l l e d b i v a r i a t e e x p e r i m e n t s , a s o r i g i n a l l y planned ( s e e C a t t e l l 1957), r e l y i n g m o s t l y on c h e c k i n g t h e v a l i d i t i e s of 1 6 PF. However, d e s p i t e a l l t h e above m e r i t s o f t h e 16 PF, nam e l y i n v a r i a n c e and v a l i d i t y , t h e y c a n n o t be t a k e n a s

a complete p r o o f of C a t t e l l ' s c l a i m (see C a t t e l l , Eber and T a t s u o k a 1 9 7 0 , p . 1 3 ) t h a t he d i s c o v e r e d t h e "uniquel y d e f i n a b l e " s o u r c e t r a i t s ; and even l e s s s o c a n t h e y p r o v i d e a n argument f o r c a u s a l i n t e r p r e t a t i o n o f f a c t o r s . Indeed, o b j e c t i o n s a g a i n s t f a c t o r a n a l y s i s , notably s u b j e c t i v e b i a s and r e l a t i v i t y o f t h e r e s u l t s t o t h e

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i n i t i a l s e t of v a r i a b l e s , as w e l l as u n c l e a r ( f r o m t h e measurement t h e o r y v i e w p o i n t ) s t a t u s of t h e s e v a r i a b l e s , a p p l y t o t h e C a t t e l l ' s 1 6 f a c t o r s . Even assuming t h a t t h e v a r i a b l e s chosen as t h e " p e r s o n a l i t y s p h e r e " i n f a c t exhaust a l l t h e r e l e v a n t a s p e c t s of p e r s o n a l i t y and have unambiguous s t a t u s , i t i s s t i l l c o n c e i v a b l e t h a t one c o u l d f i n d 1 6 , or some o t h e r number, of f a c t o r s ( w i t h i n t h e same p e r s o n a l i t y s p h e r e ) p r o v i d i n g e q u a l l y good, o r even b e t t e r , d e s c r i p t i o n o f p e r s o n a l i t y s a t i s f y i n g t h e c r i t e r i a of i n v a r i a n c e , v a l i d i t y and e x p l a n a t o r y i n t e r p r e t a t i o n . Thus, t h e u n i q u e n e s s c l a i m cannot be def'ended.

C a t t e l l a s s e r t s ( s e e C a t t e l l , Wagner and C a t t e l l 1 9 7 0 , or C a t t e l l , Eber and T a t s u o k a 19701, t h a t t h e p e r s o n a l i t y s t r u c t u r e as measured by 1 6 PF i s b e t t e r t h a n t h e a l t e r n a t i v e d e s c r i p t i o n s g i v e n b y , s a y , Eysenck o r G u i l f o r d and Zimmermann. But even i f t h e 16 PF s h o u l d g i v e a b e t t e r d e s c r i p t i o n of p e r s o n a l i t y t h a n any of t h e p r e s e n t l y e x i s t i n g q u e s t i o n n a i r e s , i t would s t i l l p r o v i d e no l o g i c a l argument f o r t h e u n i q u e n e s s of Cattell's structure. Now, i f t h e s t r u c t u r e i s n o t u n i q u e , one cannot a t t a c h t o i t any " a b s o l u t e " meaning. Thus, a l l t h a t i s shown by i n v a r i a n c e and v a l i d i t y i s t h a t i n d i f f e p e n t popul a t i o n s t h e r e e x i s t similar p a t t e r n s i n answering t h e same s e t o f i t e m s , and t h a t u s i n g t h i s f a c t , one can c o n s t r u c t a s e t of v a r i a b l e s s e r v i n g f a i r l y w e l l f o r p r e d i c t i n g b e h a v i o u r . T h i s shows t h e p r a g m a t i c v a l u e of t h e 1 6 P F . Moreover, one ought t o p o i n t o u t t h a t C a t t e l l ' s f a c t o r s p r o v i d e a r i c h and i n t e r e s t i n g desc r i p t i o n o f p e r s o n a l i t y , a r e c o n s i s t e n t w i t h psycholog i c a l knowledge and i n t u i t i o n , and t h e i r i n t e r p r e t a t i o n

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r e v e a l s g r e a t p s y c h o l o g i c a l i n s i g h t and e r u d i t i o n of t h e i r author .

2.4.2.

V e r i f i c a t i o n of t h e b a s i c h y p o t h e s i s of C a t t e l l .

A s a l r e a d y mentioned, C a t t e l l d i d n o t succeed i n a com-

plete

v e r i f i c a t i o n o f h i s fundamental h y p o t h e s i s o f

i d e n t i t y of s t r u c t u r e s i n t h e t h r e e s o u r c e s of d a t a : o b s e r v a t i o n s ( L ) , s e l f - o b s e r v a t i o n s (Q), and o b j e c t i v e tests (T). The f a i l u r e o f t h e v e r i f i c a t i o n o f t h i s h y p o t h e s i s may be r e g a r d e d as one of t h e v e r y v a l u a b l e and i n t e r e s t i n g r e s u l t s of C a t t e l l . I n t u i t i v e l y , C a t t e l l ' s h y p o t h e s i s may a t f i r s t g l a n c e a p p e a r q u i t e p l a u s i b l e : 5.t a s s e r t s t h a t i d e n t i c a l f a c t o r s t r u c t u r e s can be found i n a l l t h r e e c l a s s e s o f p e r s o n a l i t y v a r i a b l e s . I n o t h e r words, t h e phenomenon, though viewed from d i f f e r e n t s i d e s , remains e s s e n t i a l l y t h e same. I d e a l l y , t h e n , i t s h o u l d be p o s s i b l e t o c o n s t r u c t t h r e e t o o l s of measurement of t h e same common f a c t o r s t r u c t u r e i n d a t a Q , L and T ; a q u e s t i o n n a i r e , a method of s c o r i n g o f s u i t a b l e o b s e r v a t i o n a l d a t a , and a method of s c o r i n g t h e r e s u l t s o f o b j e c t i v e t e s t s . These t o o l s o f measurement, when a p p l i e d t o a g i v e n i n d i v i d u a l , s h o u l d p r o v i d e t h r e e i d e n t i c a l ( w i t h i n t h e e r r o r s of measurement) " p e r s o n a l i t y p r o f i l e s " . I f t h i s were t h e c a s e , t h e n whatever p r e d i c t i o n s c o u l d be made on t h e ground of a p e r s o n a l i t y p r o f i l e d e r i v e d from one domain of d a t a , t h e y c o u l d a l s o be made on t h e ground of prof i l e s d e r i v e d f r o m t h e o t h e r two domains;

moreover,

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as ( i d e a l l y ) t h e s e p r o f i l e s ought t o be i d e n t i c a l , i n

p r a c t i c a l s i t u a t i o n s one s h o u l d be a b l e t o p r e d i c t a p r o f i l e o f a g i v e n p e r s o n i n one domain, knowing h i s p r o f i l e i n a n o t h e r domain. I n s h o r t , t h e r e s u l t s of e m p i r i c a l v e r i f i c a t i o n o f C a t t e l l ' s h y p o t h e s i s were as f o l l o w s ( s e e C a t t e l l 1 9 5 7 , p p . 324-326; a l s o S c h e i e r and C a t t e l l 1958, and S c h a i e (1962) : (1) F o r e a c h c l a s s o f p e r s o n a l i t y v a r i a b l e s , one can

f i n d f a c t o r s which have no c o r r e s p o n d i n g f a c t o r s i n other classes; ( 2 ) sometimes t h e f a c t o r s from d i f f e r e n t c l a s s e s of

variables d i f f e r i n their generality level; for instanc e , some second o r d e r f a c t o r s i n o b j e c t i v e t e s t d a t a (T) c o r r e s p o n d t o t h e f i r s t o r d e r f a c t o r s i n t h e ques t i o n n a i r e d a t a (9). ( 3 ) regarding i d e n t i f i c a t i o n , Cattell claims t h a t i t i s p o s s i b l e t o i d e n t i f y 1 2 f a c t o r s from two o r t h r e e c l a s s e s o f p e r s o n a l i t y v a r i a b l e s ; as a r u l e , f a c t o r s from o b s e r v a t i o n a l and q u e s t i o n n a i r e d a t a were e a s i e r t o match one w i t h a n o t h e r t h a n f a c t o r s from t h e o b j e c t i v e t e s t data.

Moreover, t h e r e e x i s t e x t e n s i v e d a t a c o n c e r n i n g t h e v a l i d i t i e s of 16 PF w i t h r e s p e c t t o v a r i o u s s o c i a l r e a l - l i f e c r i t e r i a . While p r o v i d i n g arguments for t h e e x i s t e n c e o f r e l a t i o n s between domains L and Q , t h e y n e v e r t h e l e s s l a c k t h e f o r c e of a d i r e c t v e r i f i c a t i o n . Theoretically, t h e following conditions constitute t h e

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necessary requirements f o r a p o s i t i v e v e r i f i c a t i o n of t h e h y p o t h e s i s o f i n d i f f e r e n c e o f media:

(a) t h e f a c t o r analyses of t h e t h r e e sets of data should l e a d t o e x t r a c t i o n of f a c t o r s t r u c t u r e s which c o u l d be r o t a t e d t o p o s i t i o n s a l l o w i n g i d e n t i f i c a t i o n o f t h e same f a c t o r s i n a l l t h r e e d o m a i n s . The s c o r e s i n t h e same f a c t o r i n a l l t h r e e domains s h o u l d be h i g h l y c o r -

related. Given t h a t t h e above r e q u i r e m e n t i s s a t i s f i e d , i t i s a l s o necessary t h a t : ( b ) when t h e f a c t o r a n a l y s i s i s p e r f o r m e d on t h e j o i n t m a t r i x o f i n t e r c o r r e l a t i , o n s f o r two or t h r e e d o m a i n s , i t o u g h t t o b e p o s s i b l e to e x t r a c t a s t r u c t u r e w i t h f a c t o r s c o m p r i s i n g a l l v a r i a b l e s , which a p p e a r w i t h h i g h l o a d i n g s i n t h e same f a c t o r from a l l d o m a i n s . I n a d d i t i o n , i f t h e f a c t o r s t r u c t u r e s a r e to b e i d e n t i c a l , one s h o u l d a l s o have: ( c ) e v e r y f a c t o r i n a g i v e n domain s h o u l d m a t c h w i t h e x a c t l y o n e f a c t o r i n e a c h of t h e o t h e r two d o m a i n s . I n o t h e r w o r d s , l e t u s s u p p o s e t h a t by s e p a r a t e f a c t o r a n a l y s e s o f Q , L and T d a t a , one o b t a i n s f a c t o r s , p re -

...

sumed t o b e i d e n t i c a l , m e a s u r e d by v a r i a b l e s q 1 , q 2 , i n Q d a t a , b y v a r i a b l e s 11, 12, i n L d a t a , a n d by

...

...

variables tl, t2, i n T d a t a . It o u g h t t h e n to b e p o s s i b l e t o c o n s t r u c t l i n e a r combinations of v a r i a b l e s q l , q2,..., 11' 12'". and t l ' t 2 , w h i c h would b e

...,

h i g h l y c o r r e l a t e d . M o r e o v e r , when t h e j o i n t d a t a m a t r i x

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i t ought t o be p o s s i b l e t o do i t i n s u c h a way as t o i n c l u d e a l l v a r i a b l e s q 1 , q 2 , . ,11’12 ’ ,tl,t2, i n one f a c t o r .

i s factorized,

...

...

..

The e m p i r i c a l r e s u l t s do n o t s a t i s f y t h e s e r e q u i r e m e n t s : a n a l y s i n g t h e m a t r i c e s o f i n t e r c o r r e l a t i o n s between var i a b l e s i n d i f f e r e n t domains, one n o t e s t h a t t h e c o r r e l a t i o n s between v a r i a b l e s m e a s u r i n g t h e same f a c t o r a r e s u r p r i s i n g l y low ( h i g h e s t o f t h e o r d e r 0 . 2 0 , o f t e n as low as 0 . 0 2 ; s e e C a t t e l l 1957, Appendix). I n t e r e s t i n g l y enough, such v a r i a b l e s w i t h almost z e r o c o r r e l a t i o n appear o f t e n w i t h h i g h e s t l o a d i n g s i n a given f a c t o r , hence p r o v i d e i t s supposedly b e s t measure. I n view of t h e above, i t would a p p e a r t h a t t h e hypothes i s cannot be m a i n t a i n e d and has t o be m o d i f i e d : s i n c e t h e v a r i a b l e s i n presumably m a t c h i n g f a c t o r s have v e r y low i n t e r c o r r e l a t i o n s a c r o s s t h e domains, i t i s c e r t a i n l y i m p o s s i b l e f o r t h e f a c t o r s c o r e s on t h e same f a c t o r i n d i f f e r e n t domains t o be h i g h l y c o r r e l a t e d . T h i s make s t h e p r e d i c t i o n from one domain t o a n o t h e r p r a c t i c a l l y impossible. I n addition, t h e f a c t o r structures i n d i f f e r e n t domains a r e n o t i d e n t i c a l , as e a c h domain has some f a c t o r s s p e c i f i c f o r t h i s domain o n l y . Neverthel e s s , C a t t e l l ( s e e C a t t e l l , E b e r and T a t s u o k a 1970, p . 7 ) c l a i m s t h a t s o u r c e t r a i t s c a n be measured n o t o n l y by a q u e s t i o n n a i r e , b u t a l s o by o b j e c t i v e l a b o r a t o r y kind o f t e s t s . I n view of t h e above f i n d i n g s , c o n t r a r y t o C ’ a t t e l l ’ s c l a i m , i t seems t h a t t h e b e s t one c o u l d s a y i s t h a t t h e f a c t o r s t r u c t u r e s i n t h e t h r e e domains r e v e a l a c e r t a i n d e g r e e of s i m i l a r i t y .

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2.4.3.

C a t t e l l ' s e x p l a n a t i o n of d i f f i c u l t i e s i n v e r i f y -

ing h i s hypothesis.

I n s p i t e o f t h e above d e s c r i b e d

r e s u l t s o f e x p e r i m e n t s , C a t t e l l m a i n t a i n s t h a t h i s hypot h e s i s i s t r u e ; he e x p l a i n s t h e p a r t i a l f a i l u r e o f ver i f i c a t i o n by t h e p r e s e n c e o f d i s t u r b i n g f a c t o r s , and p r e d i c t s t h a t i n t h e f u t u r e one w i l l be a b l e t o c o n s t r u c t s p e c i a l e x p e r i m e n t s and t e c h n i q u e s which would e i t h e r e l i m i n a t e t h e d i s t u r b i n g f a c t o r s , or a l l o w one t o evaluate t h e i r influence, leading ultimately t o tools which would y i e l d t h e p o s i t i v e v e r i f i c a t i o n o f h i s hypothesis. Among f o u r s o u r c e s of d i s t u r b a n c e mentioned by C a t t e l l ( s e e C a t t e l l 1 9 6 1 ; C a t t e l l and Digman 1 9 6 4 ) t h e most i n t e r e s t i n g seem t o be i n s t r u m e n t a l f a c t o r s , and d i f f e r ences i n "density" of representation o f v a r i a b l e s ( t h e q u o t a t i o n mark i s used by C a t t e l l t o s t r e s s t h e i n t u i t i v e meaning of t h e term)- C a t t e l l c l a i m s t h a t i n s t rumental f a c t o r s play a c r u c i a l r o l e i n d i s t u r b i n g t h e i d e n t i f i c a t i o n of f a c t o r s from t h e Q , L and T d a t a . I n most g e n e r a l terms, t h e s e f a c t o r s a r e c o n n e c t e d w i t h t h e form o f t o o l , i . e . t h e y are s p e c i f i c f o r t h e t o o l

as a whole. I n t u i t i v e l y , due t o a f o r m a l s i m i l a r i t y o f v a r i a b l e s c o n s t i t u t i n g a g i v e n tool, t h e i n s t r u m e n t a l f a c t o r d i s t o r t s u n i f o r m l y a l l p e r s o n a l i t y f a c t o r s meas u r e d by t h i s t o o l . Thus, f o r e v e r y f o r m a l l y homogeneous c l a s s of p e r s o n a l i t y v a r i a b l e s , one c a n e x p e c t t h e e x i s t e n c e of i n s t r u m e n t a l f a c t o r s c h a r a c t e r i s t i c f o r t h i s c l a s s o f v a r i a b l e s . Such f a c t o r s can be d i s c o v e r e d by comparing t h e f a c t o r s t r u c t u r e s from f o r m a l l y d i f f e r e n t t o o l s w i t h t h e same p s y c h o l o g i c a l c o n t e n t . Regarding t h e " d e n s i t y " of r e p r e s e n t a t i o n , C a t t e l l maint a i n s t h a t t o e n s u r e t h e adequacy o f measurement o f t h e

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same t r a i t i n d i f f e r e n t c l a s s e s o f p e r s o n a l i t y v a r i a b l e s , t h e measures ought t o b e based on d i f f e r e n t numb e r s of v a r i a b l e s . I n d e e d , i t i s known t h a t q u e s t i o n n a i r e d a t a have a h i g h e r l e v e l o f g e n e r a l i t y ( t h e subj e c t answers how he u s u a l l y b e h a v e s , i n a b s t r a c t i o n from a s p e c i f i c s i t u a t i o n ) t h a n t h e L d a t a ( t h e o b s e r v e r r e c o r d s what he s e e s ) , and i n p a r t i c u l a r t h a n t h e T data ( t h e t e s t r e f l e c t s t h e behaviour a t a given moment )

.

C o n s e q u e n t l y , a l r e a d y a r e l a t i v e l y small number o f variables i n Q i s s u f f i c i e n t t o d e f i n e a given t r a i t , w h i l e i n o r d e r t o d e f i n e i t i n terms o f v a r i a b l e s i n L , or e s p e c i a l l y i n T , one needs t o use more v a r i a b l e s . I n C a t t e l l ’ s words, t h e v a r i a b l e s i n Q have h i g h e r “dens i t y ” t h a n t h e v a r i a b l e s i n L or T . According t o C a t t e l l t h i s e x p l a i n s why f o r matching f a c t o r s i t was n e c e s s a r y t o i d e n t i f y t h e second o r d e r f a c t o r s from t h e T d a t a w i t h t h e f i r s t o r d e r f a c t o r s from t h e Q d a t a : t o a t t a i n t h e l e v e l o f g e n e r a l i t y of t h e l a t t e r , i t was n e c e s s a r y t o go t o a h i g h e r l e v e l o f a b s t r a c t i o n w i t h t h e t e s t d a t a , s u b p l i e d by t h e second o r d e r f a c t o r a n a l y s i s .

2 . 5 . D i s c u s s i o n o f h y p o t h e s i s o f i n d i f f e r e n c e of media The above e x p l a n a t i o n o f C a t t e l l a p p e a r s q u i t e convinci n g and i s i n l i n e w i t h t h e contemporary t e n d e n c i e s towards s t u d y i n g t h e f a c t o r s d i s t u r b i n g t h e measurements by p s y c h o m e t r i c t o o l s . However, by f a r t h e most obvious e xpl anation of t h e p a r t i a l f a i l u r e i n v e r i f i c a t i o n o f t h e h y p o t h e s i s i s simply t h a t t h e h y p o t h e s i s

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may b e f a l s e , a t l e a s t r e g a r d i n g t h o s e o f i t s a s s e r t -

i o n s which f a i l e d t o be v e r i f i e d ( e . g . about t h e i d e n t i t y of s t r u c t u r e s ; t h i s removed from t h e h y p o t h e s i s , and i t i s t h a t t h e s t r u c t u r e s a r e not i d e n t i c a l ,

the assertion r e q u i r e m e n t was now r e c o g n i z e d s i n c e e a c h con-

t a i n s f a c t o r s s p e c i f i c t o t h e g i v e n domain, s u c h as f a c t o r s Q1

- Q4).

The p o i n t o f view t h a t t h e h y p o t h e s i s may be p a r t i a l l y f a l s e i s s t r o n g l y s u p p o r t e d when one looks a t t h e matr i c e s o f i n t e r c o r r e l a t i o n s f o r v a r i a b l e s from d i f f e r e n t domains, mentioned i n t h e p r e c e d i n g s e c t i o n . I t a p p e a r s t h a t t h e covariational p a t t e r n s a r e not s u f f i c i e n t l y pronounced t o y i e l d p s y c h o l o g i c a l l y e q u i v a l e n t and h i g h l y c o r r e l a t e d f a c t o r s i n a l l t h r e e media.

C o n s e q u e n t l y , i f f a c t o r a n a l y s i s i s t r e a t e d p u r e l y as a s t a t i s t i c a l t e c h n i q u e l e a d i n g t o e x t r a c t i o n of i n t e r pretable covariational p a t t e r n s , then the hypothesis s h o u l d be l a r g e l y m o d i f i e d ( b y a s s e r t i n g merely t h e e x i s t e n c e o f some i n t e r r e l a t i o n s between v a r i a b l e s def i n a b l e i n two o r t h r e e d o m a i n s ) . I f t h i s h y p o t h e s i s i s t o be r e t a i n e d , t h e n n o t o n l y i t i s n e c e s s a r y t o p u t t h e blame f o r i n c o m p l e t e n e s s o f i t s v e r i f i c a t i o n on d i s t u r b i n g f a c t o r s , b u t one has t o t r e a t f a c t o r a n a l y s i s ( a s d o e s C a t t e l l ) as a much more powerful t o o l t h a n i t a c t u a l l y i s , namely as a t o o l which can r e a c h i n t o t h e c a u s a l s t r u c t u r e of t h e phenomenon, r e g a r d l e s s o f how d i s t o r t e d t h e c o v a r i a t i o n a l p a t t e r n may b e . The c r i t i q u e o f C a t t e l l ’ s e x p l a n a t i o n d o e s n o t , o f cou r s e , e x t e n d t o t h e c o n c e p t s which he used f o r i t . I t seems t h a t b o t h t h e i n s t r u m e n t a l f a c t o r s and t h e “dens i t y ” ( i . e . l e v e l of g e n e r a l i t y o f v a r i a b l e s ) a r e of

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p r i m a r y i m p o r t a n c e i n t h e t h e o r y o f m u l t i v a r i a t e experiments. F i n a l l y , i n connection w i t h t h e v e r i f i c a t i o n o f t h e h y p o t h e s i s , i t i s w o r t h s t r e s s i n g t h a t one can d e s i g n r e l a t i v e l y s i m p l e and s t r a i g h t f o r w a r d t e c h n i q u e s f o r a s s e s s i n g t h e e x t e n t t o which t h e h y p o t h e s i s may be r e t a i n e d . The idea i s t o u s e i n f o r m a t i o n t h e o r y and eval u a t e t h e amount o f i n f o r m a t i o n c a r r i e d by v a r i a b l e s i n one domain about t h e v a r i a b l e s i n o t h e r domains. Whatever p r e d i c t i o n s , l i n e a r o r n o t , a r e t o be made a c r o s s t h e domains, t h e i r p r e c i s i o n i s bounded above by t h e a p p r o p r i a t e amounts o f i n f o r m a t i o n . The u s e o f i n f o r m a t i o n t h e o r y has a n a d d i t i o n a l a d v a n t a g e o v e r t h e c o r r e l a t i o n methods, namely t h a t no assumptions are needed a b o u t t h e n a t u r e o f s c a l e s of measurement ( a l l measurements are t r e a t e d as i f t h e y were on nominal scale). Knowing t h e " i n f o r m a t i o n a l : s t r u c t u r e " o f t h e t h r e e domains, one c o u l d p r o c e e d w i t h b u i l d i n g a p p r o p r i a t e f u n c t i o n s o f t h e v a r i a b l e s ( l i n e a r or n o t ) s u p p l y i n g as much i n f o r m a t i o n about one a n o t h e r as p o s s i b l e ; t h e s e f u n c t i o n s would p l a y t h e r o l e o f f a c t o r s , p r o v i d ed a n i n t e r p r e t a t i o n i s a s s i g n e d t o them.

2.6.

G e n e r a l remarks about C a t t e l l ' s a p p r o a c h

Looking on C a t t e l l ' s r e s e a r c h program and h i s r e s u l t s from a somewhat more g e n e r a l p o i n t o f view, one may i n t e r p r e t i t as an a t t e m p t t o c o n s t r u c t a language f o r a domain o f psychology, namely p e r s o n a l i t y t h e o r y . The a n a l o g i e s between C a t t e l l ' s s y s t e m and l i n g u i s t i c con-

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c e p t s o r t h e o r i e s may be drawn i n s e v e r a l d i f f e r e n t w a y s . F i r s t of a l l , l o o k i n g r a t h e r s u p e r f i c i a l l y o n l y on t h e methods a p p l i e d , one may n o t e t h a t C a t t e l l ' s program f o r s t u d y i n g t h e s t r u c t u r e o f p e r s o n a l i t y i s i n many r e s p e c t s a n a l o g o u s t o t h e program o f Osgood ( s e e Osgood,

S u c i and Tannenbaum 1 9 5 7 ) f o r s t u d y i n g t h e

s t r u c t u r e o f meaning. I n some s i m p l i f i c a t i o n , one may

s a y t h a t i n b o t h c a s e s t h e dimensions o f t h e phenomenon under i n v e s t i g a t i o n were o b t a i n e d by i n t e r p r e t i n g s u i t able c o v a r i a t i o n a l p a t t e r n s i n s e l e c t e d s e t s of a d j e c t i v e s . The c o v a r i a t i o n s were o b t a i n e d by e v a l u a t i n g , i n t e r m s of t h e c o n s i d e r e d a d j e c t i v e s , t h e samples o f obj e c t s s e l e c t e d from a p p r o p r i a t e p o p u l a t i o n s ( t h e obj e c t s b e i n g i n d i v i d u a l s i n C a t t e l l ' s a p p r o a c h , or nouns i n O s g o o d ' s ) . N a t u r a l l y , t h e s e t s o f a d j e c t i v e s were d i f f e r e n t ; a l s o d i f f e r e n t were t h e d e t a i l s o f c o n s t r u c t i o n o f t h e t e c h n i q u e s o f e v a l u a t i n g an o b j e c t w i t h resp e c t t o a given a d j e c t i v e . A n a l y s i n g t h e s t r u c t u r a l p r o p e r t i e s of t e n d e n c i e s o f some a d j e c t i v e s t o a p p e a r t o g e t h e r when t h e y a r e used t o d e s c r i b e i n d i v i d u a l s , one a r r i v e s ( b y f a c t o r a n a l y s i s ) a t t h e s t r u c t u r e of p e r s o n a l i t y . On t h e o t h e r hand, a n a l y s i n g t h e s t r u c t u r a l p r o p e r t i e s o f t h e t e n d e n c i e s of some a d j e c t i v e s t o a p p e a r t o g e t h e r when t h e y are used t o d e s c r i b e nouns (more p r e c i s e l y , p r o p e r t i e s o f o b j e c t s d e s i g n a t e d by t h e s e n o u n s ) , one a r r i v e s , a l s o by f a c t o r a n a l y s i s , a t t h e s t r u c t u r e o f meaning. The g e o m e t r i c d i s t a n c e between two words i n t h e o b t a i n ed s p a c e i s t h e n r e l a t e d t o t h e a c c e p t a b i l i t y o f a comb i n a t i o n o f t h e s e two words. T h i s analogy between C a t t e l l ' s and Osgood's r e s e a r c h programs c a n n o t , however, be pursued t o o far. C a t t e l l

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c o n t i n u e s t h e r e s e a r c h towards checking t h e i n v a r i a n c e o f h i s s t r u c t u r e and i n v e s t i g a t i n g t h e v a l i d i t i e s w i t h

r e s p e c t t o v a r i o u s c r i t e r i a on t h e one hand, and on t h e o t h e r , t oward s t e st i n g t h e hypo t he s i s o f i n d i f f e r e n c e o f media. Osgoodls r e s e a r c h seems t o be d e v e l o p i n g i n a d i r e c t i o n away from f a c t o r a n a l y t i c models (which p r o v i d e s o l u t i o n s n o t v e r y a t t r a c t i v e from t h e l i n g u i s t i c v i e w p o i n t ) t o w a r d s h i e r a r c h i c a l taxonomies under which e a c h noun i s c a t e g o r i z e d w i t h r e s p e c t t o s u c c e s s i v e dependent c a t e g o r i e s . T h i s g i v e s a b e t t e r a c c e s s t o s p e c i f i c a t i o n of t h e l i n g u i s t i c r u l e s f o r s e l e c t i o n ( t h e t e c h n i q u e s i n v o l v e c a t e g o r i z a t i o n o f nouns by v e r b s , and n o t by a d j e c t i v e s ; s e e Osgood 1970, and a l s o Noordman and L e v e l t 1970). Now, l o o k i n g n o t o n l y f o r a n a l o g i e s i n methods, b u t a l s o a t t h e o b t a i n e d r e s u l t s , C a t t e l l ' s a p p r o a c h may be i n t e r p r e t e d as a n a t t e m p t t o c o n s t r u c t a c e r t a i n l a n g u a g e , o r more p r e c i s e l y , a d i c t i o n a r y , whose t e r m s ( e n t r i e s ) a r e o b t a i n e d a c c o r d i n g t o t h e r u l e s determined by t h e c h o i c e o f program o f f a c t o r a n a l y s i s . T h e scheme o f a d e f i n i t i o n o f a term i s , r o u g h l y s p e a k i n g , g i v e n by a p r o p o s i t i o n a l f u n c t i o n o f t h e form " f a c t o r

x i s t h e c o n j u n c t i o n of a , b , c ,... ' I . More p r e c i s e l y , "N has t r a i t x" means "N i s a , and N i s b, and . . . ' I .

...

Here a , b , c , need n o t be words i n t h e n a t u r a l langua g e ; t h e y may be t a k e n as c e r t a i n b e h a v i o u r s ( t r e a t i n g a s e l f - e v a l u a t i o n on a q u e s t i o n n a i r e item a l s o a s a

...

form of b e h a v i o u r ) . I f i n p l a c e of a , b , c , one subs t i t u t e s t h e w o r d - v a r i a b l e s which s e r v e d as t h e b a s i s

for f a c t o r a n a l y s i s ,

one o b t a i n s a d e f i n i t i o n o f t h e

...

one subf i r s t order factor; i f i n place of a,b,c, s t i t u t e s t h e f i r s t o r d e r f a c t o r s , one o b t a i n s t h e de-

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f i n i t i o n o f a second o r d e r f a c t o r , e t c . A s a r e s u l t , t h e t e r m s o f t h e language s o c o n s t r u c t e d form a c e r t a i n h i e r a r c h y , i n which t o t e r m s on one l e v e l t h e r e c o r r e s pond s e t s o f terms on lower l e v e l s . T e c h n i c a l l y , a l l d e f i n i t i o n s of t e r m s i n t h i s language a r e unambiguous ( a t l e a s t to t h e e x t e n t o f Unambiguity of t h e i n i t i a l terms a , b , c , t a k e n as a f o u n d a t i o n of t h e c o n s t r u c t i o n of t h e language).

...

However, t o r e l a t e terms o f t h i s l a n g u a g e w i t h t h e p r e s e n t p s y c h o l o g i c a l knowledge and i n t u i t i o n , C a t t e l l o f t e n a s s i g n e d t o f a c t o r s ( i . e . s e t s of e l e m e n t s a , b , c , . . . ) names o f t r a i t s e x i s t i n g i n psychology. T h i s may o c c a s i o n a l l y l e a d t o a c e r t a i n a m b i g u i t y , due t o t h e p o s s i b l e d i f f e r e n c e s i n o p i n i o n as t o t h e c o r r e c t n e s s o f s p e c i f i c c h o i c e s . Thus, w h i l e a t r a n s l a t i o n o f t e r m s o b t a i n e d by C a t t e l l i n t o t h e e x i s t i n g l a n g u a g e of psychology i n c r e a s e s t h e M r e a d Z & b i l i t y o" f C a t t e l l I

s

t e r m s , a t t h e same t i m e it t a k e s o u t some o f t h e p r e c i s i o n o f u s a g e , i n view o f t h e s u r p l u s meaning u s u a l l y connected w i t h t h e s e t e r m s . The d e f i n i t i o n s r e m a i n , o f c o u r s e , p r e c i s e and unambiguous; i t i s o n l y t h a t t h e t e r m s which are b e i n g d e f i n e d have a l r e a d y - e s t a b l i s h e d i n t u i t i v e meanings, a l t h o u g h n o t n e c e s s a r i l y p r e c i s e o n e s , or i d e n t i c a l f o r d i f f e r e n t p s y c h o l o g i s t s . To use an example, when C a t t e l l d e f i n e s t e r m s s u c h a s , s a y , P r e m s i a , t h e s e a r e new t e r m s , and everyone who u s e s them must do s o i n t h e s e n s e as d e v i s e d by C a t t e l l . When, however, C a t t e l l u s e s for h i s f a c t o r s t e r m s l i k e " a n x i e t y " o r "superego" , t h e n -- i n d e p e n d e n t l y o f t h e s u g g e s t e d d e f i n i t i o n s -- t h e p s y c h o l o g i s t s u s i n g Catt e l l ' s t e s t s would o f t e n r e l y on t h e i r own i n t u i t i o n s

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concerning these concepts. Nevertheless, t h e very e x i s t ence o f r i g i d d e f i n i t i o n s o f c o n c e p t s r e d u c e s t h e l o o s e n e s s o f t h e i r usage, t h u s l e a d i n g towards g r e a t e r p r e c i s i o n of p s y c h o l o g i c a l language. Now, c r e a t i n g a s t r u c t u r e d d i c t i o n a r y o f p s y c h o l o g i c a l t e r m s , w h e t h e r new or p r e v i o u s l y known, i . e . a c e r t a i n system o f i n t e r r e l a t e d c o n c e p t s , amounts t o e s t a b l i s h i n g a s e m a n t i c s o f t h e language o f p e r s o n a l i t y . I f one a g r e e s t o t r e a t t h e u n d e r l y i n g v a r i a b l e s i n t h e domains Q , L and T as s e m a n t i c components ( i n t h e s e n s e of B i e r w i s c h 1969; s e e a l s o Katz and Fodor 1963), t h e n C a t t e l l ' s d i c t i o n a r y of p e r s o n a l i t y p r o v i d e s each l e x i c a l e n t r y t o g e t h e r w i t h i t s semantic decomposition. These e n t r i e s form a h i e r a r c h i c a l s t r u c t u r e , i n t h e s e n se t h a t e n t r i e s on t h e lower l e v e l o f a b s t r a c t i o n form p a r t s o f e n t r i e s on h i g h e r l e v e l s of a b s t r a c t i o n . Thus, " p e r s o n a l i t y " i s t h e e n t r y whose meaning i s d e f i n a b l e i n t e r m s o f h i g h e s t o r d e r t r a i t s ( s e c o n d , o r even t h i r d order f a c t o r s ) ; these i n t u r n a r e definable i n terms of lower o r d e r t r a i t s ( f i r s t o r d e r f a c t o r s ) , and s o o n , down t o t h e l e v e l o f b e h a v i o u r a l v a r i a b l e s , t h a t i s , t h e s e m a n t i c components t o g e t h e r w i t h t h e r e l a t i o n s between them g i v e n by t h e m a t r i x o f i n t e r c o r r e l a t i o n s . The fundamental t y p e of p r o p o s i t i o n s which can be formu l a t e d i n t e r m s o f t h e language c r e a t e d o r f o r m a l i z e d by C a t t e l l , a r e t h o s e which a s s e r t t h e d e g r e e t o which f a c t o r x may be a s c r i b e d t o a given' p e r s o n N . The i n t e n s i o n a l i t y i s d e t e r m i n e d by a p p r o p r i a t e measurement procedures, thus providing t h e p o s s i b i l i t y of expressi n g i t much more a c c u r a t e l y t h a n w i t h t h e use o f i n t e n s i o n a l f u n c t o r s such as "very", " h i g h l y " , e t c . i n t h e

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n a t u r a l l a n g u a g e . The b a s i c t y p e s o f p r o p o s i t i o n s have t h e form "N h a s t r a i t x i n d e g r e e d " ; a c c o r d i n g t o C a t t e l l , t h e c o n j u n c t i o n of 16 such p r o p o s i t i o n s , f o r t h e

16 p e r s o n a l i t y t r a i t s ( t h e s o - c a l l e d p e r s o n a l i t y p r o f i l e ) c o n t a i n s most o f t h e p s y c h o l o g i c a l l y r e l e v a n t i n f o r m a t i o n about t h e s u b j e c t ; t h i s , i n t u r n , a l l o w s one t o make p r e d i c t i o n s about t h e f u t u r e b e h a v i o u r o f t h e subject. I n t h e c o n s i d e r e d l i n g u i s t i c i n t e r p r e t a t i o n , t h e fundamental h y p o t h e s i s o f i n d i f f e r e n c e of media a s s e r t s t h a t i f one t a k e s as i n i t i a l v a r i a b l e s a , b , c , the variabl e s from t h e domains Q , L and T , t h e o b t a i n e d language

...

w i l l c o n t a i n synonyms. More p r e c i s e l y , t h i s h y p o t h e s i s

a s s e r t s t h a t for e x p r e s s i o n s o b t a i n e d by s u b s t i t u t i n g i n p l a c e of a , b , c t h e v a r i a b l e s from t h e c l a s s Q , t h e r e e x i s t e x p r e s s i o n s w i t h t h e same meaning o b t a i n e d by s u b s t i t u t i n g t h e v a r i a b l e s from c l a s s e s L o r T . Equiv a l e n t l y , one may s a y t h a t t h e e x p r e s s i o n s whose s e -

,...

m a n t i c components i n v o l v e t h e v a r i a b l e s from one domain only may be p a r a p h r a s e d t o e x p r e s s i o n s i n v o l v i n g t h e components i n o t h e r domains. A s shown e m p i r i c a l l y , however, n o t a l l terms i n s u c h

a language have synonyms: t h e r e e x i s t e x p r e s s i o n s char a c t e r i s t i c f o r one domain of v a r i a b l e s o n l y , which have no c o r r e s p o n d i n g e x p r e s s i o n s from o t h e r domains. I n c o n n e c t i o n w i t h t h e above i n t e r p r e t a t i o n o f C a t t e l l ' s r e s u l t s as a l a n g u a g e o f p e r s o n a l i t y , one c a n pose a q u e s t i o n t o which e x t e n t t h e m u l t i v a r i a t e methodology and e s p e c i a l l y f a c t o r a n a l y s i s , c o n t r i b u t e d t o c r e a t i n g a u n i v e r s a l b e h a v i o u r a l language o f d e s c r i p t i v e psychol o g y . The b a s i c r e s u l t s a r e i n t h e domains s u c h as p e r -

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s o n a l i t y and m o t i v a t i o n ( C a t t e l l 1957; C a t t e l l and Schei e r 1961; G u i l f o r d 1959; Eysenck 1953) and i n t e l l i g e n ce ( G u i l f o r d 1956, 1 9 7 1 ) . There a r e a l s o a p p l i c a t i o n s t o a p t i t u d e s , a t t i t u d e s , s o c i a l behaviour, learning, p e r c e p t i o n , motor s k i l l , e t c . C e r t a i n a t t e m p t s o f a s y n t h e s i s were g i v e n by French (1951, 1953) and Royce ( 1 9 7 0 ) . A s y n t h e s i s from a m e t a s c i e n t i f i c p o i n t o f view ( o f t h e o r i e s of m o t i v a t i o n , n o t o n l y b a s e d on f a c t o r a n a l y s i s ) was g i v e n b y Madsen ( 1 9 6 8 ) . Here one s h o u l d a l s o mention t h e program o f Royce ( 1 9 6 5 ) who wanted t o c o n s t r u c t a u n i f y i n g m u l t i - f a c t o r t h e o r y o f b e h a v i o u r a l v a r i a b i l i t y , comprising not only s o c i a l and c u l t u r a l , b u t a l s o g e n e t i c and b i o l o g i c a l d e t e r m i n a n t s . H e wanted t o l i n k t h e m u l t i p l e f a c t o r t h e o r y o f p e r s o n a l i t y w i t h t h e m u l t i p l e f a c t o r theory of g e n e t i c s . When one a t t e m p t s t o s y n t h e s i s e t h e r e s u l t s o f v a r i o u s a u t h o r s , or ( a s i n t h e c a s e of Royce 1965) when one s t a r t s i n a s e n s e a f r e s h w i t h a l a r g e number o f i n i t i a l v a r i a b l e s c o v e r i n g a wide domain, t h e b a s i c d i f f i c u l t y l i e s i n t h e problem o f i d e n t i f i c a t i o n o f f a c t o r s w i t h i n one domain or a c r o s s d i f f e r e n t subdomains. A s shown on t h e example o f C a t t e l l ' s h y p o t h e s i s o f i n d i f f e r e n c e o f media, s u c h a m a t c h i n g , b e i n g i n f a c t e q u i v a l e n t t o c o n s t r u c t i n g a s y s t e m o u t of more or l e s s l o o s e l y connected concepts, i s not easy t o achieve. By means o f f a c t o r a n a l y s i s , one o b t a i n s a h i e r a r c h i c a l

system, t h e h i e r a r c h y b e i n g induced by t h e r e l a t i o n of p a r t i a l o r d e r i n g ( " i s a n element o f " ) . The matching c o n s i s t s o f l o c a t i n g nodes w i t h i d e n t i c a l meanings i n t h e o b t a i n e d graph. N o w , a l l nodes of t h i s h i e r a r c h i c a l t r e e , e x c e p t t h o s e

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a t t h e l o w e s t l e v e l of o b s e r v a t i o n a l v a r i a b l e s , r e p r e s e n t e i t h e r i n t e r v e n i n g v a r i a b l e s ( i f c l o s e enough t o t h e l e v e l of o b s e r v a t i o n s ) , or c o n s t r u c t s (Royce 1 9 6 7 ) . The p a r t o f t h e t r e e below a g i v e n node r e p r e s e n t i n g

a c o n s t r u c t provides a decomposition of t h i s c o n s t r u c t i n t h e domain of v a r i a b l e s l o c a t e d a t t h e l o w e s t l e v e l o f t h i s s u b t r e e ( s e r v i n g , i n f a c t , as a p o s s i b l e d e f i n i t i o n o f t h i s c o n s t r u c t ) . A s e a c h c o n s t r u c t becomes t h u s r e p r e s e n t e d i n form o f one o r more t r e e s ( i f d i f f e r e n t domains a r e u s e d ) one can e a s i l y o b t a i n a taxonomy of c o n s t r u c t s ( o r , more p r e c i s e l y , o f t h e i r r e p r e s e n t a t i o n s ) i n t e r m s o f c e r t a i n p a r a m e t e r s ( d i m e n s i o n s ) . Thus, f o r example, one can t h i n k o f "depth" o f a c o n s t r u c t , d e f i n e d as t h e number o f l e v e l s s e p a r a t i n g i t from t h e l e v e l of observable v a r i a b l e s ( t h i s corresponds t o t h e o r d e r o f f a c t o r s , t h a t i s , t o t h e i r l e v e l of g e n e r a l i i t y ) . Another p o s s i b l e parameter may be t h e "width" o f t h e c o n s t r u c t , d e f i n e d as t h e number o f nodes a t t h e f i r s t l e v e l of t h e t r e e , c o u n t i n g from t h e t o p ( t h i s may p e r h a p s be connected w i t h t h e scope o f meaning o f t h e c o n s t r u c t ) . F i n a l l y , s t i l l a n o t h e r p a r a m e t e r may be t h e number o f nodes a t t h e l o w e s t l e v e l . Depending on t h e r e l a t i v e l e v e l s of g e n e r a l i t y o f t h e u n d e r l y i n g v a r i a b l e s used f o r t h e r e p r e s e n t a t i o n of t h e same cons t r u c t i n d i f f e r e n t domains, t h e s e numbers may v a r y ; i t e x p l a i n s why t h e same f a c t o r s i n d i f f e r e n t domains i n C a t t e l l ' s r e s u l t s had b o t h d i f f e r e n t " d e p t h s " and " d e n s i t i e s " , t h e l a t t e r c o r r e s p o n d i n g t o t h e numbers o f nodes a t t h e l o w e s t l e v e l . C a t t e l l ' s e x p e r i e n c e shows t h a t t h e above dimensions

of c o n s t r u c t s p l a y an importabt r o l e i n matching f a c t o r s a c r o s s t h e domains; t h u s , t h e y can be o f some method-

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o l o g i c a l i n t e r e s t f o r t h e m u l t i v a r i a t e approach t o psychology. A s a l r e a d y p o i n t e d out, t h e main g o a l i n t h e d i s c u s s e d

" f a c t o r i a l g e n e r a t i o n " of a language o f a g i v e n domain

i s to c h a r a c t e r i z e and a n a l y s e i t s s e m a n t i c s . Here t h e t e r m "language" i s i n t e r p r e t e d l i t e r a l l y as a c o l l e c t i o n o f words whose i n t e r r e l a t e d meanings a r e e x p r e s s i b l e i n b e h a v i o u r a l t e r m s . One c a n , however, u s e t h e l i n g u i s t i c techniques d i r e c t l y f o r t h e analysis of t h e b e h a v i o u r i t s e l f ( s e e c h a p t e r on language o f a c t i o n s ) , by t r e a t i n g t h e b e h a v i o u r as a c e r t a i n "langua g e " , and a p p l y i n g t h e methods o f m a t h e m a t i c a l l i n g u i s t i c s to a n a l y s e i t s s y n t a x and s e m a n t i c s . A s d e f i n e d f o r m a l l y , language i s a s e t of f i n i t e s e -

quences ( s t r i n g s ) formed out of e l e m e n t s o f a c e r t a i n " a l p h a b e t " , o r "vocabulary". These e l e m e n t s may be l e t t e r s , words, o r a b s t r a c t symbols ( a s i n t h e c a s e o f l a n g u a g e s of some l o g i c a l s y s t e m s , o r computer l a n g u a g e s ) . The s t r i n g s i n t h e language a r e c a l l e d "words", " s e n t e n c e s " , " w e l l formed f o r m u l a s " , "programs", e t c . , d e p e n d i n g on t h e chosen i n t e r p r e t a t i o n . Roughly, by a n a l y s i s o f a s y n t a x o f a g i v e n l a n g u a g e ( s e e , f o r i n s t a n c e , Chomsky 1963, o r Ginsburg 1966), one means t h e c o n s t r u c t i o n o f a f i n i t e s e t o f w e l l def i n e d r u l e s of g e n e r a t i o n , s u c h t h a t by a p p l y i n g t h e s e r u l e s one o b t a i n s a l l s t r i n g s of t h i s l a n g u a g e and o n l y t h o s e s t r i n g s . Languages may t h e n be c l a s s i f i e d a c c o r d i n g to t h e c o m p l e x i t i e s o f grammars which gener a t e them. A l t e r n a t i v e l y , one may c l a s s i f y t h e langua g e s a c c o r d i n g t o t h e d e g r e e o f c o m p l e x i t y o f automata which are c a p a b l e o f d i s t i n g u i s h i n g s t r i n g s i n t h e

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language from s t r i n g s o u t s i d e of i t ( t h e s e two approac h e s a r e , i n a s e n s e , d u a l one t o a n o t h e r ) . The a n a l y s i s o f s e m a n t i c s c o n s i s t s o f s t u d y i n g t h e p r o p e r t i e s e x p r e s s i b l e i n t e r m s of t h e r e l a t i o n s conn e c t i n g s t r i n g s i n t h e language w i t h e l e m e n t s o f t h e s p a c e o f meanings ( s e e Marcus 1 9 6 8 ) . I n m a t h e m a t i c a l l i n g u i s t i c s t h e r e i s no r e s t r i c t i o n on t h e i n t e r p r e t a t i o n of t h e e l e m e n t s o f t h e a l p h a b e t ; c o n s e q u e n t l y , these e l e m e n t s may a l s o be i n t e r p r e t e d as some u n i t s o f b e h a v i o u r . The s t r i n g s would t h e n c o r r e s p o n d t o s e q u e n c e s of s u c h u n i t s , performed one a f t e r a n o t h e r . T h i s i d e a , c a r r i e d o u t i n Nowakowska 1973, l e a d s t o c r e a t i o n o f t h e t h e o r y of l a n g u a g e s of a c t i o n s . It i s w o r t h w h i l e t o mention here t h a t i n t h e f i n a l p a r t of t h e i r p a p e r , M i l l e r and Chomsky (1963), w r o t e : "It i s p r o b a b l y no a c c i d e n t t h a t a t h e o r y of grammatical s t r u c t u r e s can b e s o r e a d i l y and n a t u r a l l y g e n e r a l i z e d t o a schema for t h e o r i e s of o t h e r ( t h a n l i n g u i s t i c ) k i n d s o f c o m p l i c a t e d human b e h a v i o u r "

.

I n Nowakowska ( 1 9 7 3 ) , n o t o n l y t h e s y n t a x of a c t i o n l a n g u a g e s was a n a l y s e d , b u t a l s o t h e i r s e m a n t i c s , where t h e "meaning" o f a s t r i n g of a c t i o n s was i d e n t i f i e d w i t h t h e outcome o f t h i s s t r i n g . I n psychology , under t h e l i n g u i s t i c i n t e r p r e t a t i o n , t h e p s y c h o l o g i c a l t r a i t s would no l o n g e r be d e f i n e d as f a c t o r s : b e i n g c e r t a i n g e n e r a l p r o p e r t i e s of b e h a v i o u r , t h e y would a c q u i r e a n i n t e r p r e t a t i o n i n t e r m s o f synt a c t i c or s e m a n t i c p r o p e r t i e s of t h e " b e h a v i o u r a l l a n guage" o f a g i v e n p e r s o n . Such a way of d e f i n i n g t r a i t s i s , a t l e a s t a t p r e s e n t , d i f f i c u l t t o apply t o t r a i t s

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on h i g h e r o r d e r o f a b s t r a c t i o n . The p r o j e c t i s n o t ent i r e l y u n r e a l i s t i c , t h o u g h , as s u g g e s t e d b y Suppes (1963), who p r o p o s e s t o taxonomize v a r i o u s l e v e l s o f " t r a i t s " s u c h a s grammar a c q u i s i t i o n ( o r : t h e complexi t y o f t a s k s which v a r i o u s s p e c i e s o f a n i m a l s a r e capabl e o f p e r f o r m i n g ) , b y a n a l y s i n g t h e t y p e s o f automata which r e c o g n i z e t h e a p p r o p r i a t e language ( o f a c h i l d , o r of t h e behaviour of animals of a given s p e c i e s ) . One can hope t h a t a p p l i c a t i o n o f m a t h e m a t i c a l l i n g u i s t i c s t o psychology w i l l p r o v i d e n o t o n l y a d i f f e r e n t a p p r o a c h t o o l d problems, b u t ( a s was t h e c a s e w i t h o t h e r m a t h e m a t i c a l models) a l s o a s t i m u l u s l e a d i n g t o new c o n c e p t s o r even new a r e a s o f r e s e a r c h . The l i n g u i s t i c i n t e r p r e t a t i o n o f f a c t o r i a l t h e o r i e s o f

p e r s o n a l i t y may be v a l u a b l e f o r q u a n t i t a t i v e l i n g u i s t i c s , i n p a r t i c u l a r t e x t t h e o r y , and a l s o f o r t h e o r i e s o f v e r b a l and n o n - v e r b a l communication, where t h e r e l a t i o n s between v a r i o u s t y p e s o f d a t a a r e of g r e a t t h e o r e t i c a l and p r a c t i c a l i m p o r t a n c e .

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HARMAN, H.H. 1976 Modern Factor Analysis. Chicago. Univ. Press. HAYS, W.L., BENNETT, J. 1961 Multidimensional Scaling. Determining Configuration for Complete Rank Order Preference Data. Psychometrika. 26; 221-238 HENRYSSON, S. 1957 Applicability o f Factor Analysis in the Behavioral Sciences. Stockholm. Almqvist & Wiksell. HORST, P. 1965 Factor Analysis of Data Matrices. New York. Holt, Rinehart & Winston. JONES, L.R., YOUNG, F. 1972 The Structure of a Social Environment. A Longitudinal Individual Difference Scaling of the L.L. Thurstone Psychometric Laboratory. J. of Personality and Social Psychology. 24;108-121. JURESKOG, K.G. 1967 Some Contributions to Maximum Likelihood Factor Analysis. Psychometrika. 32; 443-482. KATZ, J.J., FODOR, J. 1963 The Structure of Semantic Theory. Language, 39; 170-210. KRANTZ, D.H., LUCE, R.D., SUPPES, P., TVERSKY, A. 1971 Foundations of Measurement. New York. Academic Press. KRUSKAL, J.B. 1964a Multidimensional Scaling by Optimizing Goodness of Fit to Nonmetric Hypothesis. Psychometrika. 29; 1-28.

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196413 N o n m e t r i c M u l t i d i m e n s i o n a l S c a l i n g . A N u m e r i c a l Method. P s y c h o m e t r i k a . 29; 115-130. LEE, SIK-YUM

1 9 8 1 A B a y e s i a n Approach t o C o n f i r m a t o r y F a c t o r Anal y s i s . P s y c h o m e t r i k a . 4 6 ; 153-157.

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1958 C o n f i r m a t i o n of O b j e c t i v e T e s t F a c t o r s a n d Assessment of T h e i r R e l a t i o n s h i p s t o Q u e s t i o n n a i r e F a c t o r s . A F a c t o r A n a l y s i s of 113 R a t i n g , Q u e s t i o n n a i r e a n d O b j e c t i v e T e s t Measurements i n P e r s o n a l i t y . J o u r n a l of M e n t a l S c i e n c e . 1 0 4 ; 608-624. SHEPARD, R . N .

1962a The A n a l y s i s of P r o x i m i t i e s . M U l t i d i m e n s i o n a l S c a l i n g w i t h a n Unknown D i s t a n c e F u n c t i o n . I . Psychometrika. 27;

124-140.

1962b The A n a l y s i s of P r o x i m i t i e s . M u l t i d i m e n s i o n a l S c a l i n g w i t h a n Unknown D i s t a n c e F u n c t i o n . 1 1 . Psychometrika. 27; 219-246.

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A t ' t e n t i o n and t h e Metric S t r u c t u r e o f t h e S t i mulus S p a c e . J o u r n a l

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1958 An I n t e r - B a t t e r y Method o f F a c t o r A n a l y s i s . P s y c h o m e t r i k a . 2 3 ; 111-136. WEEKS, D . G . ,

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86; 349-354.

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SOME PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION AND APPLICATION OF Q U E S T I O N N A I R E S

T h i s c h a p t e r w i l l d e a l w i t h two main t o p i c s : (1) t h e p s y c h o l o g i c a l mechanisms u n d e r l y i n g r e s p o n s e s t o i t e m s and s t a t i s t i c a l c o u n t e r p a r t s o f t h e s e mechanisms, ( 2 ) c e r t a i n new i d e a s i n c o n s t r u c t i o n o f t e s t s , where t h e o b s e r v e d s c o r e i s d e f i n e d i n a way o t h e r t h a n t h e t r a d i t i o n a l sum (or l i n e a r c o m b i n a t i o n ) o f s c o r e s i n p a r t i c u l a r i t e m s , and ( 3 ) some a p p l i c a t i o n s o f t e s t s .

1. MODELS OF RESPONSE TO A Q U E S T I O N N A I R E ITEM

We b e g i n w i t h p s y c h o l o g i c a l mechanisms i n v o l v e d i n answering q u e s t i o n s , e s p e c i a l l y t h e t y p e s o f q u e s t i o n s which a p p e a r i n p s y c h o l o g i c a l or s o c i o l o g i c a l q u e s t i o n naires. The t h r e e main o b s e r v a b l e v a r i a b l e s which c o u l d p r o v i d e a n a c c e s s t o p s y c h o l o g i c a l mechanisms i n v o l v e d i n q u e s t i o n a n s w e r i n g a r e (1) t h e answer, ( 2 ) i t s l a t e n c y , and ( 3 ) i t s v a r i a b i l i t y under r e p e t i t i o n . I n case of questionnaire items, latency i s typically n o t o b s e r v e d . Consequently, t h e a n a l y s i s o f t h e p r e s e n t

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s e c t i o n w i l l c o n c e r n o n l y t h e answers and t h e i r v a r i a b i l i t y . The main o b j e c t i v e w i l l be t o a n a l y s e psycholo-

g i c a l mechanisms a c c o u n t i n g f o r v a r i a b i l i t y , s u c h as (among o t h e r s ) a m b i g u i t y and d i f f i c u l t y o f q u e s t i o n s . The concept o f a m b i g u i t y a c q u i r e d c o n s i d e r a b l e i n t e r e s t a f t e r t h e a p p e a r e n c e o f t h e t h e o r y o f f u z z y s e t s o f Zadeh ( 1 9 6 5 ) : t h i s t h e o r y p r o v i d e s formal t o o l s of prec i s e a n a l y s i s o f vague c o n c e p t s ( t h e s e t o p i c s w i l l be t a k e n up i n n e x t c h a p t e r , where t e s t measurement w i l l be suggested f o r c e r t a i n t y p e s of fuzzy c o n c e p t s ) . One of t h e problems d i s c u s s e d i n t h i s s e c t i o n w i l l conc e r n p s y c h o m e t r i c paradox: a n e g a t i v e r e l a t i o n between s t a b i l i t y ( h e n c e a l s o l a c k o f a m b i g u i t y ) o f i t e m s , and t h e i r d i s c r i m i n a t i n g power (which e x p r e s s e s how w e l l t h e i t e m s e r v e s as a measure o f t h e c o n c e p t ) . Next, some e m p i r i c a l r e s u l t s w i l l be p r e s e n t e d concerni n g r e l a t i o n s between v a r i a b i l i t y o f i t e m s and t h e i r perception by respondents. Analysis of t h e s e data l e a d t o a c o n c e p t u a l model o f r e s p o n s e t o an i t e m , and f i n a l l y , t o a f o r m a l i z a t i o n o f t h i s model i n t e r m s o f e x p l i c i t assumptions a l l o w i n g t h e o r e t i c a l a n a l y s i s and s i m u l -

ation. The aim of t h i s s e c t i o n i s n o t s o much t o e x p l i c a t e t h e

s t a t i s t i c a l techniques of item s e l e c t i o n , but rather t o show some p s y c h o l o g i c a l phenomena which a c c o u n t f o r s t a t i s t i c a l f e a t u r e s of item behaviour.

1.1. V a r i a b i l i t y and a m b i g u i t y

I n most o f t h i s s e c t i o n t h e a n a l y s i s w i l l c o n c e r n b i n a r y

PSYCHOLOGICAL PROBLEMS OF CONSTRUCTION

185

i t e m s , i . e . i t e m s which may be answered "Yes" o r rrNol'. Sometimes we s h a l l a l s o a l l o w t h e i n t e r m e d i a t e answer "I d o n ' t know", a b b r e v i a t e d D K . A s shown by numerous e m p i r i c a l s t u d i e s (most of them

i n t h e 5 0 ' s and 6 0 r s ) , t h e i n t r a - s u b j e c t v a r i a b i l i t y o f answers t o q u e s t i o n n a i r e i t e m s i s c o n s i d e r a b l e , sometimes as h i g h as 35 % ( s e e Goldberg 1963, o r Goldberg and J o n e s 1 9 6 4 f o r a r e v i e w o f s t u d y of v a r i a b i l i t y ) . A l l psychological explanations of v a r i a b i l i t y , includ-

i n g t h e model p r e s e n t e d i n t h i s s e c t i o n , a r e f o r m u l a t e d w i t h i n t h e g e n e r a l c o n c e p t u a l framework of t h e model o f p a i r w i s e d i s c r i m i n a t i o n o f Thurstone ( 1 9 2 7 a , b ) . I n T h u r s t o n e ' s model o f c o m p a r a t i v e judgment, t h e r e s ponse ( o r , more p r e c i s e l y , t h e p r o b a b i l i t y o f a g i v e n t y p e o f r e s p o n s e ) i s d e t e r m i n e d by comparison o f two p r o c e s s e s , r e p r e s e n t i n g " l o c a t i o n s " o f two s t i m u l i on some continuum. I n c a s e o f q u e s t i o n a n s w e r i n g , one proc e s s t y p i c a l l y r e p r e s e n t s t h e p e r c e i v e d v a l u e o f some a t t r i b u t e x of an o b j e c t 0 , while t h e o t h e r process i s t h e "boundary" o f t h e q u e s t i o n , s a y x Typically, the q q u e s t i o n i s r e d u c i b l e (however o r i g i n a l l y p h r a s e d ) t o

.

t h e form "Is 0 such-and-such?". The d e s c r i p t o r "suchand-such" i s r e p r e s e n t a b l e as a s e m i - i n f i n i t e s e t on a continuum, hence i s d e t e r m i n e d by t h e p o i n t x The q answer t o t h e q u e s t i o n s h o u l d be ( s a y ) "Yes" i s x i s t o t h e l e f t of x and "No" o t h e r w i s e . N a t u r a l l y , t h e 9' o b j e c t 0 may be t h e r e s p o n d e n t h i m s e l f , as i n t h e i t e m such as "DO you c o n s i d e r y o u r s e l f s h y ? " , e t c .

.

G e n e r a l l y , b o t h d i f f i c u l t y and a m b i g u i t y o f t h e i t e m a r e i n t e r p r e t e d as t h e p r o p e r t y t h a t x and x a r e c l o s e q

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one t o a n o t h e r ( t h e c o n c e p t s o f d i f f i c u l t y and a m b i g u i t y , though c o n c e p t u a l l y v e r y much d i f f e r e n t , c a n n o t be d i s t i n g u i s h e d w i t h i n t h e model, w i t h o u t r e f e r e n c e t o t h e s p e c i f i c c o n t e n t of t h e i t e m ) . I n s u b s e q u e n t p a r t s o f t h i s s e c t i o n , a model w i l l be p r e s e n t e d which s p e c i f i e s t h e above framework, by i n c l u s i o n o f v a r i o u s b i a s e s . A t t h i s p l a c e , one s h o u l d ment i o n t h e work o f Goldberg (1963), who p o s t u l a t e d s p e c i f i c d i s t r i b u t i o n o f x and x and assumed t h a t t h e r e i s q' a c o n s t a n t A , such t h a t p e r s o n s w i t h / x - x A will q5 change t h e i r answers upon r e p e t i t i o n . Given t h e e m p i r i c a l f r e q u e n c i e s o f answers r r y e s r ' , and t h e f r a c t i o n o f changes under r e p e t i t i o n , one c a n t h e n e s t i m a t e t h e v a l u e o f A ( c a l l e d Ambdex, o r a m b i g u i t y i n dex). One s h o u l d mention h e r e t h e f o l l o w i n g laws and p r e d c i c t i o n s , which c o n n e c t a m b i g u i t y , d i f f i c u l t y and v a r i a b i l i t y of i t ems.

a ) The q u e s t i o n s whose b h u n d a r i e s are p e r c e i v e d by t h e s u b j e c t s as c l o s e t o t h e l o c a t i o n o f t h e o b j e c t ( e . g . themselves, i n self-evaluation items) are: -- o f t e n answered by "I d o n ' t know"; -- w i t h l o n g l a t e n c y ; -- e v a l u a t e d as d i f f i c u l t t o answer. Consequently, t h e c l o s e r i s x t o x

i n the perception of q t h e r e s p o n d e n t , t h e l e s s s t a b l e w i l l be h i s answer.

From t h e p o i n t o f view o f t h e o b s e r v e d answer, we may say that :

--

i t e m s w i t h answer D K w i l l be r e l a t i v e l y l e s s s t a b l e ;


--

--

187

i t e m s w i t h l o n g e r l a t e n c y w i l l be l e s s s t a b l e ; i t e m s e v a l u a t e d as d i f f i c u l t w i l l b e l e s s s t a b l e .

Next, l e t u s assume t h a t t h e d i s t r i b u t i o n o f v a l u e s o f x i n t h e p o p u l a t i o n i s normal. Hence i f x i s f a r from t h e c e n t r e o f t h e d i s t r i b u t i o n , t h e r e w i l l be few p e r s o n s who w i l l p e r c e i v e t h e v a l u e s o f x c l o s e t o t h e boundary o f t h e q u e s t i o n . C o n s e q u e n t l y , items w i t h h i g h agreement on t h e answer ( s o - c a l l e d unbalanced i t e m s ) w i l l be seldom p e r c e i v e d a s d i f f i c u l t . Thus:

--

balanced items w i l o f t e n t h a n unbalanced -- b a l a n c e d i t e m s w i l -- b a l a n c e d i t e m s a r e t h a n unbalanced o n e s .

l be p l a c e d i n c a t e g o r y DK more

items; l have l o n g e r l a t e n c y ; more o f t e n e v a l u a t e d as d i f f i c u l t

A s r e g a r d s t h e v a l u e s of Ambdex, i t s h o u l d c o r r e l a t e

p o s i t i v e l y w i t h v a r i a b i l i t y of a n s w e r s . To sum up, s t a b i l i t y o f r e s p o n s e i s c o n n e c t e d w i t h

--

p e r c e i v i n g o n e ' s own p o s i t i o n as f a r from t h e boundar y o f t h e q u e s t i o n , and -- u n b a l a n c e d n e s s of t h e i t e m , i . e . p e r c e i v i n g t h e bounda r y o f i t e m a t one o f t h e e n d p o i n t s o f t h e continuum; On t h e o t h e r hand, v a r i a b i l i t y of r e s p o n s e i s c o n n e c t e d with:

-- a m b i g u i t y o f t h e i t e m ; -- d i f f i c u l t y o f t h e i t e m ; -- u n d e c i s e v e n e s s . Now, one of t h e s h o r t c o m i n g s o f G o l d b e r ' s a s s u m p t i o n s i s t h a t p e r s o n s from t h e a m b i g u i t y band change t h e i r answer upon r e p e t i t i o n o f i t e m , w h i l e t h e o t h e r s do n o t .

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It i s e a s y t o s e e why a c c e p t i n g s u c h a r e s t r i c t i v e a s -

sumption would l e a d i n t o l o g i c a l d i f f i c u l t i e s . Imagine namely t h a t t h e q u e s t i o n i s r e p e a t e d n o t t w i c e , b u t t h r e e t i m e s . Then t h e r e s h o u l d b e no p e r s o n s whose answer p a t t e r n would be llYes'r - ''No" - "No", s i n c e t h e f i r s t change means t h a t t h e p e r s o n i s i n t h e a m b i g u i t y band, hence s h o u l d change t h e answer a g a i n . The f o l l o w i n g s i m p l e model shows t h e way i n which t h i s d i f f i c u l t y may b e overcome. Suppose t h a t t h e s u b j e c t s may be d i v i d e d ( w i t h r e s p e c t t o a g i v e n i t e m ) i n t o t h r e e c a t e g o r i e s : t h o s e , who a l w a y s answer "Yes", t h o s e who

a l w a y s answer "No" , and t h o s e u n d e c i d e d . About t h e l a s t c a t e g o r y we assume t h a t t h e y g i v e answers "Yes" o r "No" a t random, i n d e p e n d e n t l y from t r i a l t o t r i a l , w i t h prob a b i l i t y $. L e t p , q and z d e n o t e t h e p r o p o r t i o n s o f s u b j e c t s i n e a c h o f t h e above t h r e e c a t e g o r i e s , s o t h a t p t q t z = 1. C l e a r l y , two o f t h e s e numbers may b e t a ken as p a r a m e t e r s of t h e q u e s t i o n , s a y p and z . T h e obj e c t i s t h e n t o e s t i m a t e t h e s e parameters from t h e s t a t i s t i c a l d a t a on a n s w e r s , c o l l e c t e d from a d o u b l e t r i a l on t h e same q u a t i o n . Thus, t h e d a t a have t h e form of a t r i p l e t ( Y , N , C ) , where Y , N and C d e n o t e t h e numbers o f p e r s o n s i n t h e sample who answered "Yes" on b o t h times, who answered ''No" on b o t h t i m e s , and who changed t h e i r answer. F u r t h e r m o r e , l e t n = Y + N + C , and a s s u me that C

5

min ( 2 N , 2 Y ) .

The l a s t c o n d i t i o n i s j u s t i f i a b l e i n view of t h e assumed model of random answers from c a t e g o r y o f " c h a n g e r s " : on t h e a v e r a g e , t o e v e r y 2 p e r s o n s from t h i s group who answered "Yes" on t h e f i r s t t r i a l , one p e r s o n w i l l


189

change h i s answer, w h i l e t h e o t h e r w i l l n o t , and t h e

.

same h o l d s f o r a n s w e r ''No" T o p u t i t d i f f e r e n t l y , t h e p r o b a b i l i t y of a c h a n g e of a n s w e r i s 3 , w h i l e t h e p r o b a b i l i t y o f two i d e n t i c a l a n s w e r s ( b y a p e r s o n f r o m t h e " u n d e c i d e d " g r o u p ) i s $ for a n s w e r "Yes" a n d $ for a n swer "No". L e t u s now t r u t o f i n d t h e maximum l i k e l i h o o d e s t l m a t a n d z . The p r o b a b i l i t y o f

ors o f p r o b a b i l i t i e s p , q

a n s w e r "Yes" i s p t { z , p r o b a b i l i t y o f a n s w e r "No" i s q t $z = 1

-

p

-

T3 z , a n d p r o b a b i l i t y o f a c h a n g e o f

answer i s gz. Consequently, t h e l i k e l i h o o d f u n c t i o n i s

which g i v e s

+ c

log z

- c

log 2.

D i f f e r e n t i a t i o n w i t h r e s p e c t t o p and z y i e l d s , a f t e r some s i m p l i f i c a t i o n s , t h e e q u a t i o n s (N

+

Y)p t g ( N

+ 3Y)

= Y,

Cp t $(2N t 3 C ) = C . S o l v i n g , w e o b t a i n f i n a l l y f o r p , q and z p = (2Y

-

C ) / 2 n , q = ( 2 N - C ) / 2 n , z = 2C/n

where n = Y t N

+

C.

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I f one now a d d s some d i s t r i b u t i o n a l a s s u m p t i o n s ,

one may u s e t h e above f o r m u l a s t o g e t i n f o r m a t i o n a b o u t t h e l o c a t i o n o f c e r t a i n p o i n t s on t h e co n tin u u m of t h e t r a i t . Suppose t h a t t h e d i s t r i b u t i o n o f v a l u e s of t h e t r a i t i n t h e p o p u l a t i o n i s n o r m a l , w i t h known mean m and known s t a n d a r d d e v i a t i o n 6. F u r t h e r , s u p p o s e t h a t t h e p e r s o n s who w i l l a n s w e r "Yes" a r e c h a r a c t e r i z e d b y t h e i r v a l u e s x b e i n g l e s s t h a n some t h r e s h o l d a . S i m i l a r l y , t h o s e who w i l l a nsw er "No" a r e c h a r a c t e r i z e d by t h e c o n d i t i o n b c: x , w h i l e t h o s e u n d e c i d e d an d a n s w e r i n g randomly sat i s f y the condition a < x < b. Then t h e v a l u e s a an d b may be d e t e r m i n e d f r o m e q u a t i o n s

where

( I \ ( x) i s t h e s t a n d a r d n o r m a l d i s t r i b u t i o n .

The d i f f e r e n c e b-a may be c a l l e d t h e w i d t h o f t h e un-

c e r t a i n t y area ( e x p r e s s e d i n u n i t s e q u a l :i, w i t h z e r o a t m ) . If w e r e t u r n t o t h e i n t e r p r e t a t i o n a c c o r d i n g t o whic h e a c h p e r s o n (or o b j e c t ) i s c h a r a c t e r i z e d by t h e v a l u e x , t h e n t h e u n c e r t a i n t y area w i l l c o n s i s t o f a l l those x with a < x L b. The f r a c t i o n o f c h a n g e s o f a n s w e r C d e p e n d s n o t o n l y on

t h e w i d t h o f t h e u n c e r t a i n t y a r e a , b u t a l s o on t h e dens i t y o f t h e n o r m a l d i s t r i b u t i o n o v e r t h i s area; t h u s ,

i t d e p e n d s on t h e i n t e g r a l b

One may f o r m u l a t e t h e f o l l o w i n g h y p o t h e s i s r e l a t i n g

191


t h e w i d t h of t h e u n c e r t a i n t y a r e a b-a and i t s l o c a t i o n on t h e a x i s ( c e n t r a l , i . e . c l o s e t o m y v s . e x t r e m a l , i . e . i n t h e t a i l of t h e d i s t r i b u t i o n ) .

i s small, t h e n t h e v a r i a b i l i t y of answers i s low, r e g a r d l e s s of t h e l o c a t i o n o f t h e u n c e r t a i n t y a r e a . The q u e s t i o n i s b a l a n c e d o r n o t , depending on t h e l o c a t i o n of t h i s a r e a . If t h e w i d t h b-a

On t h e o t h e r hand, i f t h e w i d t h b-a i s l a r g e , t h e v a r i a b i l i t y i s l a r g e i n c a s e o f c e n t r a l l o c a t i o n , and smalle r i n c a s e of extreme l o c a t i o n of t h e u n c e r t a i n t y a r e a .

1 . 2 . A f u z z i f i c a t i o n o f t h e model

The above model n e e d s t o b e f u z z i f i e d , s i n c e i n t h e

p r e s e n t form i t p o s t u l a t e s d i s c o n t i n u i t i e s a t t h e p o i n t s a and b y i n t h e s e n s e t h a t a l l s u b j e c t s w i t h t h e v a l u e x of t h e t r a i t l y i n g t o t h e l e f t from a behave i n a d i f f e r e n t way t h a n t h o s e w i t h a G ' X b y and t h o s e i n t u r n behave d i f f e r e n t l y from s u b j e c t s w i t h x < b .

.r-

0

a,

Q X

a,

k

TI

a,

0 t-r:

a,


t 0

.r- .d k oaa,

a,ba

a h x

m aa,

d

+--

0

21 1

‘ 1

M

c Q

(d

.Q

---I

I_

4

/“

212

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schemes d o m i n a t e s , t h e s u b j e c t chooses p a t h 2 , where t h e r e s p o n s e i s d e t e r m i n e d by f a c t o r

V ( S o c i a l approv-

a l ) . On t h e o t h e r hand, i n c a s e o f a b s e n c e of s u c h sche-

mes, or when t h e i r s t r e n g t h w i l l be small, t h e s u b j e c t w i l l choose p a t h 1, where t h e r e s p o n s e i s d e t e r m i n e d by s u b s e q u e n t f a c t o r s : I V (Value o f t h e i t e m ) and I

( e m o t i o n a l and m o t i v a t i o n a l a t t i t u d e ) . It s h o u l d be n o t e d , t h a t here t h e response i s c o n t r o l l e d by feedback mechanism, r e f l e c t e d i n t h e s u b j e c t i v e p r o b a b i l i t y t h a t t h e answer i s c o r r e c t .

F i n a l l y , i t i s p o s s i b l e t h a t p a t h 3 w i l l be c h o s e n , where t h e v a r i a n t s of answers s e r v e as s t i m u l i , and t h e answer i s d e t e r m i n e d by s t a t i s t i c a l p r e f e r e n c e s t o p a r t i c u l a r c a t e g o r i e s of a n s w e r s . Each o f t h e t h r e e p a t h s (or some o f them) s u p p l y a t r i a l r e s p o n s e ; t h e s e a r e t h e n combined i n t o t h e f i n a l response R , c o n t r o l l e d a g a i n by t h e f e e d b a c k mechanism t o f a c t o r 111 ( o f i n t e l l e c t u a l e v a l u a t i o n ) .

The c o n t r o l o f t h e answer, e x p r e s s e d by a f e e d b a c k t o f a c t o r 111, o p e r a t e s , i n a s e n s e , on two l e v e l s : o f t r u t h and u t i l i t y . On t h e f i r s t l e v e l , o f t r u t h , t h e c o n t r o l r e f l e c t s d e t e r m i n i n g t h e r e s p o n s e R1 ( n o t neces s a r i l y d i s c l o s e d ) , which r e p r e s e n t e s t h e " i n n e r t r u t h " o f t h e s u b j e c t . The c o n t r o l c o n s i s t s on s t a b i l i z i n g t h e u n d e r s t a n d i n g o f t h e i t e m , and t h e n s t a b i l i z i n g t h e r e s p o n s e under g i v e n u n d e r s t a n d i n g . The second l e v e l of c o n t r o l i s t h a t o f u t i l i t y . Here t h e t r i a l f i n a l answer R , d e r i v e d from R1, R2 and R ; t h e 3 l a t t e r r e s p o n s e i s assumed t o i n t e r v e n e o n l y i n t h e c a s e when t h e r e s p o n s e s R1 and R2 are m i s s i n g , o r a r e


213

v e r y vague. Now, t h e r e s p o n s e s R1 and R2 may be i n agreement or n o t . I n t h e f i r s t c a s e , t h e f i n a l answer c o i n c i d e s w i t h them. I n t h e second c a s e , t h e f i n a l answer must be i n c o n s i s t e n t w i t h e i t h e r R1 or w i t h R2 ( i f o n l y answers "yes" and "no" a r e a l l o w e d ) . The c o n t r o l o f u t i l i t y of answer c o n s i s t s o f p r e d i c t i o n

o f t h e e f f e c t s o f d i s c l o s i n g t h e r e s p o n s e . The r e s p o n s e i n c o n s i s t e n t w i t h R 1 ( l y i n g ) may l e a d t o punishment i n form o f a lowered s e l f - e s t e e m ( " g u i l t y c o n s c i e n c e " ) , w h i l e a r e s p o n s e i n c o n s i s t e n t w i t h R2 ( n o n - c o n f o r m i s t ) may l e a d t o a d i s a p p r o v a l o f t h e group; i t may a l s o b r i n g some r e w a r d s , e . $ . i n form o f a t t r a c t i n g a t t e n t ion, e t c . I n b o t h c a s e s of control', i t l a s t s u n t i l t h e t r i a l answers s t a b i l i z e . The scheme of a n s w e r i n g s u g g e s t e d i n t h i s s e c t i o n w i l l s e r v e as a b a s i s f o r a s i m u l a t i o n model d i s c u s s e d i n s e c t i o n 1 . 6 . B e f o r e p r e s e n t i n g t h i s model, however, i t i s w o r t h w h i l e t o show c e r t a i n r e s u l t s c o n c e r n i n g change s o f answers.

1.5.3. P e r c e p t i o n o f i t e m s and v a r i a b i l i t y o f a n s w e r s . A s opposed t o t h e a n a l y s i s o f t h e p r e c e d i n g s e c t i o n , which was aimed a t d e t e r m i n i n g p s y c h o l o g i c a l f a c t o r s u n d e r l y i n g t h e p r o c e s s of answering, t h i s s e c t i o n w i l l be d e v o t e d t o p s y c h o l o g i c a l d e t e r m i n a n t s of v a r i a b i l i t y o f answers t o q u e s t i o n n a i r e i t e m s ( s e e Nowakowska

1973 ) .

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The d a t a which p r o v i d e d i n f o r m a t i o n about t h e d e t e r m i n a n t s of v a r i a b i l i t y were t h e same e v a l u a t i o n s on 17 s c a l e s which s e r v e d f o r f a c t o r a n a l y s i s and subseq u e n t c o n s t r u c t i o n o f t h e model o f a n s w e r i n g t o q u e s t ionnaire items. A s already explained i n Section 1 . 4 ,

each subject responded t o a n i t e m , and t h e n e v a l u a t e d t h e i t e m and h i s r e s p o n s e on 17 s e v e n p o i n t s c a l e s . T h i s p r o c e d u r e was r e p e a t e d a f t e r t h e p e r i o d of 1 4 d a y s . Regarding t h e answers t o t h e i t e m s , t h e r e were o n l y 4 p o s s i b i l i t i e s , namely YY, YN, NY and NN ( o r 9 p o s s i b i l i t i e s , i f t h e answer ? was a l l o w e d , namely YY, Y ? , YN, ?Y, ? ? , ?N, NY, N?, NN). If one i s i n t e r e s t e d i n v a r i a b i l i t y o f r e s p o n s e s , and i s w i l l i n g

t o a b s t r a c t from

t h e d i r e c t i o n o f changes, t h e n i n e i t h e r c a s e , t h e r e s p o n s e s t o i t e m s may be c a t e g o r i z e d i n t o + ( a change i n r e s p o n s e ) , and

-

(no change i n r e s p o n s e ) .

The r e l a t i o n between v a r i a b i l i t y o f answers and t h e e v a l u a t i o n o f t h e i t e m on s c a l e s was d e t e r m i n e d b y t h e u s e o f o p t i m a l method o f e s t i m a t i o n o f v a r i a b i l i t y on t h e b a s i s o f e v a l u a t i o n s . The problem was t r e a t e d as one o f d i s c r i m i n a t i o n : t h e v a r i a b l e d i s c r i m i n a t e d i s t o r - (change o f answer o r i t s l a c k ) , and t h e d i s c r i m i n a t o r i s t h e e v a l u a t i o n o f an i t e m on a g i v e n s c a l e ( a c t u a l l y , a p a i r of s u c h e v a l u a t i o n s ) . The p r o b a b i l i t y o f c o r r e c t d i s c r i m i n a t i o n under t h e o p t i m a l p r o c e d u r e e v a l u a t e d from t h e d a t a y i e l d t h e e s t i m a t e s o f t h e corresponding population value. To b e t t e r u n d e r s t a n d t h e problem o f d i s c r i m i n a t i o n o f

v a r i a b i l i t y on t h e b a s i s of e v a l u a t i o n s on a g i v e n sca-


215

l e y one may t h i n k m e t a p h o r i c a l l y of t h e e m p i r i c a l d a t a as f o r m i n g two s e t s o f p a i r s o f e v a l u a t i o n s , one from s u b j e c t s o f " t y p e t" ( t h o s e who changed t h e r e s p o n s e t o t h e i t e m ) , and one from s u b j e c t s o f t y p e -, t h e s e s e t s o f d a t a b e i n g p l a c e d i n two u r n s , marked r e s p e c t i v e l y t and A p a i r o f e v a l u a t i o n s i s chosen a t r a n dom from one o f t h e u r n , and t h e o b j e c t i s t o g u e s s from which u r n t h e p a i r was c h o s e n .

-.

B e f o r e p r e s e n t i n g t h e method and t h e c o m p u t a t i o n a l scheme, i t i s w o r t h t o e x p l a i n : (1) why t h e method o f d i s c r i m i n a t i o n f u n c t i o n was c h o s e n , i n s t e a d o f t r a d i t i o n a l method o f s t u d y i n g dependence by c o r r e l a t i o n methods ; ( 2 ) w h y , i n s t e a d of b u i l d i n g one d i s c r i m i n a t i o n f u n c t i o n

based on a l l 17 p a i r s of e v a l u a t i o n s , t h e problem was s p l i t i n t o s e p a r a t e d i s c r i m i n a t i o n problems, one f o r each e v a l u a t i o n s c a l e . The answer t o t h e f i r s t q u e s t i o n i s t h e f o l l o w i n g : t h e v a r i a b l e which s e r v e d as a b a s i s f o r d i s c r i m i n a t i o n ( a p a i r o f e v a l u a t i o n s ) c o u l d n o t be l i n e a r l y o r d e r e d i n a m e a n i n g f u l w a y , which made i t i m p o s s i b l e t o u s e even the rank correlation c o e f f i c i e n t . The answer t o t h e second q u e s t i o n i s t h a t ( a p a r t from t h e o b v i o u s t e c h n i c a l d i f f i c u l t y o f d i s c r i m i n a t i o n on 17 t h e b a s i s o f a v a r i a b l e which may assume ( 7 x 7 ) , i . e . about 5 . 4x102* v a l u e s ) , s e p a r a t e a n a l y s i s o f s c a l e s p r o v i d e d more p s y c h o l o g i c a l i n s i g h t i n t o t h e r e l a t i o n s between changes o f answers and e v a l u a t i o n s . The g e n e r a l problem o f d i s c r i m i n a t i o n o f a dichotomous

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v a r i a b l e may be f o r m u l a t e d as f o l l o w s . We o b s e r v e a c e r t a i n v a r i a b l e x , chosen a c c o r d i n g t o one o f t h e two p r o b a b i l i t y d i s t r i b u t i o n s , s a y p - ( x ) and p t ( x ) , c o r r e s ponding t o t h e v a l u e s - and t of t h e d i s c r i m i n a t e d v a r i a b l e . Knowing x we have t o d e c i d e whether t h e v a l u e o f t h e d i s c r i m i n a t e d v a r i a b l e ( t h e " u r n " from which x was selected) i s + o r Any r u l e which t e l l s u s how t o a s s i g n a v a l u e t o r - t o t h e observed v a l u e x , i s by d e f i n i t i o n , a d i s c r i m i n a t i o n f u n c t i o n . The o p t i m a l d i s c r i m i n a t i o n f u n c t i o n i s t h a t one which l e a d s t o t h e l e a s t number o f e r r o r s .

-.

I n g e n e r a l , a d i s c r i m i n a t i o n f u n c t i o n a s s i g n s t o each o b s e r v e d v a l u e x t h e p r o b a b i l i t y w i t h which one s h o u l d a s s e r t t h a t the value of t h e discriminated variable i s +. Thus, i f f ( x ) i s t h i s p r o b a b i l i t y , t h e n w i t h probab i l i t y l - f ( x ) one s h o u l d (upon o b s e r v i n g x ) a s s e r t t h a t t h e va l ue of t h e d i s c r i m i n a t e d v a r i a b l e i s

-.

<
0 , nUv = 0 t h e r a t i o i s t a k e n t o be i n f i n i t e , and t h e p o i n t s ( u , v ) w i t h mUv = n = 0 are d i s r e g a r d e d . uv

...,

K, where K L e t t h i s o r d e r i n g be ( u i , v i ) , i = 1,2, i s t h e t o t a l number o f p a i r s ( u , v ) w i t h a t l e a s t one o f n o t z e r o . Thus, we have t h e numbers muv,n uv

D e f i n e now

L e t r be t h e smallest i n d e x s u c h t h a t C r

5 D 1” and

CHAPTER 3

218

Drtl. Such a n i n d e x e x i s t s s i n c e t h e sequence i n c r e a s e s from 0 t o 1, w h i l e t h e sequence D k d e c r e a -

Crtl) C

k

s e s from 1 t o 0 . A f t e r f i n d i n g r , we d e f i n e t h e o p t i m a l d i s c r i m i n a t i o n * n f u n c t i o n f ( u , v ) as f o l l o w s . F i r s t l y , we p u t f (ui,vi) n = 0 f o r i = 1, r . Next, i f C r = D r , w e p u t f (u , v . ) j J = 1 f o r a l l r e m a i n i n g j. I n c a s e o f C r Dr and C r t l ) * Drtl, we p u t f (u , v . ) = 1 f o r j = r t 2 K.

..., j

v , + ~ )+

P-(U,+~,V,+~).

n It i s easy t o s e e t h a t t h e v a l u e f ( u r t l , v r t l )

is well

d e f i n e d , s i n c e Q ? 0 , and l i e s between 0 and 1. The p r o b a b i l i t y o f c o r r e c t d i s c r i m i n a t i o n under t h e n optimal discrimination function f equals

*

We s h a l l now p r e s e n t t h e p r o o f t h a t t h e f u n c t i o n f def i n e d above i s i n d e e d o p t i m a l i n t h e minimax s e n s e . For an a r b i t r a r y d i s c r i m i n a t i o n f u n c t i o n f , l e t a ( f ) and b ( f ) be t h e p r o b a b i l i t i e s o f e r r o n e o u s d i s c r i m i n a t i o n , f o r d i s t r i b u t i o n s p - and p,, t h a t i s ,

a ( f ) = p r o b a b i l i t y t h a t t h e d e c i s i o n r e a c h e d on t h e b a s i s o f f i s t when i n f a c t t h e o b s e r v e d v a l u e x was chosen from t h e d i s t r i b u t i o n p - ,

219


and s i m i l a r l y f o r b ( f ) . THEOREM. The d i s c r i m i n a t i o n f u n c t i o n f

*

i s optimal i n t h e minimax s e n s e , ( f o r loss f u n c t i o n d e f i n e d as t h e p r o b a b i l i t y o f e r r o n e o u s d e c i s i o n ) , t h a t i s , f o r anx d i s c r i m i n a t i o n f u n c t i o n f we have

To p r o v e t h e theorem l e t u s o b s e r v e t h a t ( b y c o n s t r u c t -

*

i o n of f ) we have a ( f

*

*

*

*

= b ( f ) . Let a ( f ) = b ( f ) = c .

We have t o show t h a t f o r any d i s c r i m i n a t i o n f u n c t i o n f we have

i n o t h e r words, we have t o p r o v e t h e i m p l i c a t i o n if a ( f )

4 c , then

b(f)

2

C.

L e t u s c o n s i d e r t h e r e f o r e an a r b i t r a r y d i s c r i m i n a t i o n

f u n c t i o n f , and l e t a ( f ) C c . F o r t h e p r o o f , l e t u s t r e a t t h e d i s c r i m i n a t i o n f u n c t i o n f as a s t a t i s t i c a l t e s t o f t h e h y p o t h e s i s Ho: d i s t r i b u t i o n i s p - ( x ) , a g a i n s t t h e a l t e r n a t i v e H1: d i s t r i b u t i o n i s p + ( x ) . The c o n s i d e r e d d i s c r i m i n a t i o n f u n c t i o n , t r e a t e d a s a s t a t i s t i c a l t e s t , has t h e s i g n i f i c a n c e l e v e l a ( f ) and. power 1 - h(f). F o r s i m p l i c i t y o f n o t a t i o n s , p u t a ( f ) = a , and l e t u s c o n s i d e r t h e most p o w e r f u l t e s t o f hypot h e s i s Ho a g a i n s t t h e a l t e r n a t i v e HI a t t h e s i g n i f i c a n ce l e v e l a . Suppose t h a t s u c h a t e s t i s f a . By t h e Neyman-Pearson

220

CHAPTER 3

lemma ( s e e Lehmann, 1 9 6 0 ) such a most p o w e r f u l t e s t e x i s t s , and i s based on t h e r a t i o p + ( x ) / p ( x ) i n a manner -* similar t o t h e d i s c r i m i n a t i o n function f constructed above, e x c e p t t h a t t h e c r i t i c a l r e g i o n for t h e t e s t f a i s such t h a t t h e s i g n i f i c a n c e l e v e l i s a (as d i s t i n c t n from t h e c o n s t r u c t i o n of f , where i t was r e q u i r e d t h a t t h e s i g n i f i c a n c e l e v e l e q u a l s t h e p r o b a b i l i t y of t h e e r r o r of t h e second k i n d ) . For t h e most p o w e r f u l t e s t f

a

w e have

The second i n e q u a l i t y f o l l o w s from t h e f a c t t h a t f i s a t h e most p o w e r f u l t e s t a t t h e s i g n i f i c a n c e l e v e l a . It remains t o prove t h e r e f o r e t h a t t h e i n e q u a l i t y a ( f )

r

(1.14)

Then

where v l , v 2 a r e i n d e p e n d e n t random v a r i a b l e s d i s t r i and a , b a r e some p o s i t i v e b u t e d u n i f o r m l y on [O,l], constants, representing the thresholds f o r tendencies towards and towards "no". Alternative

i n t e r p r e t a t i o n h e r e i s as f o l l o w s : t h e

t e n d e n c i e s towards " y e s r r and "no" a r e e q u a l v1 and v 2' b u t s t a r t t o o p e r a t e o n l y i f t h e y exceed t h e i r r e s p e c t i v e t h r e s h o l d s a and b ( a r e " p e r c e i v e d " by t h e respondent)

.

C o n s e q u e n t l y , t h e v a l u e s o f a and b w i l l c a u s e some bias i n responses.

1.6.3. The f i n a l answer. A s a l r e a d y mentioned, t h e f i n a l answer, t o be d e n o t e d by t ' , i s based on t r i a l answers. N a t u r a l l y , depending on t h e q u e s t i o n , t h e r e s p o n d e n t may t a k e up a l o n g t i m e t o answer, r e v i e w i n g i n h i s mind a l l p o s s i b i l i t i e s , and presumably " t e s t i n g " t h e t r i a l answers. I n t h e model o f t h i s s e c t i o n , it w i l l be assumed t h a t t h e f i n a l answer i s based on j u s t two t r i a l a n s w e r s , t l and t 2 ; v e r s i o n s o f t h e model w i t h h i g h e r number of t r i a l answers a r e e a s i l y c o n s t r u c t e d .

230

CHAPTER 3

To be more p r e c i s e , t h e f i n a l answer w i l l depend on t h e v a l u e s x1 = x t

)2

rl

and x2 = x

+

r,,

where

il

and a r e sampled i n d e p e n d e n t l y a c c o r d i n g to t h e r u l e s de-

s c r i b e d i n t h e p r e c e d i n g s e c t i o n . Observe t h a t xI and

x2 are n o t i n d e p e n d e n t , even though

tl

r

are i n d e p e n d e n t , because o f t h e p r e s e n c e o f common component x. and

j2

One c o u l d t h i n k h e r e o f a v a r i e t y o f r u l e s , Some o f them have been t r i e d i n s u c c e s s i v e v e r s i o n s of t h e p r e s e n t model, which i s g e n e r a l l y termed MASIA. The s i m p l e s t v e r s i o n , c a l l e d MASIA 1, was d e f i n e d s o t h a t t h e g i v e n answer t 1 was " y e s " i f b o t h t l and t 2 were " y e s " , t I was rrno" i f b o t h tl and t 2 were "no" , and t 1 i s "I d o n ' t know" ( D K ) , i f t l # t 2 . A l g e b r a i c a l l y , i f t h e answer DK i s d e s i g n a t e d by 0 , MASIA 1 o p e r a t e d a c c o r d i n g t o the rule

t ' = (tI t t p 2 . S u c c e s s i v e models were d e s i g n e d to overcome c e r t a i n s h o r t c o m i n g s o f t h e p r e c e d i n g models. What i s p r e s e n t e d h e r e i s t h e v e r s i o n c a l l e d MASIA 7 . The i d e a i s t o l o o k a t t h e d i f f e r e n c e s p - x1 and p - x2; h i g h p o s i t i v e v a l u e s i n d i c a t e " s t r o n g y e s " , and h i g h n e g a t i v e v a l u e s i n d i c a t e " s t r o n g no". T h i s s u g g e s t s t o d e t e r m i n e t h e f i n a l answer on t h e b a s i s of t h e d i f f e r e n ce between t h e s e v a l u e s , namely

u

= (p

-

XI)

Here U = 0 , i . e . x

- (x*

-

p ) = 2p - (xl t x 2 >

(1.16)

1 t x2 = 2p i n d i c a t e s t h a t x1 and x2 are a t t h e same d i s t a n c e from p , l y i n g on o p p o s i t e s i d e s o f i t , i . e . i f one t r i a l answer i s " y e s " , t h e o t h e r i s


23 1

, i n t h e same d e g r e e . T h i s s u g g e s t s t h a t i n such a c a s e , t h e answer s h o u l d be D K , i f i t i s t o b e based on j u s t t h e two t r i a l a n s w e r s . llno"

Somewhat more g e n e r a l l y , t h e g i v e n answer i s D K , i f t h e sum x1 t x2 i s c l o s e enough t o 2 p . F o r m a l l y , M A S I A

7 operates according t o the r u l e if

x1

+ x2

t 1 = ? DK

if

2p

-

-1

if

2p t k

/'

\

1

k

$

C_

- k

2p

c x1 x1

t x t

2

2p t k

(1.17)

x2.

Here k i s a p a r a m e t e r r e p r e s e n t i n g t h e ( u n ) i m p o r t a n c e o f a p p e a r i n g d e c i s i v e : i f k i s s m a l l , t h e n x1 t x 2 must l i e v e r y c l o s e t o 2p t o e l i c i t t h e r e s p o n s e D K . G e n e r a l l y , t h e a r e a s "Yes", D K , and "No" are shown on F i g . 1, as s e t s of p o i n t s on t h e ( x l , x 2 ) - p l a n e , which c o r r e s p o n d t o t h e marked r e s p o n s e s . The w i d t h o f t h e DK area r e p r e s e n t s h e r e t h e ( u n ) i m p o r t a n c e of a p p e a r i n g decisive. The a s s u m p t i o n s u n d e r l y i n g t h e model may now be summari z e d a s f o l l o w s . The r e s p o n d e n t i s i n f l u e n c e d , t o a varyi n g d e g r e e , b y s e v e r a l t e n d e n c i e s , s u c h as " y e s - s a y i n g " , "no-saying", which i n t u r n may r e s u l t from o t h e r t e n d e n c i e s , s u c h as "need of a c c e p t a n c e " , "need o f c o n f o r m i t y " , and s o on. These t e n d e n c i e s a f f e c t t h e r e s p o n s e o n l y i f t h e y exceed t h e i r r e s p e c t i v e t h r e s h o l d s . I f these t e n d e n c i e s o p e r a t e , t h e n t h e i r i n t e n s i t y w i l l be h i g h e r when g i v i n g t h e t r u e answer w i l l be s e e n as

less important. I n g i v i n g t h e f i n a l answer, t h e r e s p o n d e n t f o r m u l a t e s

232

CHAPTER 3

f i r s t two t r i a l a n s w e r s , based on t h e t r a i t v a l u e d i s t o r t e d by t h e above two t e n d e n c i e s and by t h e import-

ance o f t h e q u e s t i o n . The f i n a l answer i s t h e n g i v e n

X

2 A

2P

,

1 I 2p-k

"Yes '\,

area \ '

'\\\

1

DK a r e a IINoII

\.,\,

2p-k

2p ,

2 p t, k

>

X

1

Fig. 1

a c c o r d i n g to t h e r u l e (1.17) as i l l u s t r a t e d on t h e F i g u r e above, where k i s t h e p a r a m e t e r measuring t h e ( u n ) i m p o r t a n c e of a p p e a r i n g d e c i s i v e .

1.6.4. Theoretical analysis.

Suppose now t h a t we have

t h e d a t a from some i n t e r v i e w , and we know n o t o n l y t h e answers g i v e n by t h e r e s p o n d e n t s , b u t a l s o t h e i r t r u e a n s w e r s . Such d a t a would have t h e form of a t a b l e w i t h s i x e n t r i e s , as p r e s e n t e d below.


233

answer given ( t ' )

'-.

answer

(t)

I i

n

I

1

i

n

2

i

n

3

no

T a b l e 1. Form o f t h e d a t a F o r i n s t a n c e ( s e e van d e r Zouwen, Nowakowska and D i j k s t r a 1979) t h e d a t a on q u e n t i o n "Voted i n 1946 Congressi o n a l E l e c t i o n ? " are g i v e n i n T a b l e 2 .

4-1

I

total

I

i

Table 2 The problem i s t h e n t o e s t i m a t e t h e v a l u e s o f t h e param e t e r s a J b J u and k for t h i s q u e s t i o n , p r o v i d i n g t h u s some i n s i g h t i n t o t h e p s y c h o l o g i c a l p r o c e s s e s i n v o l v e d i n t h e answer.

234

CHAPTER 3

A r e l a t i v e l y s i m p l e a p p r o a c h t o t h i s problem i s t o s i -

m u l a t e t h e a n s w e r i n g p r o c e s s f o r any v a l u e s of paramet e r s , and t h e n u s e t h e t r i a l - a n d - e r r o r method t o f i n d t h e p a r t i c u l a r combination o f p a r a m e t e r s which would produce a t a b l e o f data s u f f i c i e n t l y c l o s e t o t h e emp i r i c a l l y g i v e n t a b l e . Such a n a p p r o a c h was used ( f o r a somewhat d i f f e r e n t v e r s i o n o f M A S I A model) i n van d e r Zouwev, Nowakowska and D i j k s t r a ( 1 9 7 9 ) . One c o u l d , however, u s e t h e maximum l i k e l i h o o d method i n t h e following way. Denote by F ( z ) = F

a,b

(z) t h e p r o b a b i l i t y d i s t r i b u t i o n

f u n c t i o n o f t h e random v a r i a b l e w = z - - z

+

= Ib(V1)

- Ia(v2),

(1.18)

where v1 and v 2 a r e i n d e p e n d e n t random v a r i a b l e s uniformly d i s t r i b u t e d on [O,l],

and I r ( x ) i s d e f i n e d b y

t h e formula ( 1 . 1 4 ) . C l e a r l y , f o r a n a p p l i c a t i o n o f maximum l i k e l i h o o d met h o d , we need t o have t h e o r e t i c a l p r o b a b i l i t i e s o f v a r i o u s c o m b i n a t i o n s o f answers ( t , t ' ) as i n T a b l e 1, as f u n c t i o n s of p a r a m e t e r s o f t h e model, t h a t i s , a , b , u and k . Denote t h e s e p r o b a b i l i t i e s by p,, . . . , p 6 , w i t h i n d i c e s c o r r e s p o n d i n g t o T a b l e 1. For i n s t a n c e , p1 = P ( t r u e and g i v e n answer a r e b o t h " y e s " )

235


so t h a t P p1

-

+

PIWl

,

W2 < ( 2 p

-

k

- 2x)/u]dx

(1.19)

0 w h e r e W1 a n d W2 a r e i n d e p e n d e n t random v a r i a b l e s w i t h

F.

distribution

-

S u b s t i t u t i n g (2p

k

-

2x)/u = y , w e g e t dx =

-

U

dy,

(1.20)

1 t W2 L y ) i s t h e d i s t r i b u t i o n f u n c t i o n o f t h e sum of two i n d e p e n d e n t random v a r i a b l e s ,

where G ( y ) = P(W

each with d i s t r i b u t i o n F. I n a s i m i l a r way we o b t a i n p2 = P ( x

= P[x

< p

, t h e p r o b a b i l i t y o f c h o o s i n g t h e f o r m e r i s (4-3)/(4-1) = 0 . 3 3 , and s o on.

For e a c h of t h e v a l u e s x = 1, 1 . 5 , 2 and 2 . 5 t h e r e were 10 s i m u l a t e d t r i a l s , each c o n s i s t i n g of choosing 6 o u t o f 1 2 s t a t e m e n t s from e a c h o f t h e c o r r e s p o n d i n g s e t s of s t i m u l i .

A t y p i c a l c h o i c e , for x

=

2 . 5 , was

254

CHAPTER 3

12 1-5

13 1-5

23

14 3-3

24 4-2

3-3

34

5-1

s o t h a t t h e v e c t o r c o r r e s p o n d i n g t o t h e g r a p h o f domination i s [-1, -1, 0 , 0 , 0 , 1 3 . An easy check shows t h a t t h e d i s t a n c e s from t h e v e c t o r s

c h a r a c t e r i s t i c for t h e s e t s A1, B 1 2 , . . . , A4 a r e 7 , 6, 4 , 1, 2 , 5 s o t h a t t h e e s t i m a t e d v a l u e e q u a l s t o t h e

t r u e value x = 2.5. I n g e n e r a l , t h e r e s u l t s o f s i m u l a t i o n s were as f o l l o w s : t r u e value

estimated value ( # of c a s e s )

1.5

2

5

4.5

0.5

1

7 4

2

1

1 1.5 2

2.5

3 1

2.5

3

3 9

The t r i a l s f o r v a l u e s x = 3 , 3.5 and 4 were n o t r u n because o f t h e symmetry o f t h e problem. The n o n i n t e g e r v a l u e s a p p e a r e d because o f t i e s i n t h e e s t i m a t e d val u e s , i . e . t h e minimum o c c u r r i n g a t more t h a n one v a l u e . T h i s t a b l e shows, f o r i n s t a n c e , t h a t when t h e t r u e va-

l u e was x = 1, t h e e s t i m a t e d v a l u e was a l s o 1 i n 5 o u t o f 10 c a s e s ; i n f o u r c a s e s i t was e q u a l 1 . 5 , and i n one c a s e t h e minimum was a t p o i n t s 1 . 5 and 2.


255

S i m i l a r l y , when t h e t r u e v a l u e was x = 1 . 5 , t h e e s t i m a t ed v a l u e was a l s o 1.5 i n 7 o u t o f 10 c a s e s . I n one cas e i t was 1, and i n two c a s e s i t was 2 . I t may be s e e n t h a t t h e w o r s t r e s u l t s were o b t a i n e d for

x = 2 : t h e f u z z i n e s s o f t h e e s t i m a t e s was l a r g e s t . S t i l l , i n no c a s e t h e error exceeded 0 . 5 . These r e s u l t s a r e , o f c o u r s e , o n l y p r e l i m i n a r y , and f a r from b e i n g c o n c l u s i v e , b e c a u s e o f t h e small s i z e o f t h e sample, and b e c a u s e one needs a l s o o t h e r v a l u e s o f t h e p a r a m e t e r s . N e v e r t h e l e s s , t h e y i n d i c a t e t h a t t h e method of e s t i m a t i o n by l o o k i n g a t t h e " c l o s e s t graph" may be more p r e c i s e t h a n t h e method of s i m p l e s c o r i n g t h e numb e r o f c h o i c e s from e a c h o f t h e s t i m u l i . A t t h e c o n c l u s i o n , it i s worth t o p o i n t o u t t h a t t h e

c o n t e x t u a l t e s t s are c h a r a c t e r i z e d by t h e f a c t t h a t t h e y make a d e l i b e r a t e u s e n o t o n l y o f t h e c o n t e x t , b u t a l s o o f f u z z i n e s s . I n d e e d , i n t h e u s u a l c h o i c e between s t i m u l i , t h e s u b j e c t ' s d e c i s i o n i s e i t h e r b i n a r y or t e r n a r y , i f t h e t i e s are a l l o w e d . Here, because o f t h e complex n a t u r e o f t h e s t i m u l u s , t h e c h o i c e may be r e g a r d e d as f u z z y , I n d e e d , i n c a s e o f a s p l i t s u c h a s 4 - 2 , one can s a y t h a t t h e c h o i c e was made i n 2/3 f o r one s t i m u l u s and l/3 f o r t h e o t h e r s t i m u l u s . S u r e l y , f o r t h e c o n s t r u c t i o n o f t h e g r a p h , i t was nec e s s a r y t o a d o p t a c o n v e n t i o n which removes f u z z i n e s s , b u t s t i l l t h e c o n t e x t u a l t e s t s show how f u z z i n e s s may be u s e f u l l y employed ( b e s i d e randomness) t o i n c r e a s e t h e p r e c i s i o n o f measurement, and a l s o t o a l l o w r i c h e r psychological i n t e r p r e t a t i o n s of the r e s u l t s .

256

2.2.

CHAPTER 3

Dynamic q u e s t i o n n a i r e

T h i s s e c t i o n w i l l be d e v o t e d t o p r e s e n t a t i o n o f a n o t h e r

measurement t o o l i n p s y c h o l o g y , namely t h e dynamic ques t i o n n a i r e . I t d i f f e r s from t h e t r a d i t i o n a l q u e s t i o n n a i r e by system o f a s k i n g q u e s t i o n s . I n c l a s s i c a l ques t i o n a i r e , t h e r o l e of experimenter i s r a t h e r passive: u s u a l l y he would o n l y d i s t r i b u t e t h e forms, r e a d t h e i n s t r u c t i o n s , and c o l l e c t t h e r e s u l t s , p e r h a p s o c c a s i o n a l l y a n s w e r i n g some q u e r i e s and c l e a r i n g some a m b i g u i t i e s . A s opposed t o t h a t , i n dynamic q u e s t i o n n a i r e s , t h e experimenter’s r o l e i s a c t i v e i n asking questions, o r p r e s e n t i n g i t e m s , i n an o r d e r depending on t h e previ o u s answers o f t h e s u b j e c t . C o n s e q u e n t l y , e v e r y p e r s o n may answer o n l y some o f t h e q u e s t i o n s . To u s e an example, a t r a d i t i o n a l q u e s t i o n n a i r e may b e compared t o a v i s a a p p l i c a t i o n form, which would u s u a l l y cont a i n q u e s t i o n s s u c h as a b o u t maiden name, e t c , whether o r n o t t h e form i s f i l l e d b y males o r f e m a l e s . The dynamic q u e s t i o n n a i r e may be compared t o a c l i n i c a l i n t e r v i e w , where n e x t q u e s t i o n depends l a r g e l y on t h e r e p l i e s t o t h e preceding ones.

The f o r m a l scheme i n g e n e r a l c a s e . The g e n e r a l scheme o f a dynamic q u e s t i o n n a i r e may be d e s c r i b e d b y s p e c i f i c a t i o n o f t h e n e x t d e c i s i o n (which may be e i t h e r t h e n e x t q u e s t i o n t o b e a s k e d , or t h e d e c i s i o n t o s t o p 2.2.1.

q u e s t i o n i n g and t o a c c e p t f i n a l s c o r e , n o t n e c e s s a r i l y one d i m e n s i o n a l . To d e s c i r i b e i t f o r m a l l y , assume f o r s i m p l i c i t y t h a t e a c h q u e s t i o n may be answered o n l y “ y e s ” o r “no” ( t h i s

257


i s n o t a s e v e r e r e s t r i c t i o n of g e n e r a l i t y ; t h e model may be e a s i l y a d a p t e d t o t h e c a s e o f m u l t i p l e p o s s i b i l i t i e s i n answers.) 4

Suppose t h a t a l l q u e s t i o n s a r e l a b e l e d , and l e t K be t h e s e t o f a l l l a b e l s . G e n e r a l l y , a l a b e l may be a number,

or a v e c t o r ( e . g . i n c a s e t h e q u e s t i o n s a r e p a r t -

i t i o n e d i n t o groups and numbered w i t h i n e a c h g r o u p ; i n t h i s c a s e a l a b e l may be a p a i r ( k , r ) , where k i s t h e number o f t h e group and r i s t h e number o f i t e m i n t h e group, e t c . )

.

L e t p k ( x ) be t h e p r o b a b i l i t y of answer " y e s " t o t h e q u e s t i o n l a b e l e d k b y a p e r s o n w i t h v a l u e x of t h e

trait. A dynami.c q u e s t i o n n a i r e must s p e c i f y

( a ) t h e l a b e l , k l , o f t h e q u e s t i o n t o be asked f i r s t ; ( b ) t h e r u l e , which t o e v e r y p a r t i a l h i s t o r y ( a n s w e r s t o q u e s t i o n s asked t h u s f a c r ) a s s i g n s e i t h e r t h e l a b e l o f t h e n e x t q u e s t i o n , or t h e f i n a l s c o r e .

...

L e t il, i 2 ,

s t a n d f o r answers t o s u c c e s s i v e q u e s t -

i o n s , w i t h in = 1 s i g n i f y i n g t h e answer " y e s " ,

and in

= -1 s i g n i f y i n g t h e answer "no".

Then a " p a r t i a l h i s t o r y " i s a s t r i n g o f t h e form (kl,il; where kl, and i l , i

k2,.

2'"'

k2,i2;

...,

kn > i n )

.,

a r e l a b e l s of c o n s e c u t i v e q u e s t i o n s , a r e answers t o them.

If D stands f o r the set of a l l p o s s i b l e f i n a l scores

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CHAPTER 3

t h e n a dynamic q u e s t i o n n a i r e may be i d e n t i f i e d

with

a s y s t e m o f t h e form

where k l

K i s t h e l a b e l of t h e i n i t i a l q u e s t i o n t o

be a s k e d , and f n : Hn

-+

K

LJ

D

w i t h Hn b e i n g t h e c l a s s of a l l h i s t o r i e s s a t i s f y i n g t h e following conditions:

a . They a r e of l e n g t h n; b.

then f o r every r

I f ( k l Y i l ; , . . . , k ,i-.) Hn,

(which i m p l i e s t h a t ( k l , i l , . . c. I f

(kl,il,...,kn,in)

i and j

.,kr,ir)

n

E Hr);

t h e n ki # k

Hn,

, 21. a based on e i t h e r d l o r d 2 ) . THEOREM 1.

I f A =a

€3

g& B

=b

( h e r e h may be

c , then

A =a+b- 1 c

.

I n d e e d , l e t f A , f and f c be t h e membership f u n c t i o n s B o f A , B and C . Then we have di(A,B) 1 - a and d i ( B , C ) 5 1 - b . The a s s e r t i o n now f o l l o w s from t h e t r i a n g l e i n e q u a l i t y f o r the metric di.

1-c.

t h e n d(A',A")

Since d(A',B')c

1 - a a n d d ( A " , B " ) 6 1-b f o r a l l B ' c< F ' and B "

>

w e must have d ( B ' , B " ) l-c-(l-a)-(l-b) which i s e q u i v a l e n t t o t h e a s s e r t i o n .

S_

F",

= 1-Cc-(atb)],

We have a l s o

THEOREM b-bundle

5. Let

F'

, F"

be r e p s e c t i v e l y an a-bundle

i n F . I f t h e r e e x i s t s A ' & F' and A"

t h ' a t d(A1,A")

c

F" s u c h c , t h e n F l u F" f o r m s a n ( a t b t c ) - b u n d l e .

I n d e e d , f o r any B '

4

F ' a n d B" C F " we may w r i t e , u s i n g

t h e t r i a n g l e i n e q u a l i t y , d(B' ,B")

+

and

d(A",B")

('

d(B' , A f ) t d(A' ,A")

a t b t c , which p r o v e s t h e a s s e r t i o n .

We may now f o r m u l a t e t h e f o l l o w i n g c o r o l l a r y . COROLLARY.

If t h e a - b u n d l e F 7 a n d b - b u n d l e F" a r e n o t

disjoint, then Ff

F" i s a n ( a t b ) - b u n d l e .

F o r t h e p r o o f i t s u f f i c e s t o n o t e t h a t w e may t h e n t a k e i n t h e preceding theorem c = 0 .

3.2. Structure of concepts L e t now L b e a s e t , whose e l e m e n t s w i l l be i n t e r p r e t e d

as l a b e l s , or names, o f c o n c e p t s . We assume t h a t e l e m e n t s o f L may b e n e g a t e d , a n d j o i n e d w i t h c o n j u n c t i o n

269


and d i s j u n c t i o n . The s e t L need n o t , however, be c l o s e d under t h e s e o p e r a t i o n s , i . e . t h e c o n d i t i o n s u,v C L do L or u v v L. n o t , i n g e n e r a l , e n t a i l -u 6 L , u A v T h i s means t h a t a c o n c e p t formed as a c o n j u n c t i o n of two c o n c e p t s b e a r i n g a name need n o t i t s e l f b e a conc e p t b e a r i n g a name. Next, l e t T: F +L

(3.6)

where @ L i s a s p e c i a l symbol s i g n i f y i n g “no name”. The r e l a t i o n T ( f ) = @ means t h a t t h e f u z z y s e t w i t h memb e r s h i p f u n c t i o n f has no name, s i m p l e or composite. It w i l l be a s s u j e d t h a t t h e f u n c t i o n T s a t i s f i e s t h e

following conditions.

L2

I f T ( f ) = u and

L3

If T ( f i )

= ui,

t h e n T[min(fl,

L4

If T(fi)

= ui,

-u E L , t h e n T ( 1 - f )

,..., n ,

i = 1

..., f n ) l =

ulJ\

i = l,...,n,

t h e n T[max ( f l , . . . , f n ) l =

and u 1 4

... A

and ulv ulV

... v

= -u,

...

A

u

n C L,

un ’

... v

un C L ,

u . n

C o n d i t i o n L 1 a s s e r t s t h a t e a c h c o n c e p t w i t h l a b e l from L i s r e p r e s e n t a b l e by a t l e a s t one f u z z y s e t i n F , whil e c o n d i t i o n s L2 - L4 a s s e r t a c e r t a i n c o n s i s t e n c y between o p e r a t i o n s on l a b e l s i n L and s e t - t h e o r e t i c a l

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CHAPTER 3

o p e r a t i o n s on f u z z y s e t s . Given a s e t

AC' F , i t s

d i a m e t e r i s d e f i n e d as

(3.7) We have t h e n THEOREM

6 . Suppose t h a t u, -u L L . Then f o r i

= 1,2

The i d e n t i t y ( 3 . 8 ) f o l l o w s i m m e d i a t e l y from t h e i n v a r i a n c e o f e i t h e r o f t h e m e t r i c s d o r d 2 under n e g a t i o n , 1 as a s s e r t e d i n Theorem 1. If u

/,

v E L, define

Tmin -1 ( U , A V ) =

f

h : h = min ( f , g ) , f 6 T - ' ( u ) , g

and s i m i l a r l y , i f u

-1

TmaX(uv v ) =

1h:

v

C_

CT-'(v)j

L , we put

h = max (f, g )

,

ft

T-'(u)

, g 6 T-l(v)?

From L 3 and L4 i t follows t h a t w e have THEOREM

7.

If u

A

v, u

L.'

v C L,

then

For t h e c o n j u n c t i o n and d i s j u n c t i o n , w e have ( i n c a s e of m e t r i c d l ) , t h e f o l l o w i n g t h e o r e m , c o r r e s p o n d i n g t o

PSYCHOLOGICAL PROBUMS OF CONSTRUCTION

271

Theorem 3 . THEOREM 8 .

Let u ,

v

and

Similarly, i f u v v E L,

u,,

v b e l o n g t o L . Then

then

Proof. We s h a l l prove t h e f i r s t i n e q u a l i t y , t h e p r o o f o f t h e second b e i n g a n a l o g o u s .

Observe f i r s t t h a t -1 ( u n v ) i s w e l l d e f i n e d , i n view of t h e assumption Tmin t h a t u A v (i L . By L 3 we have

Let D [T-'(u)] = a , D,[T-'(v)] = b . Then for a l l f , f ' -1 i n T ( u ) and a l l x we have f ' ( x ) - a g f ( x ) 5 f ' ( x ) t a , and s i m i l a r l y , g ' ( x ) - b e g ( x ) : g ' ( x ) t b f o r 1 a l l x and a l l g , g ' < T- ( v ) . We can t h e r e f o r e w r i t e min [ f ( x ) , g ( x ) ] L min [ f ' ( x ) t a , g ' ( x ) t b J

I n a s i m i l a r way we g e t

which y i e l d s lmin [ f ( x ) , g ( x ) l

-

min [ f ' ( x ) , g ' ( x ) ] ' < m a x ( a , b ) .

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CHAPTER 3

Taking supremum f i r s t w i t h r e s p e c t t o x , and t h e n w i t h r e s p e c t t o f , f t ,g,gl we o b t a i n t h e a s s e r t i o n . L e t now u t L and l e t us c o n s i d e r a l l d e c o m p o s i t i o n s

u = u

fi

u2

1 any k w i t h 2

A

) .

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CHAPTER 4

One c a n p r o v e t h a t t h e c l a s s o f a l l maximal masks det e r m i n e s u n i q u e l y t h e c l a s s o f a l l masks ( f o r t h e p r o o f , s e e Nowakowska 1 9 7 3 ) . The c h o i c e o f a mask i n a c t u a l s i t u a t i o n d e t e r m i n e s t o some e x t e n t t h e " l e v e l o f t r u t h " t h a t may be approa c h e d , and t h e r e f o r e a mask a l s o imposes some c o n s t r a i n t s on t h e d e r i v e d c o n c e p t s of f a i t h f u l n e s s and e x a c t n e s s . T h i s t y p e o f i n t e r r e l a t i o n o f c o n c e p t s prov i d e s d e e p e r i n s i g h t i n t o t h e f o u n d a t i o n o f knowledge f o r m a t i o n and b a s i c f u n c t i o n s o f s c i e n c e . A t t h e end, i t i s w o r t h w h i l e t o c o n s i d e r s t i l l a n o t h e r

c l a s s of s i g n s , i n t h e c l a s s i f i c a t i o n o f P e i r c e c a l l e d i n d e x e s . Here t h e i s s u e c o n c e r n s t h e c a u s a l r e l a t i o n between o b j e c t s . When o b j e c t A i s a c a u s e o f o b j e c t B , t h e n B i s a s i g n o f A . From t h e f o r m a l p o i n t o f v i e w , t o some p a i r s o f o b j e c t s c o n n e c t e d by c a u s a l laws, there a r e assigned explanatory sentences. Thus, for e a c h e x p l a n a t o r y s e n t e n c e ( a l a w ) , one may s p e a k o f a s e t o f o b j e c t s which s a t i s f y t h i s l a w . T h i s g i v e s t h e domain o f a l a w , and a l s o i t s i n v e r s e domain, and t h e i n t e r s e c t i o n o f t h e l a t t e r c o r r e s p o n d s t o a c o n f i g u r a t i o n o f o b j e c t s . I n o t h e r words, a n o b j e c t may be d e s c r i b e d by many laws; more p r e c i s e l y , b y a l l t h o s e laws, t o whose domains i t b e l o n g s . Coherent s e t s o f laws form t h e o r i e s o f some s e t s o f o b j e c t s , s u c h as s i t u a t i o n s or phenomena. The d e c i s i o n whether or n o t an o b j e c t b e l o n g s t o a

domain o f some law depends i n many c a s e s on t h e o b s e r v a b i l i t y r e s t r i c t i o n s , which were d i s c u s s e d above. O b s e r v a t i o n of an o b j e c t i s r e l a t i v e t o a s i t u a t i o n ,

FORMAL SEMIOTICS

3 03

t o measurement p r o c e d u r e , and mappings i n t h e n u m e r i c a l and l i n g u i s t i c domains, t h a t a r e t h e p r e r e q u i s i t e f o r f o r m u l a t i n g t h e o b s e r v a t i o n a l judgments. The l a t t e r mappings a l l o w u s t o r e p l a c e t h e o p e r a t i o n s on o b j e c t s b y o p e r a t i o n s on symbols -- e i t h e r i n m a t h e m a t i c a l domain, o r i n l i n g u i s t i c s p a c e o f c l a s s i f i c a t i o n s o r d e s c r i p t i o n s . T h i s mapping -- and i t s c h a r a c t e r -i s d e t e r m i n e d b y t h e c l a s s o f t r a n s f o r m a t i o n r u l e s of o b s e r v a t i o n a l s y s t e m s , t h a t impose c o n s t r a i n t s on i t s adequacy and i n v a r i a n c e . I n o t h e r words, one may d e a l w i t h t h e whole c l a s s o f d e s c r i p t i o n s o r n u m e r i c a l a s s i g n m e n t s , from which one chooses onl y t h o s e t h a t are c h a r a c t e r i z e d by s u f f i c i e n t l e v e l of i n v a r i a n c e . An i m p o r t a n t problem here i s t h e c h o i c e o f a d e q u a t e r e p r e s e n t a t i o n o f o b s e r v a t i o n s of o b j e c t s a r e r e l a t i o n s between t h e m . Both f o r t h e r e p r e s e n t a t i o n i n n u m e r i c a l s p a c e and i n l i n g u i s t i c s p a c e , t h e c r i t e r i o n o f adequacy i s v a l i i d i t y , and a l s o parsimony: s i m p l i c i t y , e f f i c i e n c y , e t c . The s y s t e m o f c a u s a l r e l a t i o n s between o b j e c t s , i n t r o -

duced p r i m a r i l y f o r o b j e c t s , must b e e x t e n d e d t o s e t s of o b j e c t s t h a t c o n s t i t u t e s i t u a t i o n s , t h e c a u s a l rel a t i o n s between networks of o b j e c t s , s e t s o f s e n t e n c e s d e s c r i b i n g f a c t s o r laws, c l a s s e s o f p r o c e d u r e s , num e r i c a l and l i n g u i s t i c mappings, and c o n d i t i o n s f o r invariance. The s y s t e m s o g e n e r a l i z e d a l l o w s US t o i n t r o d u c e t h e

g e n e r a l i z e d d e s c r i p t i o n s o f o b j e c t s , i n form of laws or theories. T h i s i s a dynamical s y s t e m , n o t o n l y i n t h e s e n s e of

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changes o f o b j e c t s and t h e c o r r e s p o n d i n g changes of d e s c r i p t i o n s , p o s s i b i l i t i e s of continuous c o l l e c t i o n o f i n f o r m a t i o n , b u t a l s o b e c a u s e of some m o d i f i c a t i o n s o f t h e o b s e r v a t i o n a l system -- f o r i n s t a n c e due t o i n t r o d u c t i o n o f new measurement tools, or improvement o f t h e r e p r e s e n t a t i o n s ( e . g . by i n t r o d u c i n g a new model) t h a t i n d u c e t h e e n r i c h e d and improved d e s c r i p t i o n s o f r e l a t i o n s between o b j e c t s . I n consequence, i t l e a d s t o e l i m i n a t i o n o f e r r o r s , e x t e n s i o n o f o l d and i n t r o d u c t i o n o f new t h e o r y , f o r t h e e x t e n d e d s e t of observational sentences. To sum up, t h e dynamics of r e l a t i o n s between o b s e r v a t i o n s and c o l l e c t e d knowledge o f o b j e c t s on t h e one hand, and p r o d u c t i o n o f t h e i r c o p i e s on a g i v e n l e v e l o f r e p r e s e n t a t i o n , shows t h e m e t h o d o l o g i c a l and i n t e g r a t i v e r o l e of semiotic conceptions of t h e foundation of cognition.

2. MULTIMEDIAL C O M M U N I C A T I O N I n t h i s s e c t i o n , t h e a n a l y s i s w i l l c o n c e r n t h e problems of communication c a r r i e d o u t s i m u l t a n e o u s l y on s e v e r a l media, s u c h as v e r b a l medium, medium o f f a c i a l e x p r e s s i o n s , medium o f g e s t u r e s , and s o f o r t h . These t o p i c s were f i r s t f o r m a l l y a n a l y s e d i n Nowakowska (1977, 1981, 1983). Here an e x t e n d e d and e n r i c h e d model

1978,

o f s i t u a t i o n w i l l be shown, c a p t u r i n g t h e i n h e r e n t f u z z i n e s s o f b o t h t h e communication u n i t s , as w e l l as t h e meanings of t h e s e u n i t s .

305

FORMAL SEMIOTICS

2.1.

Formal a p p r o a c h t o s y n t a x o f m u l t i m e d i a 1 communic-

a t i o n languages We s h a l l c o n s i d e r communication c a r r i e d o u t s i m u l t a n e o u s l y on a number of media, l a b e l e d m l , m 2 , ,mr. The c h o i c e o f media f o r a n a l y s i s depends on t h e g o a l

...

o f s t u d y . For i n s t a n c e , one may r e g a r d t h e arm movements as a s e p a r a t e medium; however, f o r a d e s c r i p t i o n o f I n d i a n d a n c e s , s a y , one s h o u l d c o n s i d e r movements o f e a c h f i n g e r s e p a r a t e l y as c a r r i e d on some medium. With e a c h medium we a s s o c i a t e s e t o f s t a t e s ( o f t h e r e l e v a n t p h y s i c a l s y s t e m , s u c h as arm, f a c e , e t c . ) L e t S . be t h e s e t o f s t a t e s c h a r a c t e r i s t i c f o r medium mi. 1

We s h a l l assume t h a t t h e s t a t e s a r e d e f i n e d i n a d i s j u n c t i v e and e x h a u s t i v e w a y , s o t h a t a t e a c h t i m e e x a c t l y one o f t h e s t a t e s t a k e s p l a c e . The s t a t e s on i - t h medium mi w i l l g e n e r a l l y be d e n o t e d b y s i , or s i ( t ) , i f w e need t o s p e c i f y t h e s t a t e a t t i m e t . F u r t h e r , w i t h e a c h medium m we s h a l l a s s o c i a t e i t s i (fuzzy) vocabulary V i n form o f a f a m i l y of f u z z y i’ s u b s e t s of Si. The e l e m e n t s o f V . w i l l be d e n o t e d by 1 v i1, v i2, and t h e same symbol w i l l be udes t o d e n o t e

...,

k v i ( s ) , s C Si, i s k t h e d e g r e e t o which t h e s t a t e s b e l o n g s t o t h e s e t vi.

t h e membership f u n c t i o n , s o t h a t

L e t u s now f i x some q w i t h $ 4 q 5 1. Given si

let

and l e t

S

i’

3 06

CHAPTER 4

n

i ,q

( s i ) = # o f elements i n B

i ,q

(si)

.

(2.2)

Thus, B ( s . ) i s t h e s e t of a l l e l e m e n t s o f t h e vocai,q 1 b u l a r y Vi o f medium m which a r e e x p r e s s e d by s i i n i degree at l e a s t q. k If max v . ( s . )

4

remains n e u t $ a l (on l e v e l q ) , and t h a t i t t h e n e x p r e s s e s t h e 1

p a u s e It

1

q , we s h a l l s a y t h a t medium m

i

#i.

We have h e r e t h r e e p o s s i b l e c a s e s : t h e s e t Bi (si) ,q may be e m p t y ; i t may c o n t a i n j u s t one e l e m e n t ; and i t may c o n t a i n more t h a n one e l e m e n t . I n t h e f i r s t c a s e , i n t h e second c a s e , w e s h a l l medium m e x p r e s s e s # i i’ s a y t h a t it i s unequivocal (on l e v e l q ) , w h i l e i n t h e t h i r d c a s e , i t i s ambiguous ( o n l e v e l 9 ) .

-

L e t u s now c o n s i d e r t h e v e c t o r 2 = ( s l,.

..,sr)

of s t a t e s o f c o n s e c u t i v e media, o c c u r r i n g a t some t i m e t. Some o f t h e s e media w i l l e x p r e s s # i’ w h i l e some o t h e r s w i l l e x p r e s s c e r t a i n elements o f t h e i r r e s p e c t i v e vocabulark i e s . L e t u s w r i t e B i ,q ( s i ) = i f max v i ( s i ) < q , 1 k and l e t u s a s s i g n t o 2 t h e s e t

F#.l

The number o f e l e m e n t s i n Q(g) i s t h e n t h e p r o d u c t o f t h e s i z e s of components, i . e . we have

If n

(5;)

g -

= 1, t h e s t a t e

2 i s u n e q u i v o c a l o r unambiguous,

FORMAL SEMIOTICS

307

( a t t h e l e v e l q ) , s i n c e e a c h medium must e i t h e r e x p r e s s #. or e x p r e s s some unique element o f t h e v o c a b u l a r y V i ' 1 On t h e o t h e r hand, i f n (s) > 1, t h e n a t l e a s t one q -

medium e x p r e s s e s more t h a n one element o f i t s vocabulary.

The number o f e l e m e n t s i n t h e s e t Q ( s ) i n c r e a s e s r a p i d q l y when more and more media become ambiguous. T h i s phenomenon, a t l e a s t i n t h e o r y , would make communication i m p o s s i b l e i n c a s e o f even moderate a m b i g u i t y on some o f t h e media. However, w e s h a l l show t h a t s e m a n t i c s e l i m i n a t e s most o f t h e a m b i g u i t y .

D e f i n i t i o n of m u l t i m e d i a 1 l a n g u a g e . We may now i n t r o d u c e f o r m a l l y t h e system which w i l l a l l o w u s t o d e f i n e f u r t h e r n o t i o n s . Let 2.1.1.

be t h e c l a s s o f a l l s t a t e - v e c t o r s .

By

a s y n t a c t i c s t r u c t u r e o f m u l t i m e d i a 1 communication,

w e s h a l l mean a system

h

where S i s a c l a s s o f s t r i n g s 2 ( 1 ) , 2 ( 2 ) , . . . , 2 ( T ) , T = 1 , 2 , ..., and g ( n ) f5S f o r n = 1,..., T.

*

with

I n t u i t i v e l y , e l e m e n t s o f S w i l l be t h e a d m i s s i b l e s t r i n g s o f s t a t e s , as o b s e r v e d a t some f i x e d t i m e s , s a y t = 1,2,...,T f o r some T .

3 08

CHAPTER 4

Now, w i t h e a c h q and e a c h element o f S

*

we may a s s o c i a -

t e a c l a s s o f s t r i n g s o f v e c t o r s o f elements o f t h e v o c a b u l a r i e s o f media. F o r m a l l y , l e t V; = Vi

V

=

V'1 n

...

s o t h a t e l e m e n t s or where vi:

€

u

and l e t

V;

v

(2.7) k

,. . . ,vpr)

a r e v e c t o r s v_ = ( v k l l

Vl.

n Given q w i t h $ 4 q f 1 and a s t r i n g s = ( ~ ( l. ) . . ,, g (T)) w e a s s i g n t o i t a s e t o f s t r i n g s o f t h e form

where

(2.9) with

viki(n) 6 Bi

,q

(s.(n)). -1

The c l a s s o f a l l s t r i n g s o b t a i n e d i n t h i s way w i l l b e

c a l l e d t h e m d t i m e d i a l l a n g u a g e o f communication a c t ions. To b e t t e r v i s u a l i z e t h e c o n s t r u c t i o n , one may p r o c e e d

*

as f o l l o w s . The s t r i n g s = ( g ( l ) ,. . . , g (T)) h a s v e c t o r s of s t a t e s on p a r t i c u l a r media as i t s e l e m e n t s . It may t h e r e f o r e b e r e p r e s e n t e d as a m a t r i x

3 09

FORMAL SEMIO’ITCS

I .

...

(2.10)

where s . ( n ) i s t h e s t a t e on medium m 1

i

at t i m e t = n.

If now q i s f i x e d , t h e n w i t h t h e s t a t e s i ( n ) on i - t h

medium t h e r e i s a s s o c i a t e d one or more e l e m e n t s o f t h e v o c a b u l a r y V’ i . e . t h e o r i g i n a l v o c a b u l a r y , e n r i c h e d i’ by the pause #i. Thus, e a c h element o f m a t r i x ( 2 . 1 0 ) may b e r e p l a c e d by t h e a p p r o p r i a t e element o f V 1 * t h i s i’ yields a matrix

...

vk 1l ( T )

...

vk 2 2( T )

(2.11)

Whenever t h e r e i s a m b i g u i t y for some s t a t e s . ( n ) , t h e 1

c o r r e s p o n d i n g element may b e f i l l e d w i t h any o f t h e * e l e m e n t s of t h e s e t B ( s i ( n ) ) . Thus, a s t r i n g s iq w i l l t y p i c a l l y have s e v e r a l m a t r i c e s o f t h e form ( 2 . 1 1 ) assigned t o it. Any column which may a p p e a r i n t h e m a t r i x ( 2 . 1 1 ) w i l l be c a l l e d a f e a s i b l e u s i t o f m u l t i m e d i a 1 communication a c t i o n s , and t h e c l a s s of a l l p o s s i b l e i - t h rows i n m a t r i c e s ( 2 . 1 1 ) w i l l be c a l l e d t h e language Li of i - t h medium.

CHAPTER 4

310

A s a n example of a u n i t , c o n s i d e r " G r e e t i n g " , performed

s i m u l t a n e o u s l y on v e r b a l medium, medium of f a c i a l expr e s s i o n s , body movements, and s o on. N e x t t o i t , w e g i v e a n example o f an i n a d m i s s i b l e u n i t ( n o t b e l o n g i n g t o t h e language of a c t i o n s on communication m e d i a ) . medium

I'

verbal

s a y i n g "Hello"

Say i n g " H e 1l o

body movements

bow

Say i n g " H e 11o I'

gestures

t i p p i n g t h e hat

tipping the hat

facial e x p r e s si o n s

smile

tipping the hat

......

Greet i ng

....

inadmissible unit

....

Here t h e i n a d m i s s i b l e u n i t has simply i m p o s s i b l e a c t i o n s on v a r i o u s media, i . e . a c t i o n s a s s o c i a t e d w i t h d i f f e r e n t media. The i n t u i t i o n s of language of communication a c t i o n s

c o n c e r n s a d m i s s i b i l i t y n o t of u n i t s , but o f s t r i n g s o f u n i t s , even if e a c h s t r i n g s e p a r a t e l y i s a d m i s s i b l e . A s t r i n g ( m a t r i x of a c t i o n s , or a l t e r n a t i v e l y , a bunch o f s i m u l t a n e o u s s t r i n g s on v a r i o u s media) i s a d m i s s i b l e , i f i t i s p h y s i c a l l y p o s s i b l e t o p e r f o r m , and a l s o i f i t r e p r e s e n t s a fragment o f b e h a v i o u r which i s meani n g f u l , s o c i a l l y a c c e p t a b l e , and s o o n . I t i s n o t n e c e s s a r y t o g i v e h e r e more e x a c t i n t e r -

FORMAL SEMIOTICS

31 1

p r e t a t i o n , mainly because f l e x i b i l i t y of t h i s i n t e r p r e t a t i o n g i v e s t h e model more a p p l o c a b i l i t y . F o r i n s t a n c e , i f one w a n t s t o d e s c r i b e and a n a l y s e some communication a c t i o n s of s p e c i a l t y p e , s u c h as r o y a l c o u r t p r o t o c o l , J a p a n e s e t e a ceremony, e t c . , t h e s e t o f a l l a d m i s s i b l e a c t i o n s , a nd h en ce a l s o t h e s e t o f s t r i n g s a d m i s s i b l e on p a r t i c u l a r media ( l a n g u a g e s L i ) h a v e t o be a p p r o p r i a t e l y d e f i n e d i n each context (as conforming t o c o u r t protocol, etc.). The f a c t t h a t n o t a l l m a t r i c e s ( 2 . 1 1 )

are o b t a i n a b l e makes i t p o s s i b l e t o d e f i n e v a r i o u s from s t r i n g s t y p e s o f c o n s t r a i n t s b etween e l e m e n t s ( a c t i o n s on t h e same o r v a r i o u s m e d i a ) . Th u s, two s u c h r e l a t i o n s , o f e n f o r c i n g a nd e x c l u s i o n , may be d e f i n e d a s f o l l o w s .

* s

F i r s t l y , w he nev er we h av e v ki( n ) = x i

=+

k.

v J(n) j

z

y

for a l l n,

t h e n a c t i o n x on medium mi e x c l u d e s s i m u l t a n e o u s a c t i o n y on medium m Similarly, i f j' k.

ki

vi

v J ( n ) = y f o r some n, j

(n) = x

t h e n a c t i o n x on medium mi e n f o r c e s s i m u l t a n e o u s a c t i o n y o n medium m j'

These r e l a t i o n s c o n c e r n t h e same t i m e , i . e . t h e y r e f e r

t o t h e same u n i t . I n a s i m i l a r way, o n e may s a y t h a t the condition v ki( n ) = x i

+

k. v J ( n t c ) = y f o r some n j

312

CHAPmR 4

means t h a t a c t i o n x on medium m i e n f o r c e s a c t i o n y on medium m a t c u n i t s o f t i m e l a t e r . S i m i l a r c o n d i t i o n j h o l d s f o r e x c l u s i o n on t h e same medium a c r o s s t i m e . F i n a l l y , one may have e n f o r c i n g or e x c l u s i o n a c r o s s b o t h media and t i m e , e . g .

means t h a t a c t i o n x on medium m medium m

j

i

e x c l u d e s a c t i o n y on

a t c u n i t s of time later.

It i s c l e a r t h a t t h e r e l a t i o n o f e n f o r c i n g i s t r a n s i -

t i v e , b u t n o t n e c e s s a r i f y symmetric, w h i l e t h e r e l a t i o n o f e x c l u s i o n i s symmetric, b u t n o t n e c e s s a r i l y transitive. Somewhat more g e n e r a l l y , w e may c o n s i d e r t h e c o n c e p t o f p a r a s i t i c i t y , b o t h w i t h i n a u n i t o f communication a c t i o n s , and w i t h i n a s t r i n g of s u c h u n i t s . I n c a s e o f u n i t , a p a r t i c u l a r combination o f a c t i o n s on media i s p a r a s i t i c , i f t h e unit i s not admissible regardless of a c t i o n s on t h e r e m a i n i n g media. O b v i o u s l y , i f two a c t i o n s e x c l u d e one a n o t h e r , t h e n t h e y b o t h j o i n t l y are parasitic S i m i l a r l y , p a r a s i t i c i t y may be d e f i n e d f o r s t r i n g s o f

u n i t s : a s t r i n g u i s p a r a s i t i c , i f vuw does n o t b e l o n g t o t h e language ( i . e . i s n o t a d m i s s i b l e ) f o r any s t r i n g s v and w ( t h e s t r u c t u r e o f t h i s d e f i n i t i o n i s t h e same f o r t h e c a s e o f m u l t i m e d i a 1 and u n i m e d i a l strings). P a r a s i t i c i t y p r o v i d e s u s e f u l i n f o r m a t i o n about t h e

313

FORMAL SEMIOTICS

s y n t a c t i c c o n s t r a i n t s i n t h e language; i n t u i t i v e l y , t h e more p a r a s i t i c s t r i n g s , t h e " r i c h e r " i s t h e s t r u c t u r e , i . e . t h e more s t r i n g s a r e i n a d m i s s i b l e . P r o b a b i l i s t i c i n t e r p r e t a t i o n . Thus f a r , we used * t h e c o n c e p t of t h e s e t S of s t r i n g s o f s t a t e v e c t o r s * * t o d e f i n e a mapping which t o e a c h s t r i n g s i n S a s s i g n s a s e t o f m a t r i c e s o f t h e form ( 2 . 1 1 ) . 2.1.2.

*

I n f a c t , v a r i o u s e l e m e n t s o f S o c c u r w i t h v a r i o u s prob a b i l i t i e s . T h i s means t h a t some s t r i n g s o f m u l t i m e d i a 1 * a c t i o n s o c c u r more f r e q u e n t l y t h a n o t h e r s . L e t P ( s ) * be t h e p r o b a b i l i t y o f o c c u r r e n c e o f t h e s t r i n g s

.

Denote g e n e r a l l y by f t h e mapping which which a s s i g n s

*.

t h e s e t of m a t r i c e s o f t h e form ( 2 . 1 1 ) t o s t r i n g s s

Then t o e v e r y c l a s s M of m a t r i c e s ( 2 . 1 1 ) one may assign the probability (2.12)

that i s , (2.13)

P(M) = P [ f - ' ( M ) ] .

T y p i c a l l y , t h e s e t M of i n t e r e s t w i l l be c l a s s e s o f m a t r i c e s ( 2 . 1 1 ) which s a t i s f y some c o n d i t i o n s , e . g . have some s p e c i f i c c o m b i n a t i o n s o f a c t i o n s on s p e c i f i c media. Moreover, t h e s t r i n g s o f a c t i o n s on p a r t i c u l a r media, i . e . e l e m e n t s o f l a n g u a g e s L i , may b e r e g a r d e d

a s s t o c h a s t i c p r o c e s s e s of some s o r t , w i t h p r o b a b i l i t y d i s t r i b u t i o n s indhced b y t h e p r o b a b i l i t y d i s t r i b u t i o n * P ( s ) . T h u s , a s t r i n g v i n L w i l l have t h e p r o b a b i -i

lity

i

CHAPTER 4

3 14

P ( ~ . )=

?(M:

i - t h row of M i s

1

Xi),

e t c . N a t u r a l l y , t h e p r o c e s s e s o f d i f f e r e n t media a r e strongly interrelated. I n f a c t , these interrelations, t h a t i s , a t e n d e n c y o f j o i n t a p p e a r e n c e o f some comb i n a t i o n s o f a c t i o n s on v a r i o u s media, a r e o f t e n s o s t r o n g t h a t t h e y may s e r v e as i n d i v i d u a l c h a r a c t e r i s t i c s o f a p e r s o n . T h i s s e r v e s as a b a s i s f o r i m i t a t i o n s , and a c c o u n t f o r t h e f a c t t h a t i t i s g e n e r a l l y p o s s i b l e t o r e c o g n i z e who i s b e i n g i m i t a t e d . From t h e d e s c r i p t i o n o f u n i t s g i v e n t h u s f a r i t f o l l o w s t h a t t h e y are some s o r t o f " m u l t i - w o r d s " , which form f u n c t i o n a l e n t i t i e s . The a d m i s s i b i l i t y i n a g i v e n l a n guage a l l o w s us t o b u i l d grammatical s t r i n g s o f u n i t s . One can c o n s i d e r , i n a d d i t i o n to t h e i n h e r e n t s t r u c t u r e of s u c h m u l t i - w o r d s , some p r o p e r t i e s which may be obs e r v e d i n a s t r i n g . Thus, i n s i t u a t i o n s o f f o r m i n g mult i - e x p r e s s i o n s ( s t r i n g s ) some a c t i o n s may a p p e a r c y c l i c a l l y , w h i l e o t h e r s may r e p r e s e n t p e r s e v e r a n c e o f a c t i o n s o f some t y p e s on c e r t a i n media. T h i s may o c c u r , i f some a c t i o n i s k e p t c o n s t a n t t h r o u g h s e v e r a l u n i t s , w h i l e a c t i o n s on o t h e r media change. T h i s means some c o u p l i n g o f u n i t s . F o r m a l l y , two n e i g h b o u r i n g u n i t s

aEe c o u p l e g , i f t h e r e e x i s t s i n d e x i such t h a t we have v i ( n ) = v i ( n t l ) for soge n . For c y k l i c a l a p p e a r e n c e , i i t h e c o n d i t i o n would be v i(n+km) = v i i ( n ) f o r k = 1,2, i . . and some rn and n .

.

The p r o c e s s o f c o n t r o l o f

performance c o n s i s t s o f v a r i o u s o p e r a t i o n s on u n i t s of "conventional" c h a r a c t e r , s u c h a s o m i s s i o n o r i n s e r t i o n o f some a c t i o n s , i n t e r c h a n g e of a c t i o n s , e t c .

FORMAL SEMIOTICS

315

S i n c e l a n g u a g e s o f communication have o f t e n l o c a l char a c t e r , r e s t r i c t e d t o some s i t u a t i o n , and g i v e a desc r i p t i o n o f r e g u l a r i t i e s of a c t i o n s t r u c t u r e i n such s i t u a t i o n s , f o r an a c t o r i t i s i m p o r t a n t t o know how t o p a s s from one language t o a n o t h e r , t h u s combining s t r i n g s from v a r i o u s a c t i o n l a n g u a g e s . Such c o m b i n a t i o n s may i n v o l v e embedding ( s i m u l t a n e o u s performance o f s t r i n g s from two or more l a n g u a g e s ) , and c o n c a t e n a t i o n s , i . e . s t a r t i n g new s t r i n g o n l y a f t e r c o m p l e t i n g t h e p r e c e d i n g one. C o n c a t e n a t i o n s o f l a n g u a g e s , s a y L1 and L2, a r e d e f i n e d f o r m a l l y as

L1L2 = f u u ' : u c L1, u '

f

LJ.

grammar, i . e . s y n t a x and l o g i c o f s u c c e s s i o n o f s t r i n g s from d i f f e r e n t l a n g u a g e s of a c t i o n s , one c o u l d e x t e n d t h e p r o b a b i l i s t i c i n t e r p r e t a t i o n t o algebra of languages, where t h e l a t t e r would p l a y t h e r o l e o f words i n t h e v o c a b u l a r y . One c o u l d t h e n form a monoid b u i l t o u t o f l a n g u a g e s . The a d m i s s i b l e s t r i n g s would form here a "super-language" of communication a c t i o n s , t h a t c o u l d be i d e n t i f i e d w i t h s e t o f h i s t o r i e s o f a c t i o n s , or dual l y , h i s t o r i e s o f e v o l v i n g meanings g e n e r a t e d by p e r forming s t r i n g s of a c t i o n s . F o r f o r m a l d e s c r i p t i o n o f some i n t e r - s t r u c t u r a l

2.2.

Semantics o f m u l t i m e d i a 1 l a n g u a g e s

Let u s now e n r i c h t h e system of t h e p r e c e d i n g s e c t i o n by i n c l u s i n g i t s s e m a n t i c s . We s h a l l d e n o t e simply by

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L t h e c l a s s o f a l l m a t r i c e s o f t h e form ( 2 . 1 1 ) which

*

a r e o b t a i n a b l e from e l e m e n t s o f S , and by P t h e probab i l i t y measure on t h e c l a s s o f t h e s e m a t r i c e s , as indu* ced by t h e measure P on S Moreover, i t w i l l be conv e n i e n t t o d e n o t e by C t h e c l a s s o f a l l p o s s i b l e columns i n m a t r i c e s ( 2 . 1 1 ) , s o t h a t e a c h element o f L i s a s t r i n g c1c2.. cT of columns.

.

.

We s h a l l now c o n s i d e r a system

where C , L and p r o b a b i l i t y measure P have a l r e a d y been d e f i n e d , Z i s t h e s e t o f a l l p o s s i b l e meanings, and g: L x Z 4C O , l l

(2.15)

i s t h e f u n c t i o n d e s c r i b i n g fuzzy semantics, i . e . a

f u n c t i o n which to e v e r y meaning z and s t r i n g v i n L a s s i g n s t h e v a l u e g ( v , z ) , r e p r e s e n t i n g t h e d e g r e e , to which t h e meaning of s t r i n g v i s z . S i n c e t h e language L o f communication a c t i o n s c o n t a i n s , among o t h e r s , u t t e r a n c e s , t h e s e t Z of a l l meanings must c o n t a i n a l l p o s s i b l e meanings o f u t t e r a n c e s . The main o b j e c t o f i n t e r e s t h e r e w i l l b e , however, t h o s e meanings which a r e c a r r i e d m o s t l y b y n o n - v e r b a l media; examples o f s u c h meanings may b e " F r i e d l i n e s s " , "Annoyance", "Sco.rn" , " D i s l i k e " , and s o on. Most o f t h e s e meanings have t h e p r o p e r t y t h a t t h e y may appear i n m o d a l i t i e s ; t h u s t h e set Z w i l l c o n t a i n meanings s u c h as " M i l d annoyance", "Extreme annoyance", and so on.

FORMAL SEMIOTICS

317

A group o f meanings (sememes) r e p r e s e n t i n g v a r i o u s

d e g r e e s o f a meaning z w i l l be c o n s i d e r e d a s f o r m i n g a l i n e a r s c a l e , i . e . t h e y w i l l be assumed t o be o r d e r e d a l o n g a c e r t a i n continuum s p e c i f i c f o r a g i v e n meaning. G e n e r a l l y , e a c h s t r i n g v may have more t h a n one meaning associated with it; thus,

i s t h e s e t o f a l l meanings z which v e x p r e s s e s i t t h e d e g r e e a t l e a s t h. S i m i l a r l y , t h e same meaning z may be o f t e n e x p r e s s e d b y more t h a n one s t r i n g of communication actions, so that

i s t h e s e t o f a l l s t r i n g s i n L which e x p r e s s z i n t h e degree a t l e a s t h.

To b e t t e r a p p r e c i a t e t h e i n t e r p r e t a t i o n o f g , assume t h a t i t a s s i g n s meanings n o t o n l y t o s t r i n g s , b u t a l s o t o u n i t s . T h i s means, t e c h n i c a l l y s p e a k i n g , t h a t we assume t h a t columns i n m a t r i c e s ( 2 . 1 1 ) a r e a l s o e l e ments of L . The assumption o f e x i s t e n c e o f f u z z y membership f u n c t i o n g f o r s i n g l e u n i t s (and n o t o n l y for s t r i n g s ) r e q u i r e s t h e c l a r i f i c a t i o n of two p o i n t s : (1) t o which e x t e n t , and how, i t narrows down t h e po-

s s i b i l i t y of choice of u n i t s ?

( 2 ) how t o i n t e r p r e t t h e c o n c e p t o f f u n c t i o n g i n t h e

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co nt ext of p r e s e n t c o n s i d e r a t i o n s ? Regarding t h e f i r s t q u e s t i o n , t h e answer i s t h a t i n o r d e r t o g e t t h e p r o p e r i n t e r p r e t a t i o n , one cannot d i s t i n g u i s h u n i t s which a r e t o o small: e a c h u n i t must b e l a r g e enough t o c a r r y some meaning. Now, t h e r e are s e v e r a l ways i n which one may i n t e r p r e t t h e a s s e r t i o n t h a t g ( c , z ) = p , where c i s some m u l t i medial communicat,ion u n i t , and z i s some meaning. F i r s t l y , i t may mean t h a t p i s t h e f r a c t i o n o f p e r s o n s

who -- when c o n f r o n t e d w i t h u n i t c -- w i l l c l a i m t h a t i t s meaning i s z . S e c o n d l y , z may be one of p o s s i b l e meanings o f c , and p i s s i m p l y t h e f r a c t i o n o f o c c a s i o n s , when i t i s used t o denote z. T h i r d l y , p may be t h e d e g r e e o f c e r t a i n t y ( o f a s p e c i -

f i c p e r s o n ) t h a t when u n i t c i s used on p a r t i c u l a r o c c a s i o n , i t s meaning was z . Finally, there i s also possible a fourth interpretation, connected w i t h t h e f o l l o w i n g c o n c e p t u a l i z a t i o n o f u n i t c . G e n e r a l l y , c c o n s i s t s o f some u t t e r a n c e s , accompani e d b y a c t i o n s on some o t h e r media, such as g e s t u r e s , f a c i a l e x p r e s s i o n s , s u c h as s m i l e , and s o on. T y p i c a l l y t h e r e i s some d e g r e e o f freedom o f p e r f o r m i n g t h e s e a c t i o n s , w i t h i n t h e c o n s t r a i n t s imposed by human body, o c c a s i o n , and s o f o r t h . We may t h e n t h i n k o f u n i t c as b e i n g i n f a c t one o f a f a m i l y o f u n i t s , s a y ex, where x i s some ( p o s s i b l y m u l t i d i m e n s i o n a l ) p a r a m e t e r . Now, d i f f e r e n t c x d i f f e r t o t h e d e g r e e w i t h which t h e y

3 19

FORMAL SEMIOTICS

e x p r e s s meaning z . I n t h i s i n t e r p r e t a t i o n , g ( c , z ) would be i n t e r p r e t e d as f r a c t i o n of u n i t s c which X c a r r y t h e meaning z . R e g a r d l e s s o f t h e i n t e r p r e t a t i o n of f u n c t i o n g , i t may now b e used t o e x p r e s s t h e c o n t r i b u t i o n t o some meaning o f a s p e c i f i c component o f t h e u n i t . Assume t h a t c and c ' a r e two u n i t s , which have a l l a c t i o n s i d e n t i c a l e x c e p t t h a t on medium m i'* l e t vi be t h e a c t i o n on medium m i n u n i t c , w h i l e l e t u n i t c l i have " n e u t r a l " a c t i o n # i on medium m Then, we s a y i' that ( i n t h e context of other a c t i o n s , i d e n t i c a l i n c and c ' ) ,

--

v

i

s u p p o r t s meaning z , i f

and i n p a r t i c u l a r , vi g e n e r a t e s meaning z , if

--

v

i

i n h i b i t s meaning z , i f

and i n p a r t i c u l a r , v i c a n c e l s meaning z,

-- vi

if

i s n e u t r a l w i t h r e s p e c t t o meaning z , if

g(c,z) = g(c',z).

CHAPTER 4

320

These n o t i o n s , which may be e a s i l y e x t e n d e d t o t h e c a s e o f s e v e r a l a c t i o n s ( i . e . combinations o f a c t i o n s on s e v e r a l m e d i a ) , d e s c r i b e b a s i c t y p e s o f i n t e r a c t i o n s between media of communication. Thus, s u p p o r t o f a meaning o c c u r s when a c t i o n v

on i medium m i n c r e a s e s t h e l e v e l o f e x p r e s s i o n o f meaning i z above t h e l e v e l which would have o c c u r r e d i f t h e n e u t r a l a c t i o n # were performed. I n p a r t i c u l a r , i f i n i t h e l a t t e r c a s e z would n o t have been e x p r e s s e d a t a l l , we have g e n e r a t i o n o f meaning. O t h e r d e f i n i t i o n s a r e b u i l t on t h e same p r i n c i p l e : f o r i n s t a n c e i n h i b i t i o n occurs i f performing v

i

causes

l e s s e r d e g r e e of e x p r e s s i o n o f z t h a t would have sappeared i n c a s e of n e u t r a l a c t i o n #i. A t y p i c a l example f o r s u p p o r t i n g a meaning s u c h as

"Friendliness" i n t h e appropriate context i s t h e smile which accompanies o t h e r a c t i o n s . To have a n example o f g e n e r a t i o n o f a meaning, i t s u f f i c e s t o c o n s i d e r r e l i g i o u s c e r e m o n i e s , or magic, where t h e i n t e n d e d meaning i s t o t a l l y a b s e n t , u n l e s s one makes t h e a p p r o p r i a t e g e s t u r e , u t t e r s some words, e t c . Such "magical" components a p p e a r a l s o o u t s i d e o f magic o r r e l i g i o n ; f o r i n s t a n c e , i n many c o u n t r i e s , a p e r son may be h a n d c u f f e d , t a k e n t o t h e p o l i c e s t a t i o n , e t c . , b u t t h e s e a c t i o n s do n o t c o n s t i t i t e " l a w f u l

a r r e s t " u n l e s s f o r s p e c i f i c f o r m u l a i s u t t e r e d by t h e officer

.

I n c a s e of mutual i n t e r r e l a t i o n s between more a c t i o n s , t h e s i t u a t i o n i s , n a t u r a l l y , more complex, b u t t h e b a s i c i d e a s remain t h e same. A s a n example, we may

FORMAL SEMIOTICS

321

.

two media, s a y m and m L e t c be a u n i t o f a c t i o n s , i j and l e t c i , c and c b e t h e u n i t s o b t a i n e d from c j ij by r e p l a c i n g r e s p e c t i v e l y i t s a c t i o n o f medium mi b y # i , i t s a c t i o n on medium m b y # j , and b o t h a c t i o n s on j media m and m by #i and # We may t h e n s a y t h a t ( i n i j j' t h e c o n t e x t o f a c t i o n s o f c ) , a c t i o n s v and v on i 5 media m and m a r e p o s i t i v e l y a s s o c i a t e d f o r meaning i 5 z , if g ( c i j , z ) ( m i n [ g ( c i , z ) , g ( c j , z ) l and a l s o max [ g ( c i , z ) , g ( c j , z ) l 4 g ( c , z ) . On t h e o t h e r hand, v catalyses v if 0 = g ( c i j , z ) = g ( c j , z ) < g ( c i , z ) . i j' These two d e f i n i t i o n s do not e x h a u s t a l l p o s s i b i l i t i e s , and one can have a l s o p o s i t i v e a s s o c i a t i o n i n i n h i b i t i n g meaning z , o r n e g a t i v e a s s o c i a t i o n f o r o r a g a i n s t t h i s meaning , e t c . P o s i t i v e a s s o c i a t i o n o c c u r s when e a c h o f t h e a c t i o n s s u p p o r t s t h e meaning, but i n c o n j u c t i o n t h e y s u p p o r t t h i s meaning s t r o n g e r t h a n e a c h o f them s e p a r a t e l y . On t h e o t h e r hand, c a t a l y s i n g means t h a t one a c t i o n has no i n f l u e n c e on t h e meaning, u n l e s s i t i s accompanied by o t h e r a c t i o n ( h e r e a g a i n r e l i g o i u s ceremonies may s e r v e as an e x a m p l e ) . The n o t i o n s which d e s c r i b e t h e r o l e o f a n a c t i o n on a medium i n g e n e r a t i n g , s u p p o r t i n g , i n h i b i t i n g , o r canc e l l i n g a meaning, may b e e x t e n d e d t o t h e c a s e of t h e r o l e s which u n i t s p l a y i n a s t r i n g o f u n i t s . However, i t i s n e c e s s a r y t o i n t r o d u c e t h e concept which i n t h i s c a s e would c o r r e s p o n d t o t h e " n e u t r a l " a c t i o n . If we have s u c h a c o n c e p t , and # s t a n d s f o r t h e " n e u t r a l " u n i t ( f o r a g i v e n meaning z ) , t h e n t h e n o t i o n s o f i n h i b i t i o n , s u p p o r t , e t c . can be d i r e c t l y a p p l i e d , b y comparison o f g ( 2 , z ) and g ( c . , z ) , where -1 c and c d i f f e r o n l y by r e p l a c e m e n t o f i - t h u n i t by -i

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t h e n e u t r a l u n i t # . We s h a l l n o t repeat t h e d e f i n i t i o n s , b u t e x p l a i n i n s t e a d how one may d e f i n e t h e n e u t r a l unit #. For d e f i n i t i o n , one c o n s i d e r s a g i v e n meaning t o g e t h e r w i t h i t s m o d i f i c a t i o n s , b o t h i n t h e p o s i t i v e and neg a t i v e d i r e c t i o n . One can v i s u a l i z a i t as a s c a l e , of t h e form

"

not-z

"

rather not-z

"

somewhat z

"

"

z

very z

"

extremely z

F o r i n s t a n c e , f o r z = F r i e n d l i n e s s , we s h a l l have meanings s u c h a s " V e r y f r i e n d l y " , "Extremely f r i e n d l y " , a s w e l l as " U n f r i e n d l y " , and even "Rude". A s t a n d a r d u n i t for e x p r e s s i n g z w i l l be d e f i n e d as

any u n i t m e e t i n g two c o n d i t i o n s : f i r s t l y , i t d o e s n o t e x p r e s s any n e g a t i o n o f z i n any d e g r e e , i . e . g ( c , - z ) = 0 , and s e c o n d l y , among u n i t s which meet t h e f i r s t c o n d i t i o n , i t e x p r e s s e s z i n h i g h e s t d e g r e e , i . e . c must s a t i s f y g ( c , z ) = max t g ( c 1 , z ) : g ( c ' , - z ) =

01.

It i s w o r t h t o

mention t h a t s t a n d a r d u n i t d o e s n o t e x p r e s s t h e h i g h e s t d e g r e e of z , i . e . "Extremely z " , b u t e x p r e s s e s z i n t h e h i g h e s t d e g r e e . For i n s t a n c e , f o r meaning s u c h as " P o l i t e n e s s " , a s t a n d a r d u n i t w i l l n o t be such b e h a v i o u r which i s " E x t r e m e l y p o l i t e " , b u t a b e h a v i o u r which i s " P o l i t e " i n t h e h i g h e s t p o s s i b l e degree ( i . e . without exaggeration, but a l s o without t r a c e s of i m p o l i t e n e s s ) . It i s c l e a r t h a t t h e n o t i o n o f s t a n d a r d u n i t i s f u z z y , and moreover, a s t a n d a r d u n i t i s n o t unique f o r a

FORMAL SEMIOTICS

323

g i v e n meaning; , , t h e r e may e x i s t s e v e r a l s t a n d a r d u n i t s .

2.3. Pragmatic semantics A s mentioned i n t h e I n t r o d u c t i o n , one o f t h e most i n -

t e r e s t i n g problems c o n n e c t e d w i t h s e m a n t i c s o f m u l t i m e d i a l l a n g u a g e s o f communication i s t h a t o f e x p r e s s i b i l i t y o f meanings when t h e v o c a b u l a r i e s on one o r more media a r e c o n s t r a i n e d ( i n extreme c a s e s , some media may b e t o t a l l y banned from u s e , e . g . v e r b a l medium i n b a l l e t , o r communication between d e a f and m u t e ) . G e n e r a l l y , l e t u s d e n o t e b y R t h e r e s t r i c t i o n s under c o n s i d e r a t i o n s , so t h a t R may c o n c e r n banning one medium, o r r e s t r i c t i n g t h e v o c a b u l a r i e s o f some media, e t c . Such a r e s t r i c t i o n r e s u l t s i n s e l e c t i n g a s u b s e t L ( R ) o f t h e l a n g u a g e L , w i t h t h e c o n d i t i o n t h a t one may u s e o n l y t h e s t r i n g s v from L(R). The e f f e c t o f r e s t r i c t i o n o f u s e o f t h e s t r i n g s i n t h e l a n g u a g e o f communication a c t i o n s t o L(R) l i e s i n l i m i t i n g t h e p o s s i b i l i t y o f e x p r e s s i n g some meanings, and a l s o i n t h e p o s s i b i l i t y o f s e p a r a t i n g two meanings.

we may a s s i g n t o i t t h e s e t b ( z ) , whose e l e m e n t s w i l l be c a l l e d s e m a n t i c markers o f z.

If z

A s a consequence, we have a f a m i l y o f mappings f a :

+ 2H*, which

t o every s t r i n g v i n L a s s i g n s t h e s e t f ( v ) o f s t r i n g s o f s e m a n t i c m a r k e r s , t o be c a l l e d a s e m a n t i c t r a c e of v .

L

Semantic t r a c e s can be c o n c a t e n a t e d , i n t h e s e n s e t h a t fa(u)fa(v) = f x : x = YZ, Y E fa(& Y € fa(V)). By s e m a n t i c language ( l a n g u a g e o f t r a c e s ) o v e r L we s h a l l now mean t h e s e t

sa ( L )

=

u V(

fa(v).

L

Moreover, i f one has an e q u i v a l e n c e r e l a t i o n on t h e s e t o f s e m a n t i c m a r k e r s , one can c o n s i d e r c l a s s e s o f a b s t r a c t i o n o f t h i s r e l a t i o n , and c o n s e q u e n t l y , langua g e s of s e m a n t i c markers of h i g h e r o r d e r , where t h e markers would be l a b e l s o f e q u i v a l e n c e c l a s s e s . Now, t h e most i m p o r t a n t c a s e s o c c u r when f a ( v ) = 0 and when f ( v ) c o n s i s t s o f a s i n g l e e l e m e n t . I n t h e a f i r s t c a s e , we s a y t h a t s t r i n g v i s m e a n i n g l e s s : t h e r e i s no s t r i n g o f s e m a n t i c markers a s s i g n e d t o i t . I f * f ( v ) c o n s i s t s of a s i n g l e s t r i n g i n H o n l y , t h e n v a has a unique meaning ( a t l e v e l a ) , or i s a-unambiguous F i n a l l y , i f f a ( v ) c o n s i s t s o f more t h a n one e l e m e n t , w e have a m b i g u i t y o f s t r i n g v . I n p a r t i c u l a r , we have language o f t r a c e s o f s e m a n t i c markers Sa(L) o v e r t h e m u l t i m e d i a 1 l a n g u a g e o f communication L. Strings of actions generate s t r i n g s of s t a t e s , which form t h e r e f o r e a s t a t e - l a n g u a g e g e n e r a t -

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ed by c o n s e c u t i v e u n i t s ; l e t t h i s language b e S

*

(this

language a g a i n may be c o n s i d e r e d on v a r i o u s l e v e l s o f a b s t r a c t i o n , induced by e q u i v a l e n c e c l a s s e s on s t a t e s ) . * One can now c o n s i d e r p o s s i b l e homomorphisms between S and s e m a n t i c l a n g u a g e . N a t u r a l l y , such homomorphisms need n o t e x i s t s i n c e t h e r e i s no a p r i o r i r e a s o n why t h e s t r u c t u r a l p r o p e r t i e s imposing c o n s t r a i n t s o n , s a y , arm movements, s h o u l d r e s e m b l e i n any way t h e s t r u c t u r -

a l p r o p e r t i e s o f e v e n t s d e s c r i b e d by words and g e s t u r e s . However, one c o u l d e x p e c t some r e l a t i o n s h i p s t o e x i s t between t h e p h y s i c a l s t r u c t u r e of s i g n s and t h e o b j e c t s d e s c r i b e d (some t r a c e s o f such r e l a t i o n s h i p s , on a low l e v e l , e x i s t i n form o f onomatopeic words, g e s t u r e s f o r " l a r g e " and " s m a l l " , e t c . ) . Such homomorphisms c o u l d be s t u d i e d for some s i m p l i f i e d r e g u l a r forms o f m u l t i m e d i a 1 l a n g u a g e s and t h e t r a c e l a n g u a g e s g e n e r a t e d i t . T h i s would l e a d t o s e a r c h f o r e q u i v a l e n c e o f r e p r e s e n t a t i o n o f m u l t i m e d i a l commun i c a t i o n language i n s e m a n t i c domain, as w e l l as i n t h e domain o f s t a t e l a n g u a g e , r e p r e s e n t e d b y i n s t r u c t i o n l a n g u a g e s . Such problems c o u l d be a n a l y s e d a l s o f o r l a n g u a g e s of h i g h e r o r d e r s . Some hope h e r e c o u l d be p r o v i d e d by s i m u l a t i o n s t u d i e s of such l a n g u a g e s . The most i m p o r t a n t would be f i n d i n g r e p r e s e n t a t i o n s o f v e c t o r s ( m u l t i m e d i a l u n i t s ) as n e t works. ,Next s e c t i o n w i l l show d e f i n i t i o n s o f multimedi a l u n i t s and l a n g u a g e s i n terms o f p r o b a b i l i s t i c networks. 2.5.

Graph r e p r e s e n t a t i o n o f a u n i t

I n t h e a p p r o a c h p r e s e n t e d a b o v e , u n i t s are t r e a t e d as

FORMAL SEMIOTICS

335

random o b j e c t s (random s i g n s ) , composed out of e l e m e n t a r y o b j e c t s . We have namely some s p a c e , s a y Q , of c e r -

t a i n o b j e c t s (which may be a c t i o n s , p h y s i c a l o b j e c t s , e v e n t s , e t c . ) ; on Q we have a b i n a r y r e l a t i o n , i n t e r p r e t e d as " b e i n g s e m a n t i c a l l y connected to". Thus, a u n i t may b e i d e n t i f i e d w i t h a g r a p h of such a r e l a t ion. The u s a g e of a g i v e n u n i t may b e , o f a t r a n s i e n t n a t u r e (examples may be happenings i n a r t , some non-conventi o n a l s i g n s i t u a t i o n s a r r a n g e d ad hoe, m e a n i n g f u l o n l y i n a g i v e n c o n t e x t , e t c . ) . One can a l s o d i s t i n g u i s h more s t a b l e o b j e c t s , t h a t e v o l v e i n time s l o w l y . T h e e v o l u t i o n h e r e i s e x p r e s s e d t h r o u g h loss o r g a i n of some nodes o f t h e r e l a t i o n , and g a i n or loss o f some e d g e s . T o d e s c r i b e s u c h s i t u a t i o n s , one may u s e a new appro.ach t o random g r a p h s ( s e e Nowakowkska 1 9 7 9 a , 1 9 8 3 ) , which u t i l i z e s t h e p o s s i b i l i t y of random a p p e a r e n c e s and random d i s a p p e a r e n c e s o f nodes and e d g e s . Such a t h e o r y w i l l be o u t l i n e d below i n t h e p r e s e n t s e c t i o n . The f o r m a l a p p r o a c h w i l l be somewhat d i f f e r ent than i n t h e c i t e d papers. F o r m a l l y , a g r a p h i s a p a i r (S,C) , where S i s t h e s e t of elememts c a l l e d n o d e s , and C i s a b i n a r y r e l a t i o n i n S , i . e . a s u b s e t C C S % S . If xCy, we s a y t h a t e l e m e n t s x and y a r e c o n n e c t e d w i t h a n edge. The e d g e s a r e d i r e c t e d , t h a t i s , one may have xCy, b u t n o t yCx. T y p i c a l l y , random g r a p h s a r e d e s c r i b e d by assuming t h a t t h e s e t of nodes i s f i x e d , and t h e edges a r e c h o s e n a t random a c c o r d i n g t o some s a m p l i n g scheme. I n t h i s s e c t i o n , a model w i l l be s u g g e s t e d which w i l l

336

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account f o r t h e p o s s i b i l i t y o f randomness a l s o i n t h e s e t S o f nodes. A c c o r d i n g l y , we s h a l l d e n o t e by S ( t ) and C ( t ) t h e s e t o f nodes p r e s e n t a t t , and t h e s e t of e d g e s p r e s e n t

a t t . N a t u r a l l y , f o r e v e r y t w e have

(2.17) To e n s u r e c o n d i t i o n (2.17) one must assume t h a t as soon

as a node d i s a p p e a r s , a l l edges l e a d i n g t o and from t h i s node a l s o d i s a p p e a r . T h i s i s a n a t u r a l r e q u i r e m e n t . The e v o l u t i o n of a random g r a p h ( S ( t ) , C ( t ) ) may be d e s c r i b e d i n terms of' two p r o c e s s e s : moments o f chang e s o f t h e s e t S ( t ) o f n o d e s , and changes of C ( t ) o c c u r i n g between changes o f S ( t ) . A c c o r d i n g l y , we s h a l l need two k i n d s o f a s s u m p t i o n s : (1) t h o s e d e s c r i b i n g t h e changes i n S ( t ) , and ( 2 ) t h o s e d e s c r i b i n g changes i n C ( t ) which o c c u r i n

between changes of S ( t ) , hence when t h e number o f nodes i s c o n s t a n t . A s r e g a r d s t h e f i r s t assumptions, i t i s n a t u r a l t o

r e q u i r e t h a t t h e s e t S ( t ) and a l s o C ( t ) form Markov p r o c e s s e s . S p e c i f i c a l l y , we assume t h a t S ( t ) s a t i s f i e s ASSUMPTION 1. Suppose t h a t t h e s e t S ( t ) has I S ( t ) (

=

n

e l e m e n t s . Then t h e p r o b a b i l i t y of a new node a p p e a r i n g between t and t t A t e q u a l s c(n)At

+

O b t ) .

(2.18)

Here c ( n ) i s some f u n c t i o n o f t h e g r a p h s i z e n . I n t h e

331

FORMAL SEMIOTICS

p a r t i c u l a r c a s e when c ( n ) = 1 f o r a l l n , we have a r r i v a l s according t o Poisson process w i t h i n t e n s i t y Another i m p o r t a n t s p e c i a l c a s e i s c ( n ) = n for a l l n , i n which c a s e t h e a r r i v a l i n t e n s i t y i s p r o p o r t i o n a l t o t h e a c t u a l s i z e o f t h e graph. A s r e g a r d s d i s a p p e a r e n c e s of nodes o f t h e g r a p h , t h e

g e n e r a l assumption may be as f o l l o w s . ASSUMPTION 2 . L e t x b e a node i n S(t), and l e t p, and q x be t h e numbers o f e d g e s l e a d i n g fram x and t o x. Then t h e p r o b a b i l i t y o f d i s a p p e a r e n c e o f t h e node x between t and t t A t e q u a l s

Moreover, t h e dwents d e s c r i b e d i n t h i s a s s u m p t i o n f o r d i f f e r e n t nodes x , and t h e e v e n t s i n Assumption 1 a r e m u t u a l l y independent. Here r ( p x , q x ) i s some f u n c t i o n , which d e s c r i b e s " d e a t h r a t e " o f n o d e s , depending on t h e " t i e s " o f t h i s node w i t h t h e r e s t , o f t h e graph. I n p a r t i c u l a r c a s e s , w e may have r ( p x , q x ) = 1, which means t h a t p r o b a b i l i t y o f d i s a p p e a r e n c e o f a node i s c o n s t a n t , and d o e s n o t depend on t h e number o f e d g e s l e a d i n g t o o r from t h i s node. I n s u c h a c a s e , t h e p r o b a b i l i t y t h a t one node w i l l disappear i n ( t , t t A t ) equals if is the t o t a l number o f nodes a t t i m e t .

rn,

Now, as r e g a r d s t h e a s s u m p t i o n s about t h e changes i n t h e s e t o f e d g e s C ( t ) , one can assume t h e f o l l o w i n g . L e t u s f i x some t i m e t , and c o n s i d e r two n o d e s , x and

CHAPTER 4

338

y , i n S ( t ) . These nodes may be c o n n e c t e d by an e d g e

o r not; denote

a

= 1 if

o

xCy,

i f n o t xcy

(and s i m i l a r l y f o r a

). YX

ASSUMPTION 3. Suppose t h a t x , y

S ( t ) , and l e t a

= 0 XY

( s o t h a t t h e r e i s no edge l e a d i n g from x t o y ) . Then t h e p r o b a b i l i t y of a p p e a r e n c e o f a new edge from x t o y i n ( t , tt A t ) equals

On t h e o t h e r h a n d , i f a

XY

= 1 ( t h e r e i s an edge l e a d -

i n g from x t o y ) , t h e n t h e p r o b a b i l i t y t h a t t h i s edge w i l l disappear i n ( t , tt A t ) i s

Moreover, t h e e v e n t s d e s c r i b e d i n t h i s a s s u m p t i o n a r e independent f o r d i f f e r e n t edges. The form of f u n c t i o n s f and g which a p p e a r i n t h i s a s s u m p t i o n depends on t h e p a r t i c u l a r phenomenon which o n e w a n t s t o d e s c r i b e . I n p a r t i c u l a r , t h e r e may be a t e n d e n c y t o t h e a p p e a r e n c e o r d i s a p p e a r e n c e of an e d g e , d e p e n d i n g on how " p o p u l a r " a r e t h e two n o d e s , Such a t e n d e n Y' cy w i l l be g e n e a r a l l y d e s c r i b e d by a p p r o p r i a t e c h o i c e i.e.

on t h e numbers p,,

of f u n c t i o n s f and g .

q x , py and q

FORMAL SEMIOTICS

339

To c o n s i d e r some s p e c i a l c a s e s , assume t h a t t h e funct i o n s c ( n ) and r ( n ) i n Assumptions 1 and 2 a r e c o n s t a n t s e q u a l 1. L e t P n ( t ) d e n o t e t h e p r o b a b i l i t y t h a t [S(t)\

=

n . Then ( s e e F e l l e r , 1957), we have t h e s y s -

tem o f e q u a t i o n s for P ( t ) : n

for n = 1,2,...

.

S i m i l a r e q u a t i o n s may be o b t a i n e d

a l s o for o t h e r c a s e s of f u n c t i o n s c ( n ) and r(n). I n t h e c a s e under c o n s i d e r a t i o n ( a s l o n g as \ S ( t ) l > 0 ) t h e t i m e X t u n t i l t h e n e x t ( a f t e r t ) a p p e a r e n c e or d i s a p p e a r e n c e o f a node, has e x p o n e n t i a l d i s t r i b u t i o n

so that

I n c a s e when \ S ( t ) ( = 0 , t h e o n l y p o s s i b i l i t y i s t h a t a new node w i l l a p p e a r , s o t h a t

Moreover, i f ‘ S ( t ) l 7 0 , t h e p r o b a b i l i t y t h a t t h e n e x t e v e n t w i l l be a n a p p e a r e n c e o f a new node i s b / ( h t p ) , w h i l e t h e p r o b a b i l i t y t h a t i t w i l l be a d i s a p p e a r e n c e o f a node i s r/(h+p).

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Now, as r e g a r d s a p p e a r e n c e and d i s a p p e a r e n c e o f e d g e s , t h e p o s s i b i l i t i e s of modeling v a r i o u s e f f e c t s ( a s a l r e a d y m e n t i o n e d ) l i e s i n t h e p r o p e r c h o i c e of f u n c t i o n s f and g . I n t h e s i m p l e s t c a s e , assume t h a t f and g a r e c o n s t a n t s , s a y e q u a l a and b . Then t h e a p p e a r e n c e and d i s a p p e a r e n c e o f e d g e s a r e a g a i n P o i s s o n p r o c e s s e s , and moreo v e r , each o f t h e edges t h a t e x i s t a t a given time h a s t h e same chance o f d i s a p p e a r i n g , and e a c h o f t h e p o s s i b l e " m i s s i n g " e d g e s h a s t h e same chance o f a p p e a r i n g . T h i s c o r r e s p o n d s t o random s a m p l i n g of e d g e s from t h e e x i s t i n g and n o n e x i s t i n g o n e s . The w a i t i n g t i m e f o r n e x t e v e n t ( a p p e a r e n c e o r d i s a p p e a r e n c e o f an e d g e ) h a s e x p o n e n t i a l d i s t r i b u t i o n , w i t h p a r a m e t e r s a and b . One can t h e r e f o r e compute t h e e x p e c t e d change i n t h e g r a p h between c o n s e c u t i v e appearences o r d i s a p p e a r e n c e s o f nodes. I n p a r t i c u l a r , i f a t some t i m e t w e have I S ( t ) l = n and I C ( t ) l

= k ( g r a p h h a s n nodes and k e d g e s ) , and

a t t one node d i s a p p e a r s , t h e n t h e e x p e c t e d number o f e d g e s which must a l s o d i s a p p e a r i s k/n ( s i n c e t h e e x p e c t e d number o f e d g e s l e a d i n g from a g i v e n node i s k / 2 n , and e x p e c t e d number o f e d g e s which l e a d t o a g i v e n node i s a l s o k / 2 n ) . A s a n example o f g r a p h ( n e t w o r k ) r e p r e s e n t a t i o n o f u n i t s o f m u l t i m e d i a 1 l a n g u a g e s , one may t a k e changes of ways o f a r t i s t i c e x p r e s s i o n i n t i m e , e . g . e v o l u t i o n

o f G o t h i c s c u l p t u r e , where some means o f e x p r e s s i o n a p p e a r and some o t h e r d i s a p p e a r . A s a n o t h e r example, one may t a k e p l a y s a d a p t e d f o r t h e a t r e ; h e r e removing

FORMAL SEMIOTICS

341

a r o l e ( n o d e ) i m p l i e s removing a l l r e f e r e n c e s t o i t , i . e . a l l e d g e s l e a d i n g t o and from t h i s node. As s t i l l a n o t h e r example, one may c o n s i d e r a p a i n t e r who works on some p a i n t i n g . When he d e c i d e s t o e r a s e some f i g u r e i n t h e background ( s a y ) , he may have t o r e p l a c e i t by some o t h e r f i g u r e o r o b j e c t ( n o d e ) , i n o r d e r t o p r e s e r v e c o m p o s i t i o n a l e q u i l i b r i u m . I n o t h e r words, w e may have h e r e n e c e s s a r y s u b s t i t u t i o n s . S i m i l a r l y , i n t h e a t r e , one may remove a fragment of m u s i c a l background, and i f i t i s c o m p o s i t i o n a l l y n e c e s s a r y , r e p l a c e i t by some o t h e r m u s i c a l f r a g m e n t . The a s s u m p t i o n s c o n c e r n i n g changes o f edges ( a s opposed t o changes of n o d e s ) c o r r e s p o n d t o laws o f changea b i l i t y o f v a r i o u s forms of composite communication u n i t s ; t h e s e laws may be d i f f e r e n t f o r d i f f e r e n t forms o f a c t i v i t y . The r e l a t i o n s h i p between t h e p r o b a b i l i t y of a p p e a r e n c e o r d i s a p p e a r e n c e of an edge and t h e c h a r a c t e r i s t i c s of t h e n o d e s , c o r r e s p o n d t o t h e p r i n c i p l e of c o n t e x t u a l i t y of change. Here we have a law o f meaning p r e s e r v a t i o n i n c a s e o f a p p e a r e n c e of an e d g e , and s i m i l a r l a w s for t h e p r o c e s s of d i s a p p e a r e n c e . Also, some e d g e s may have s m a l l p r o b a b i l i t y o f d i s a p p e a r e n c e s ( o r even z e r o ) ; i n t e r m s of m u l t i m e d i a l u n i t s , t h i s corresponds t o multimedia1 f u n c t i o n a l u n i t s , where some components a r e s i g n - n e c e s s a r y , i . e . must a p p e a r i n any s t a g e of e v o l u t i o n of s i g n . F o r a n a l y s i n g t h e e f f e c t of s u p p o r t i n g o r i n h i b i t i n g a meaning by s e m a n t i c c o n n e c t i o n w i t h a new node ( r e -

p r e s e n t i n g a fragment of random o b j e c t ) , one can i n t r o duce t h e n o t i o n o f a f i l t e r , i . e . a f u n c t i o n which changes t h e meaning of a node, and hence i n f l u e n c e s t h e meaning o f u n i t as a whole.

342

CHAPmR 4

H y p o t h e t i c a l l y , t h i s p r o c e s s may be i n t e r p r e t e d i n two w a y s . F i r s t l y , from r e c e i v e r ' s p o i n t o f v i e w , f i l t e r may be o m i s s i o n o f some p a r t s o f random o b j e c t , due t o

s e l e c t i v i t y o f p e r c e p t i o n . From t h e p o i n t o f view o f t h e c o n s t r u c t o r o f a t r a n s i e n t m u l t i m e d i a 1 u n i t (random o b j e c t ) , f i l t e r s may be i n t e r p r e t e d as c o n s c i o u s oper a t i o n s on d e g r e e s o f meaning c a r r i e d b y n o d e s , and c h o i c e o f s u c h c o m p o s i t i o n o f s u b u n i t s which would g i v e t h e r e q u i r e d l e v e l o f i n t e n d e d o v e r a l l meaning, f o l l o w i n g some p r i n c i p l e s o f s e m a n t i c c a l c u l u s .

For more c o n v e n i e n t i n t e r p r e t a t i o n of t h e p r o c e s s o f f i l t e r i n g and c o n t r o l , i t w i l l be b e t t e r t o t r e a t t h e r e l a t i o n o f s e m a n t i c c o n n e c t e d n e s s i n t e r m s o f amount o f i n f o r m a t i o n about meaning c a r r i e d by one s u b u n i t ( n o d e ) about a n o t h e r . The flow o f i n f o r m a t i o n about meaning i n t h e random o b j e c t c o r r e s p o n d s t o e i t h e r a chosen p a t h of t h e eye o v e r p a r t s o f t h e o b j e c t , o r i n c a s e o f t h e s e n d e r -- c h o i c e o f such s t r u c t u r e o f edges and nodes which imposes some t y p e o f f l o w o f i n f o r m a t i o n i n r e c e i v e r . The p r o c e s s o f c o n t r o l can

--

be i d e n t i f i e d i n b o t h c a s e s w i t h a s t r i n g o f i n s t r u c t i o n s : f o r r e c e i v e r t h e s e w i l l be t h e i n s t r u c t i o n s a b o u t t h e nodes t o o m i t , t o v i s i t , t o a s s i g n g r e a t e r importance, e t c . For t h e s e n d e r , i n s t r u c t i o n s concern t h e d e s i g n o f t h e u n i t w i t h i n t e n d e d meaning and i t s

degree. Of c o u r s e , b o t h s e n d e r and r e c e i v e r have knowledge c o n c e r n i n g s t r u c t u r e o f more c o n v e n t i o n a l communicati o n u n i t s , and knowledge about t h e w a y s o f c o n t r o l l i n g t h e p r o c e s s o f communication t h r o u g h such u n i t s . I n o t h e r words, t h e y have knowledge about some random v a r i a b l e , and p r o c e s s o f p e r c e p t i o n o r p r o c e s s o f

FORMAL SEMIOTICS

343

d e s i g n i n g and p e r f o r m i n g a u n i t may be i d e n t i f i e d w i t h r e a l i z a t i o n o f s u c h random v a r i a b l e . We s h a l l show l a t e r t h e use o f d e f i n i t i o n o f random o b j e c t f o r t h e t h e o r y o f o b s e r v a b i l i t y . Now we s h a l l show two i m p o r t a n t mechanisms o f p e r c e p t i o n , whose i n t e r a c t i o n d e t e r m i n e s t h e p a t h o f t h e eye o v e r t h e structure of t h e object.

3. CONTROL PROCESSES I N PERCEPTION The p e r c e p t u a l p r o c e s s e s i n t h e p r e s e n t e d below new

dynamic models ( s e e Nowakowska 1983) w i l l be assumed’ to b e p a r t i a l l y c o n t r o l l a b l e . The o b s e r v a b l e v a r i a b l e s w i l l be t h e i n s p e c t i o n p r o c e s s t h r o u g h c o n s e c u t i v e g l a n c e s , governed by two b a s i c mechanisms -- o f spont a n e o u s i n s p e c t i o n , and o f p u r p o s e f u l i n s p e c t i o n , t o be d e s c r i b e d b y t h e two models below. The o t h e r two models w i l l c o n c e r n t h e t o p i c o f d a t a g e n e r a t i o n t h r o u g h c o m p a r i s o n s , and f o r m i n g e v e n t r e p r e s e n t a t i o n s o f o b j e c t s i n memory.

3.1.

P r o c e s s of s p o n t a n e o u s i n s p e c t i o n s

A s r e g a r d s t h e p r o c e s s of c o n s e c u t i v e g l a n c e s , it i s

aimed a t e x p l a i n i n g t h e mechanisms which d e t e r m i n e how

l o n g a g i v e n domain ( f r a g m e n t o f t h e o b j e c t , o r a c t i o n o f a m u l t i m e d i a 1 u n i t ) w i l l be i n s p e c t e d , and which w i l l be t h e domain i n s p e c t e d as n e x t . I f , i n a d d i t i o n , one assumes p a r t i a l c o n t r o l l a b i l i t y

of t h e p e r c e p t i o n p r o c e s s , u n d e r s t o o d as a c o n s c i o u s

344

CHAPTER 4

component i n placement o f g l a n c e s , t h e n t h e jumping eye movement a l l o w s f o r economy of p e r c e p t u a l work. T h i s economy i s a t t a i n e d under some a d d t i o n a l c o n s t r a i n t s , namely under t h e c o n d i t i o n o f s e m a n t i c c o n t i n u i t y . i . e . supplementary knowledge of t h e s i t u a t i o n , which a l l o w s f o r smoothing t h e d i s c r e t e d a t a s u p p l i e d by t h e e y e , and a c c o u n t s f o r u n d e r s t a n d i n g t h e s e d a t a . To e x p l a i n t h e jumpirlg

e x p l o r a t o r y eye movement, one introduces here t h e notion of p u l l i n g force. T h i s force

r e p r e s e n t s t h e s t r e n g t h w i t h which a g i v e n domain ( o f

a f o c a l p o i n t o f a given fragment) " a t t r a c t s " t h e e y e . The p u l l i n g f o r c e d e t e r m i n e s t h e t i m e o f s t a y i n a domain, and t h e p r o b a b i l i t y o f t r a n s i t i o n t o a g i v e n o t h e r domain. G e n e r a l l y , i t w i l l be assumed t h a t t h e p u l l i n g f o r c e d e c r e a s e s when t h e domain i s i n s p e c t e d , and i n c r e a s e s when some o t h e r domain i s i n s p e c t e d . The mechanisms o f i n c r e a s e and d e c r e a s e may be chosen i n v a r i o u s ways. One o f them i s t o assume t h a t t h e p u l l i n g f o r c e a s s o c i a t e d w i t h a domain d e c r e a s e s exp o n e n t i a l l y when t h i s domain i s i n s p e c t e d , and i n c r e a ses l i n e a r l y d u r i n g i n s p e c t i o n o f some o t h e r domain ( s i m i l a r a s s u m p t i o n s u n d e r l i e t h e model o f change o f a c t i v i t y o f Atkinson and B i r c h (1972); t h e p r e s e n t model, however, d i f f e r s i n v e r y e s s e n t i a l a s p e c t s from t h a t o f Atkinson and B i r c h ) . The main i s s u e o f t h e model l i e s i n c o n n e c t i o n between t h e unobsevable p u l l i n g f o r c e s and t h e o b s e r v e d behavi o u r -- i n t h i s c a s e , c o n s e c u t i v e g l a n c e s . The model p o s t u l a t e s t h a t t h e d e c i s i o n t o t e r m i n a t e a

g l a n c e a t some domain and move t h e eye t o some o t h e r

345

FORMAL SEMIOl7CS

domain r e s u l t s from r e s o l v i n g a " c o n f l i c t " between p u l l s by v a r i o u s domains. I n a s e n s e , v a r i o u s domains "compete" f o r t h e g l a n c e , w h i l e t h e p u l l k e e p i n g t h e g l a n c e a t a g i v e n i n s p e c t e d domain d e c r e a s e s . The eye i s t h e r e f o r e " p u l l e d " i n v a r i o u s d i r e c t i o n s . The mechanisms o f t h e . c h o i c e i s assumed t o b e s t o c h a s t i c , so t h a t t h e model a l l o w s t o o b t a i n t h e p r o b a b i l i t y d i s t r i b u t i o n o f t h e d u r a t i o n o f s t a y i n a g i v e n domain and t h e t r a n s i t i o n p r o b a b i l i t i e s t o o t h e r domains. To p r o c e e d f o r m a l l y , w e assume t h a t t h e o b j e c t i n s p e c t -

ed has a number o f f o c a l p o i n t s and t h e i r domains; t h e f o c a l p o i n t s w i l l b e d e n o t e d by p l , p 2 , and The n o t i o n o f a domain t h e i r domains by D1, D2,

... .

...

here may be f u z z y , i . e . some p o i n t s may b e l o n g i n

v a r y i n g d e g r e e s t o v a r i o u s domains. However, i t w i l l be assumed t h a t a t e a c h moment t h e eye i s f o c u s e d a t some w e l l d e f i n e d domain. Assumption 1. With e a c h f o c a l p o i n t pi t h e r e i s a s s o c i a t e d i t s p u l l i n g f o r c e f i ( t ) , which may change i n t i m e . It i s assumed t h a t f . ( t ) d e c r e a s e s exponent1 i a l l y when domain o f pi i s i n s p e c t e d , and i n c r e a s e s l i n e a r l y when some o t h e r domain i s i n s p e c t e d . S p e c i f i c a l l y , i t i s assumed t h a t i f t h e domain Di

is

i n s p e c t e d between t o and t , t h e n

(3.1) if a domain p

and t

,

then

j'

w i t h j $ i , i s i n s p e c t e d between t

0

CHAPTER 4

346

Assumption 2 .

Suppose t h a t at time t t h e domain D

i s inspected,

and l e t f l y f 2 y . . . be t h e p u l l i n g f o r c e s

a t t h a t t i m e . Then t h e p r o b a b i l i t y t h a t t h e r e w i l l be a change o f domain i n s p e c t e d ( j u m p ) between t and t + h , i s f o r domains o f p l y p 2 , . . .

2

f. h

fi j # i

+

o(h).

(3.3)

J

If a jump o c c u r s , t h e n t h e p r o b a b i l i t y t h a t i t w i l l

be t o t h e domain D k ( k # i ) e q u a l s fk/

z

fj

.

j#i

One can t h e n p r o v e t h e f o l l o w i n g t h e o r e m ( s e e Nowakows k a 1983): THEOREM. & e J f l y f 2 ) . . . be t h e p u l l i n g f o r c e s a t t i m e

t , o f b e g i n n i n g o f i n s p e c t i o n o f t h e domain D i . Then, d e n o t i n g by Ui t h e time o f u n i n t e r r u p t e d i n s p e c t i o n o f D i , w e have

= exp(i-

1 [Bite fiCi

B.

cit t

cit

(2 - ~ . ) ( e c

i

1

-

1)lf

(3.5)

where =

Bi

z

bik,

Fi =

k#i

2 fk.

(3.6)

kfi

The d i s t r i b u t i o n o f t h e d u r a t i o n o f t h e v i s i t i s n o t e x p o n e n t i a l , and t h e r e f o r e t h e p r o c e s s o f v i s i t s i s n o t a Markov p r o c e s s .

FORMAL SEMIOTICS

347

To p r o v e t h e theorem, o b s e r v e t h a t

t P(Ui>t)

=

exp

[--f

u(s)ds]

(3.7)

0

where

u ( s ) i s t h e jump i n t e n s i t y a t t i m e s . We have

here u(s) =

=

rz k#i C

%

fk(S)l/fi(S)

-c. s (f,

+ biks)l/fie

1

k#i Fi c . s

- -

Bi

e ' t - s e f:

cis

.

I n t e g r a t i n g t h e i n t e n s i t y we o b t a i n t h e a s s e r t i o n o f t h e theorem. Q u a l i t a t i v e l y s p e a k i n g , a v i s i t t o a domain i s l i k e l y t o t e r m i n a t e when t h e t o t a l p u l l t o o t h e r domains exc e e d s t h e p u l l t o t h e domain i n s p e c t e d . T h i s t o t a l

pull i n c r e a s e s l i n e a r l y , w h i l e t h e p u l l t o t h e domain i n s p e c t e d d e c r e a s e s e x p o n e n t i a l l y , s o t h a t t h e jump i n t e n s i t y i n c r e a s e s t o i n f i n i t y . Also, i f a t r a n s i t i o n occurs, t h e p r o b a b i l i t y o f t h e n e x t domain depends -a g a i n , i n a s t o c h a s t i c way -- on t h e r e l a t i v e s t r e n g t h s o f p u l l i n g f o r c e s , s o t h a t a domain w i t h s t r o n g e r pull i s more 1.iitely t o a t t r a c t t h e eye for n e x t i n s p e c t i o n . It s h o u l d be p o i n t e d o u t , however, t h a t -- a s opposed t o model o f Atkinson and B i r c h ( 1 9 7 2 ) -- i t i s n o t

g e n e r a l l y t r u e t h a t t h e t r a n s i t i o n w i l l always be t o t h e domain w i t h s t r o n g e s t p u l l .

348

CHAPTER 4

The main p o i n t o f t h e theorem i s t h a t t h e d u r a t i o n s o f v i i s i t s do n o t have e x p o n e n t i a l d i s t r i b u t i o n , hence t h e p r o c e s s of v i s i t s i s n o t Markovian. I n t u i t i v e l y , t h e r e a s o n f o r non-Markovian c h a r a c t e r o f t h i s p r o c e s s i s c l e a r : f i r s t l y , t h e l o n g e r a g i v e n domain was i n s p e c t e d , t h e more l i k e l y i t i s t h a t t h e v i s i t w i l l terminate, since i t s p u l l diminishes, while other p u l l s i n c r e a s e . T h i s i m p l i e s t h a t t h e d u r a t i o n s do n o t have e x p o n e n t i a l d i s t r i b u t i o n , s i n c e t h e jump i n t e n s i t y i s not constant.

S e c o n d l y , t h e l o n g e r a domain h a s n o t been v i s i t e d , t h e more l i k e l y i t i s t h a t i t s p u l l w i l l dominate o t h e r p u l l s -- s o t h a t a v i s i t t o t h i s domain becomes more and more l i k e l y . Again , t h i s shows a non-Markovian c h a r a c t e r o f t h e p r o c e s s of v i s i t s . Actually, t h e p u l l i n g force i s not only unobservable, b u t t h e n o t i o n as such i s f u z z y , p a r t i a l l y b e c a u s e o f f u z z i n e s s of t h e n o t i o n o f t h e domain D One can exi' p l i c a t e t h i s f u z z i n e s s as f o l l o w s . Around t h e "nominal" f o c a l p o i n t pi, one can l o c a t e a number o f o t h e r p o i n t s o f i n t e r e s t , say pil,

....

pi2, A s an example imagine a p a i n t i n g , w i t h one o f t h e domains b e i n g t h e f a c e o f t h e p e r s o n on t h e p a i n t i n g . A s t h e f o c a l p o i n t from h i s domain one may t a k e , s a y , t h e t i p o f t h e n o s e , but t h e r e a r e a l s o o t h e r p o i n t s , s u c h as e y e s , mouth, e t c . i n t h e same domain. L e t hij

be t h e d e g r e e t o which p o i n t p

t h e domain Di.

belongs t o One can t h e n assume t h a t w i t h e a c h o f ij

the points there i s associated a pulling force

and t h a t t h e t o t a l p u l l t o w a r d s t h e domain Di by

gij(t) i s given

FORMAL SEMIOlTCS

f . ( t ) = max 1 j

min Chij,

349

gij(t)l.

Here t h e s p e c i f i c a s s u m p t i o n s about changes of g ( t ) ij depend m o s t l y on t h e mutual l o c a t i o n s and d i s t a n c e s may o r may n o t between p o i n t s p i j : i n s p e c t i o n of p ij p r o v i d e i n f o r m a t i o n about pik (and hence d e c r e a s e t h e p u l l i n g f o r c e t o w a r d s t h i s p o i n t ) , depending on are at a d i s t a n c e allowing seeing w h e t h e r p i j and p ik both points o r not.

3.2. Purposeful inspection A s opposed t o t h e model of t h e p r e c e d i n g s e c t i o n , i n which t h e p r o c e s s of i n s p e c t i o n was assumed t o be p u r e l y s t o c h a s t i c , and governed by v a r i o u s p u l l i n g f o r c e s , i n t h i s s e c t i o n we s h a l l s t a r t from a n o t h e r " e x t r e m a l " p o i n t o f view: t h a t t h e p e r s o n i n s p e c t s t h e domains of f o c a l p o i n t s i n a p u r p o s e f u l way, s o as t o a t t a i n t h e g o a l of r e c o g n i t i o n of t h e meaning.

Thus, we c o n s i d e r h e r e a problem somewhat d i f f e r e n t t h a n i n t h e p r e c e d i n g s e c t i o n , where r e c o g n i t i o n may o r may n o t have been t h e g o a l o f i n s p e c t i o n . F o r t h e model, we assume t h a t w i t h e a c h domain ( o r a l t e r n a t i v e l y , w i t h each f o c a l p o i n t ) , t h e r e i s a s s o c i a t e d some s e t of f e a t u r e s . L e t Fi be t h e s e t o f f e a t u r e s a s s o c i a t e d w i t h domain D:; we assume t h a t I t h e s e s e t s a r e d i s j o i n t , so t h a t F i / ) F = @ for i # j , j and l e t

3 50

CHAPTER 4

be t h e s e t o f a l l f e a t u r e s of t h e f i g u r e t o be r e c o g n i -

zed ( i d e n t i f i e d ) . A s a n example t o g u i d e t h e i n t u i t i o n , c o n s i d e r t h e

problem o f i d e n t i f y i n g t h e make and model o f a c a r from i t s p h o t o g r a p h . Then one c a n d i s t i n g u i s h v a r i o u s 'ldomains'l, more o r l e s s f u z z y , such as w h e e l s , f e n d e r s , e t c . , and t h e i r f e a t u r e s ( e . g . s h a p e s , e t c . ) . The two b a s i c a s s u m p t i o n s , s t a t e d i n a q u a l i t a t i v e w a y , a r e h e r e as f o l l o w s :

-- t h e f i g u r e may be r e c o g n i z e d b e f o r e a l l f e a t u r e s a r e ob s e r v e d ;

-- a f e a t u r e o f a domain may

b e n o t i c e d o r missed upon

i n s p e c t i o n of t h i s domain. We s h a l l now p r e s e n t f o r m a l c o u n t e r p a r t s o f t h e s e a s s u m p t i o n s , f i r s t i n a non-fuzzy c a s e , and t h e n i n d i c a t e p o s s i b l e ways o f f u z z i f y i n g them. Thus, t h e f i r s t a s s u m p t i o n may b e e x p r e s s e d as a c l a s s of a l l s u b s e t s o f F which a r e s u f f i c i e n t f o r r e c o g n i t i o n ( t o be c a l l e d r e c o g n i t i o n s e t s ) .

Let

be s u c h a c l a s s .

*

One can t h e n d e f i n e t h e c l a s s

of minimal r e c o g n i t -

i o n s e t s as f o l l o w s : G E

y*

iff G C

18

and whenever H i s a p r o p e r s u b s e t

of G , t h e n H

L

y.

351

FORMAL SEMIOTICS

I n p r a c t i c a l s i t u a t i o n s , t h e n o t i o n o f r e c o g n i t i o n set i s f u z z y , m o s t l y b e c a u s e t h e same s e t o f f e a t u r e s may o r may n o t be s u f f i c i e n t f o r r e c o g n i t i o n , depending on t h e p e r s o n and o c c a s i o n . Thus, i s i n f a c t a fuzzy c l a s s of s u b s e t s of F , s o t h a t w i t h each (nonfuzzy) s u b s e t G C F one may a s s o c i a t e t h e v a l u e r ( G ) , which c h a r a c t e r i z e s t h e d e g r e e t o which G i s a r e c o g n i t i o n

$

set. N e x t , a s r e g a r d s t h e second a s s u m p t i o n , we s h a l l t r e a t t i m e as d i s c r e t e , w i t h p o s s i b l e changes o f t h e domains of inspection being t = 1 , 2 , . . . We assume t h a t i n e a c h u n i t o f t i m e a t most one f e a t u r e may be d e t e c t e d . Moreover, i f a t t i m e t t h e domain D i i s i n s p e c t e d , and t h e s e t o f f e a t u r e s of D i which a r e a l r e a d y d e t e c t t h e n t h e p r o b a b i l i t y o f d e t e c t i o n of ed i s F ' C F i ,

.

i

t h e f e a t u r e x k Fi

-

F! between t and t t l i s 1

We may have h e r e

xGFi-F: The i n e q u a l i t y a c c o u n t s f o r t h e f a c t

t h a t i t i s po-

s s i b l e t h a t no f e a t u r e w i l l be d e t e c t e d . F o r m a l l y t h e r e f o r e , f o r e v e r y F;

c

Fi ( i n c l u d i n g t h e

c a s e F ' = 0 ) , we have a p r o b a b i l i t y d i s t r i b u t i o n on i where N s t a n d s f o r "no d e t e c t i o n t h e s e t Fi - F;g{Nl, o f any f e a t u r e " . One may now f o r m u l a t e t h e problem o f o p t i m a l p a t h o f

352

CHAPTER 4

i n s p e c t i o n , t h e o p t i m a l i t y b e i n g c h a r a c t e r i z e d by t h e minimality of r e c o g n i t i o n t i m e . A t any g i v e n t i m e t , t h e s t a t e of i n s p e c t i o n i s d e s c -

r i b e d by t h e v e c t o r

...

F i , F$,

o f f e a t u r e s of domains D1,

D2,

. . . which

have a l r e a d y

been d e t e c t e d . L e t C(Fi,F;,...)

be t h e r e s i d u a l t i m e t i l l r e c o g n i t i o n

under o p t i m a l s t r a t e g y , s t a r t i n g from t h e s t a t e F i , F ; ,

...

o f a l r e a d y d e t e c t e d f e a t u r e s . We have t h e n t h e

Bellman e q u a t i o n

.

(3.8)

C(Fi,F$,. .)

This condition expresses the f a c t t h a t i n t h e state d e s c r i b e d by F i , F;,

... we

must make t h e d e c i s i o n

a b o u t t h e domain t o be i n s p e c t e d as n e x t . I f w e d e c i d e t o i n s p e c t t h e domain Diy

then t h e expected time till

recognition equals 1 plus t h e average of recognition

times under c o n d i t i o n o f d e t e c t i n g v a r i o u s p o s s i b l e f e a t u r e s , o r not d e t e c t i n g any. The boundary c o n d i t i o n s h e r e a r e d e s c r i b e d by t h e i n c l u s i o n o f t h e s t a t e and one o f t h e r e c o g n i t i o n s e t s :

353

FORMAL SEMIOTICS

C ( F i , F:,

...) = 0 i f

(3.9) The i n i t i a l c o n d i t i o n i s F ’ = @ f o r a l l i . i The o p t i m a l c h o i c e o f t h e domain t o i n s p e c t as n e x t

i s o b t a i n e d by s o l v i n g t h e e q u a t i o n ( 3 . 8 ) , and t a k i n g a s t h e domain t o be i n s p e c t d n e x t ( i n a g i v e n s t a t e ) t h e domain w i t h i n d e x i , which g i v e s t h e minimum on t h e r i g h t hand s i d e i n ( 3 . 8 ) . The n o t i o n o f r e c o g n i t i o n s e t s a l l o w s u s t o i n t r o d u c e t h e n o t i o n of r e c o g n i t i o n weights. I n t u i t i v e l y , t h e s e a r e numbers r e p r e s e n t i n g t h e i m p o r t a n c e o f v a r i o u s features f o r the overall recognition. To d e f i n e t h e w e i g h t s f o r m a l l y , assume t h a t t h e s e t F o f a l l f e a t u r e s i s f i n i t e , and c o n s i s t s of e l e m e n t s

F = I f l yf 2 ,

...,

fN3

‘

L e t u s now c o n s i d e r a p e r m u t a t i o n

o f e l e m e n t s o f F.

fil,

fi2,.

..,f

iN

We may now a s s o c i a t e w i t h e a c h

permutation a p i v o t , according t o t h e following definition. DEFINITION. ‘il,

f

The e l e m e n t f .

. .. , f i ,. .. , f

i2’

n longs t o t h e c l a s s

f

1..

il

,..., f . 3 ’k

,..

3

, i f t h e s e t (fi .,f. beiN 1 In o f minimal r e c o n g i t i o n s e t s ,

while f o r k = l Y 2 , . . . , n - l form {f

i s a pivot i n permutation

ri

none o f t h e s e t s o f t h e

has t h i s property.

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Thus, a p i v o t i s t h e l a s t e l e m e n t i n t h e g i v e n permuta t i o n with t h e property t h a t , together with a l l elements preceding i t , t h e s e t of f e a t u r e s thus obtained i s sufficient f o r recognition. DEFINITION. The r e c o g n i t i o n w e i g h t o f t h e f r a g m e n t f i n F i s defi.ned as

# o f p e r m u t a t i o n s i n which f i s t h e p i v o t

w(f)

=

N! T h i s d e f i n i t i o n o f r e c o g n i t i o n w e i g h t s h a s been i n t r o duced by Nowakowska ( 1 9 6 7 ) ; i t i s b a s e d on t h e n o t i o n

o f v o t i n g power and minimal w i n n i n g c o a l i t i o n s o f S h a p l e y and S h u b i k ( 1 9 5 4 ) . A s an example, i m a g i n e t h a t t h e o b j e c t t o be r e c o g n i z e d

i s a word, which i s w r i t t e n i n a random o r d e r ( i n t h e s e n s e o f a p p e a r e n c e o f l e t t e r s i n a random w a y , e a c h o c c u p y i n g i t s p l a c e i n t h e w o r d ) . Assume t h a t t h e word

i s WATER, and t h a t t h e l e t t e r s a r e a p p e a r i n g i n permuta t i o n A , R, W,

T , E.

(so that the features are f l = W,

f 2 = A , and s o o n , and for t h e p e r m u t a t i o n i n q u e s t i o n , i

1

= 2, etc.)

The c o n s e c u t i v e s t a g e s a r e h e r e as f o l l o w s - A - - -

- A - - R

W A - -

R

W A T - R

C l e a r l y , t h e f i r s t t h r e e s t a g e s a r e n o t s u f f i c i e n t for r e c o g n i z i n g t h e words as WATER, w h i l e i n t h e f o u r t h s t a g e , d e s p i t e one l e t t e r s t i l l m i s s i n g , t h e word

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w i l l b e r e c o g n i z e d . T h i s means t h a t i n t h e c o n s i d e r e d permutation, the pivot i s T. By a n a l y s i n g a l l p e r m u t a t i o n s , and i d e n t i f y i n g t h e p i v o t i n e a c h c a s e , one can a s s i g n w e i g h t s t o a l l f i v e l e t t e r s , t h u s e x p r e s s i n g t h e importance o f t h e s e l e t t e r s i n r e c o g n i z i n g t h e p a r t i c u l a r word. Human e x p e r i e n c e a l l o w s a l s o t o e v a l u a t e s u c h w e i g h t s s u b j e c t i v e l y , f o r v a r i o u s o c c a s i o n s and g o a l s , and determine t h e invariance of weights f o r c l a s s e s of o b j e c t s and g o a l s , and f o r c l a s s e s o f t r a n s f o r m a t i o n s . I n t r o d u c t i o n of t h e n o t i o n o f w e i g h t s a l l o w s a l s o f o r s t u d y i n g t h e s t r a t e g i e s of t h e p a t h of t h e eye o v e r t h e o b j e c t , which a r e g e n e r a t e d by s t r u c t u r a l p r o p e r t i e s o f t h i s o b j e c t and p a r t i a l l y c o n t r o l l e d b y t h e s u b j e c t . V a r i o u s models o f d e c i s i o n s about "walking o v e r t h e tree of the object'' ( p i c t u r e ) , r e l a t i o n s with Gestalt t h e o r y , as w e l l as a t t e m p t s o f q u a n t i f i c a t i o n o f some n o t i o n s o f t h i s t h e o r y ( e . g . t h a t o f good f i g u r e ) may be found i n Nowakowska (1967). T h i s p a p e r a l s o shows a p p l i c a t i o n of c o n d i t i o n a l weights t o s e q u e n t i a l a n a l y s i s of recognition. The u s e o f dynamic programming t o t h e s t u d y of p u r p o s e -

f u l i n s p e c t i o n seems t o be o f i m p o r t a n t m e t h o d o l o g i c a l v a l u e , a l s o i n e n g i n e e r i n g and a r t i f i c i a l i n t e l l i g e n c e , for modeling p a t t e r n r e c o g n i t i o n . I n n a t u r a l s i t u a t i o n s , w e d e a l o f c o u r s e w i t h a m i x t u r e o f two proc e s s e s , o f f r e e and p u r p o s e f u l i n s p e c t i o n , and v a r i o u s c o n t r i b u t i o n s o f t h e s e p r o c e s s e s are p r o b a b l y induced by b o t h t h e g o a l o f p e r c e p t i o n and t h e s t r u c t u r e o f t h e p e r c e i v e d o b j e c t . I n o t h e r words, t h e a c t u a l g l a n c e

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p a t h may b e r e g a r d e d as a m i x t u r e o f o p t i m a l g l a n c e p a t h a n d p u r e l y random g l a n c e p r o c e s s , g o v e r n e d by pulling forces. The g l a n c e p r o c e s s may b e assumed t o i n d u c e t h e p r o c e s s o f t r a n s p o r t a t i o n and c o m p a r i s o n s . Such

comparisons,

i n t u r n , may b e r e g a r d e d a s p r e - e v e n t s , t h a t i s , t h e c a n d i d a t e s f o r e v e n t s t o b e remembered. It c a n b e p o s tulated that recognition, identification, e t c . , requir e c o l l e c t i o n o f "raw d a t a " a b o u t t h e r e l a t i o n a l s t r u c t u r e o f t h e p e r c e i v e d o b j e c t . Such raw d a t a must i n v o l v e , among o t h e r s , t h e r e s u l t s o f c o m p a r i s o n s o f values of a t t r i b u t e s of fragments, t o d e t e c t d i f f e r e n ces , symmetries , e t c. The c o m p a r i s o n o f two f r a g m e n t s

--

i f t h e y are located

s u f f i c i e n t l y f a r a p a r t -- r e q u i r e t h a t a g l a n c e a t one p o i n t o f t h e o b j e c t i n v o l v e s a l s o a " t r a n s p o r t a t i o n " o f t h e f r a g m e n t t o b e compared t o t h e l o c a t i o n o f t h e o t h e r f r a g m e n t , w i t h which t h e comparison i s t o b e made. N a t u r a l l y , s u c h a t r a n s p o r t a t i o n l e a d s t o a f u z z i f i c a t i o n , a n d a l s o t o a loss o f p r e c i s i o n i n comparison. Under a p p r o p r i a t e a s s u m p t i o n s , o n e may d e r i v e f o r m u l a s f o r the p r o b a b i l i t y of e r r o r s , t h u s o b t a i n i n g a c l u e t o e s t i m a t i o n o f p a r a m e t e r s o f t h e m o d e l . Also -- w h i c h i s more i m p o r t a n t -- t h e s e f o r m u l a s p r o v i d e means o f t e s t i n g the hypothesis of p a r a l l e l v e r s u s serial proc e s s i n g , i n case of comparisons i n v o l v i n g s e v e r a l a t t ributes. The mechanisms o f t r a n s p o r t a t i o n w i l l b e modeled i n S e c t i o n 3 . 4 , and t h o s e of changing p r e - e v e h ts

into

FORMAL SEMIOTICS

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e v e n t s , t o b e s t o r e d i n memory -- i n S e c t i o n 3 . 5 . We s h a l l s t a r t , however, w i t h t h e problems o f g e n e r a t i o n o f meaning o f a n o b j e c t ; t h e model s u g g e s t e d w i l l be aimed a t e x p l a i n i n g t h e f a c t t h a t t h e meaning a s s i g n ed may sometimes depend on t h e o r d e r o f i n s p e c t i o n .

3 . 3 . G e n e r a t i o n o f meaning under i n s p e c t i o n The a n a l y s i s of t h e p r e c e d i n g s e c t i o n was c a r r i e d o u t under t h e t a c i t a s s u m p t i o n t h a t t h e o b j e c t i n s p e c t e d has o n l y one meaning. I n many c a s e s i t may happen t h a t t h e o b j e c t i n s p e c t e d has s e v e r a l a l t e r n a t i v e meanings, and t h e problem i s t o a s s i g n one o f t h e m t o t h e o b j e c t . I n such c a s e s , t h e f i n a l d e c i s i o n may depend on t h e o r d e r o f i n s p e c t i o n ( r e g a r d l e s s whether t h e i n s p e c t i o n i s random or p u r p o s e f u l ) . I n t h i s s e c t i o n , we s h a l l p r e s e n t a simple model which a c c o u n t s f o r t h i s dependence.

From t h e knowledge o f p s y c h o p h y s i c a l p r o c e s s e s which accompany t h e i n s p e c t i o n o f f r a g m e n t s (domains) o f a f i g u r e , one may e x p e c t t h a t t h e o b s e r v e r behaves a s i f t e s t i n g t h e h y p o t h e s i s t h a t t h e meaning i s z : s u c c e s s i v e i n s p e c t e d domains p r o v i d e p r e m i s e s f o r o r a g a i n s t t h i s h y p o t h e s i s . He f i n a l l y r e a c h e s e i t h e r t h e d e c i s i o n " f o r ' ' meaning z , i n which c a s e t h e p r o c e s s of d e c i s i o n (though not n e c e s s a r i l y of i n s p e c t i o n ) t e r m i n a t e s , o r t h e d e c i s i o n " a g a i n s t " meaning z , i n which c a s e he c o n t i n u e s t e s t i n g f o r some o t h e r meaning. A t e a c h s t a g e of i n s p e c t i o n , t h e f r a g e m e n t s i n s p e c t e d

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t h u s f a r g i v e a c e r t a i n p r e m i s e f o r t h e meaning z ; t h e j o i n t e f f e c t h e r e may be r e p r e s e n t e d as a d e g r e e o f c e r t a i n t y t h a t t h e meaning i s z . T h i s d e g r e e may i n c r e a s e o r d e c r e a s e , depending on t h e fragment i n s p e c t ed as l a s t . L e t u s d e n o t e t h i s l e v e l of c e r t a i n t y a f t e r i n s p e c t i n g t h e f i r s t k f r a g m e n t s by xk ( x k w i l l , i n g e n e r a l , depend on z , b u t f o r s i m p l i c i t y , w e c o n s i d e r here t e s t i n g f o r a f i x e d meaning z ) . The problem l i e s i n def i n i n g x i n s u c h a way t h a t i t i s n o t i n v a r i a n t w i t h k respect t o the order of inspection. One o f p o s s i b l e a s s u m p t i o n s i s t h a t i f x i s s u f f i c i e n t k ly h i g h , t h e n n e x t p r e m i s e s " f o r " do n o t a f f e c t xk. S i m i l a r l y , when xk i s s u f f i c i e n t l y low, t h e p r e m i s e s " a g a i n s t " do n o t a f f e c t x k .

, fi2 ,... d e n o t e t h e 1 s u c c e s s i v e f r a g m e n t s i n s p e c t e d , and l e t q k ( z ) d e n o t e t h e s t r e n g t h of premise f o r o r a g a i n s t z , r e s u l t i n g from i n s p e c t i o n o f fragment f

For formal d e f i n i t i o n , l e t f i

ik

.

Put xo = 0 and d e f i n e xk a s

where A

B a r e some t h r e s h o l d s .

To s e e t h e consequences o f t h i s f o r m u l a , l e t u s u s e t h e f i g u r e 3 . 1 . below. T h i s f i g u r e i l l u s t r a t e s why t h e f i n a l r e s u l t may depend on t h e o r d e r o f i n s p e c t i o n .

FORMAL SEMIOTICS

X

359

k

Fig. 3.1

When xk e q u a l s t h e upper boundary B , t h e n t h e s u c c e s i v e p r e m i s e s " f o r " do n o t c a u s e any i n c r e a s e o f x k ' It i s s e e n t h a t t h e f i n a l v a l u e o f xk would be h i g h e r i f t h e p o s i t i v e v a l u e s of q k ( z ) o c c u r r e d a t some o t h e r p l a c e s , e . g . t o w a r d s t h e end o f i n s p e c t i o n . T h i s model i s , n a t u r a l l y , o n l y one o f t h e p o s s i b l e models o f i n f l u e n c e of o r d e r on t h e d e c i s i o n .

It would be o f c o n s i d e r a b l e i n t e r e s t t o d e t e r m i n e t h e p r o b a b i l i t y d i s t r i b u t i o n o f xk under random o r d e r o f i n s p e c t i o n . However, f o r s u c h a n answer t o be t h e o r e t i c a l l y s i g n i f i c a n t , one s h o u l d impose some c o n s t r a i n t s on t h e o r d e r o f i n s p e c t i o n , s i n c e n o t a l l o r d e r s a r e e q u a l l y l i k e l y , and c a l c u l a t i o n s under t h e a s s u m p t i o n o f e q u i p r o b a b l e o r d e r s a r e u n r e a l i s t i c . The problem o f v a r i o u s p r o b a b i l i t i e s of o r d e r s i s connected c l o s e l y w i t h weights, discussed i n t h e preceding section.

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It i s w o r t h t o mention h e r e a c e r t a i n c l a s s o f o b j e c t s

f o r which t h e f i n a l d e c i s i o n does n o t depend on t h e o r d e r o f i n s p e c t i o n : t h e s e a r e o b j e c t s such t h a t every fragment " s u p p o r t s " t h e meaning, s o t h a t q k ( z ) i s pos i t i v e . Then xk does n o t depend on t h e o r d e r o f i n s pection. F o r f i g u r e s w i t h some q k ( z ) b e i n g p o s i t i v e and some n e g a t i v e , t h e o r d e r o f i n s p e c t i o n p l a y s an i m p o r t a n t role.

3 . 4 . Mechanism o f t r a n s p o r t a t i o n i n comparisons G e n e r a l l y , t h i s p r o c e s s o c c u r s i n c a s e o f comparison o f v a r i o u s p a r t s of t h e p e r c e i v e d o b j e c t s , which are l o c a t e d a t d i f f e r e n t p l a c e s . One can a r g u e t h a t s u c h comparisons are i n r e a l i t y p r e - e v e n t s , t o be d i s c u s s e d i n t h e n e x t s e c t i o n . More p r e c i s e l y , some o f t h e r e s u l t s o f s u c h comparisons e n t e r t h e memory and form t h e "raw data" which l a t e r s e r v e for t h e p u r p o s e o f identification, recognition, etc. One may namely assume t h a t when a new f e a t u r e i s p e r c e i v e d i n some domain, i t i s compared w i t h some a n a l o gous f e a t u r e s i n o t h e r domains, t o form e v e n t u a l l y , t h e knowledge about t h e o b j e c t , a s a s e r i e s o f a s s e r t ions s u c h as "x i n domain D i s b r i g h t e r t h a n y i n i domain D ' I , o r " X i n domain D i s l o n g e r t h a n y i n j i domain D ", e t c . Such knowledge i s e s s e n t i a l b o t h f o r j r e c o g n i t i o n and a l s o f o r remembering t h e o b j e c t some t i m e afterwards. To show how t h e model works, and f o r e s t i m a t i o n o f

FORMAL SEMIOTICS

36 1

t h e p a r a m e t e r s o f t h e model, w e use a s i m p l e example

o f p e r c e p t i o n o f e l e m e n t a r y forms s u c h as l i n e s , which a r e compared-with r e s p e c t t o t h e i r l e n g t h i n s p e c i a l conditions. G e n e r a l l y , when one i s t o compare t h e f e a t u r e s o f two fragments t h a t a r e s u f f i c i e n t l y f a r a p a r t , so t h a t it i s i m p o s s i b l e t o i n s p e c t them b o t h s i m u l t a n e o u s l y , t h e a c t u a l comparison c o n c e r n s t h e memory t r a c e s of t h e f r a g m e n t s . Thus, t h e model below w i l l be based on some a s s u m p t i o n s about t h e e v o l u t i o n of memory t r a c e s a f t e r t h e stimulus ceases. I n o u r example, imagine t h a t t h e s u b j e c t i n s p e c t s some fragment ( s t i m u l u s S1), f o r m i n g t h e t r a c e o f t h i s s t i m u l u s i n memory. Then h i s e y e i n s p e c t s f o r some t i m e o t h e r f r a g m e n t s ( a n d t h e t r a c e o f S1 becomes g r a d u a l l y more f u z z y ) , f i n a l l y a r r i v i n g a t a p a r t S2, which i s t o b e compared i n l e n g t h w i t h S1. S t i m u l u s S2 i s i n s p e c t e d f o r a c e r t a i n p e r i o d of t i m e , and i t s t r a c e i s formed i n memory. The t r a c e s o f S and S2 1 a r e t h e n compared and d e c i s i o n i s r e a c h e d whether S1 i s o f t h e same l e n g t h as S2 o r n o t . I n e x p e r i m e n t a l s i t u a t i o n s , t h e l e n g t h s of s t i m u l i , d u r a t i o n o f t h e i r e x p o s i t i o n and o f t h e p a u s e between t h e e x p o s i t i o n s , a s w e l l as t h e l o c a t i o n s o f s t i m u l i , would be under t h e c o n t r o l o f t h e e x p e r i m e n t e r , t h u s making i t p o s s i b l e t o employ t h e t e c h n i q u e s o f e s t i mation of parameters. I n p r a c t i c a l s i t u a t i o n s , t h e t i m e s of e x p o s i t i o n s , p a u s e , and l o c a t i o n s o f s t i m u l i depend on t h e o r d e r and dynamics o f i n s p e c t i o n . To f i x t h e i d e a s , t h e model w i l l be p r e s e n t e d on an example o f a h y p o t h e t i c a l e x p e r i m e n t , i n which t h e

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362

s t i m u l i S1 and S2 a r e d i s p l a y e d on a s c r e e n , and t h e s u b j e c t i s t o d e c i d e whether t h e y a r e o f t h e same l e n g t h , and i f n o t , which o f them is l o n g e r . Assume now t h a t t h e s u b j e c t i s p r e - t r a i n e d ,

and he knows

t h e f u t u r e l o c a t i o n o f t h e s t i m u l u s on t h e s c r e e n , as w e l l as what i s e x p e c t e d o f him. There a r e f o u r s t a g e s , e a c h i n v o l v i n g a d i f f e r e n t proc e s s . These s t a g e s a r e p r e s e n t e d on t h e f i g u r e below.

Stage 1

Stage 2

Stage 3

Stage 4

Fig. 3.2.

During s t a g e 1, s t i m u l u s S1 i s d i s p l a y e d , and i t s t r a c e i s formed i n memory.

During s t a g e 2 , t h e ' s c r e e n i s b l a n k , a n d t h e t r a c e o f

S1 i s "moved" t o t h e f u t u r e l o c a t i o n o f s t i m u l u s S2. During t h i s p r o c e s s , t h e t r a c e undergoes some d i s tortions. During s t a g e 3 , t h e s t i m u l u s S2 i s d i s p l a y e d , and i t s t r a c e i s formed i n memory. A t t h e same t i m e , t h e t r a c e o f S1 undergoes s t i l l more d i s t o r t i o n s . F i n a l l y , i n s t a g e 4 , t h e s c r e e n i s b l a n k a g a i n . Both

363

FORMAL SEMIO'IICS

t r a c e s undergo d i s t o r t i o n ( f u z z i f i c a t i o n ) d u r i n g t h a t stage. I n t h e s e q u e l , we s h a l l d e n o t e by x l ( t ) and x 2 ( t ) t h e s i z e s o f t h e t r a c e s of s t i m u l i S1 and S2 i n memory a t t i m e t , and make t h e f o l l o w i n g a s s u m p t i o n s :

ASSUMPTION 1. The d e c i s i o n t h a t S1 = S 2 i s r e a c h e d , i f a t t i m e t o f comparison, w e have (3.10) where q i s some p o s i t i v e t h r e s h o l d .

ASSUMPTION 2 . Each t r a c e x i ( t ) , i = 1 , 2 i s a Wiener process with

(3.11) where mi

i s t h e t r u e s i z e o f s t i m u l u s Si.

The s u b s e q u e n t a s s u m p t i o n s c o n c e r n t h e v a r i a n c e s o f t h e two Wiener p r o c e s s e s . One c o u l d e x p e c t t h a t t h e f u z z i f i c a t i o n o f t h e t r a c e x l ( t ) i s f a s t e r when t h e a t t e n t i o n i s c o n c e n t r a t e d a t s t i m u l u s S2, i . e . d u r i n g s t a g e 3. Moreover, i t i s n e c e s s a r y t o t a k e i n t o a c c o u n t t h e f a c t t h a t t h e first process, corresponding t o s t i m u l u s S1, l a s t s l o n g e r t h a n t h e p r o c e s s x 2 ( t ) ( i n d e e d , x 1 ( t ) b e g i n s a t t h e end o f s t a g e 1, and l a s t s u n t i l t h e end o f s t a g e 4 , w h i l e t h e ' s e c o n d p r o c e s s b e g i n s only a t t h e end o f s t a g e 3 ) . We now make t h e f o l l o w i n g a s s u m p t i o n s , which d e s c r i b e

t h e v a r i a n c e s o f t h e two p r o c e s s e s .

CHAPTER 4

3 64

ASSUMPTION 3. D u r i n g s t a g e 2, i . e . p a u s e between t h e e x p o s i t i o n of t h e s t i m u l i , t h e v a r i a n c e of x ( t ) in1 creases proportionally t o a2: Var x , ( t )

= ECxl(t)

- m112

= a 2t .

ASSUMPTION 4 . During t h e e x p o s u r e o f t h e second s t i m u l u s , the variance of x ( t ) increases proportioanally 1 t o b2:

ASSUMPTION 5 . During t h e p r o c e s s o f comparison, t h e v a r i a n c e s of x ( t ) and x 2 ( t ) i n c r e a s e p r o p o r t i o n a l l y 1 t o c2:

ASSUMPTION 6 . The p r o c e s s e s x , ( t ) pendent.

and x 2 ( t ) a r e i n d e -

2

Here a 2 , b and c 2 a r e some c o n s t a n t s . I n p a r t i c u l a r , t h e c o n s t a n t c 2 may depend on t h e mutual d i s t a n c e and l o c a t i o n o f t h e s t i m u l i . T h i s dependence may be assumed t o b e t h e f o l l o w i n g ( s e e F i g . 3 . 3 ) . It may namely be c o n j e c t u r e d , t h a t t h e l e v e l o f d i f f i -

c u l t y o f comparison, as measured by t h e v a r i a n c e , i s l a r g e r when t h e a n g l e 'Q i s l a r g e r ( i . e . i t i s e a s i e s t t o compare t h e s t i m u l i i f t h e y are " l i n e d u p " ) . T h i s c o n j e c t u r e may b e e x p r e s s e d as 2

c 2 = k2R(1 t & sinf'), R2 = u2 t v where k and E a r e some c o n s t a n t s .

2

.

365

FORMAL SEMlOTICS

F i g . 3.3

Thus, v a r i a n c e depends on R , i . e . on t h e d i s t a n c e by which one s t i m u l u s h a s t o be " t r a n s p o r t e d " , and on t h e angle of i n c l i n a t i o n of t h i s t r a n s p o r t .

We have t h e n t h e f o l l o w i n g theorem. THEOREM. The p r o b a b i l i t y t h a t t h e s t i m u l i S1

and

w i l l be p e r c e i v e d as i d e n t i c a l e q u a l s

where

A = m

1

-

m2 = t r u e d i f f e r e n c e between s t i m u l i

S2

3 66

CHAPTER 4

where * t

-

d u r a t i o n o f p a u s e between s t i m u l i , T I - t i m e o f exposure o f s t i m u l u s S 2 , T c - t i m e between t h e end o f e x p o s u r e o f S and 2t h e end o f s t a g e o f comparison, R - d i s t a n c e between S, and S2,

\4 -

I

a n g l e o f i n c l i n a t i o n between p o s i t i o n s o f s t i m u l i S, and S 2 . I

I n t h i s theorem ion function.

6 stands

f o r s t a n d a r d normal d i s t r i b u t -

From t h e e x p e r i m e n t a l v i e w p o i n t , t h e above theorem p r o v i d e s u s w i t h t h e p o s s i b i l i t y o f t e s t i n g t h e model o f t r a n s p o r t a t i o n . I n d e e d , t h e unknown p a r a m e t e r s a r e h e r e a , b , c and q (or i n s t e a d o f c , w e may t a k e k and t). The c o n t r o l l e d v a r i a b l e s a r e t h e e x p o s i t i o n t i m e s and l e n g t h o f p a u s e , as w e l l as t h e d i f f e r e n c e between * The t h e t r u e l e n g t h s o f s t i m u l i , t h a t i s , t , TI and A v a r i a b l e s o b s e r v e d a r e t h e d e c i s i o n ( " e q u a l " v s . "une q u a l " ) , and t h e d e c i s i o n t i m e T

.

C

.

By s e t t i n g v a r i o u s v a l u e s o f t h e c o n t r o l v a r i a b l e s , i t

i s possible t o observe t h e frequencies of various types o f d e c i s i o n s , and t h e d e c i s i o n t i m e s , and i n t h i s w a y , o b t a i n a n a c c e s s t o e s t i m a t i o n o f t h e unknown parameters. F o r t h e p r o o f o f t h e t h e o r e m , l e t X and Y d e n o t e t h e s i z e s of t r a c e s o f S,I and S, a t t h e t i m e o f comparison. We o b t a i n t h e r e f o r e f o r t h e d i s t r i b u t i o n s o f X and Y : L

Y-

N(m2,

2

c Tc) = N(m2,

B2 ) .

361

FORMAL SEMIOTICS

We may t h e r e f o r e w r i t e

P(S1 j u d g e d i d e n t i c a l t o S 2 ) =

P ( \ X - Y \ < q ) = P(Y

-

q

c: x c

Y

+

q)

-(u-m2) 2 / 2 B 2 du.

Substituting u = m 2 t h e theorem.

+

Bz

we o b t a i n t h e a s s e r t i o n o f

L e t u s o b s e r v e t h a t t h i s theorem might p r o v i d e some

means o f t e s t i n g t h e h y p o t h e s i s o f p a r a l l e l v e r s u s s e r i a l p r o c e s s i n g . To i l l u s t r a t e t h i s p o s s i b i l i t y , imagine now t h a t s t i m u l i S1 and S2 a r e c h a r a c t e r i z e d by s e v e r a l a t t r i b u t e s , and t h a t t h e d e c i s i o n t h a t t h e y a r e i d e n t i c a l occurs i f t h e perceived values of t h e c o r r e s p o n d i n g a t t r i b u t e s a r e w i t h i n some t h r e s h o l d d i s t a n c e ( i n t h e s i m p l e s t c a s e , each o f t h e s t i m u l i may b e o f t h e shape o f l e t t e r L , s o t h a t t h e two a t t r i b u t e s t o b e compared a r e t h e l e n g t h s o f v e r t i c a l and horizontal p a r t s ) . I f t h e s e comparisons o c c u r s i m u l t a n e o u s l y , t h e probab i l i t y o f t h e d e c i s i o n o f i d e n t i t y may be o b t a i n e d by

t h e same method a s f o r t h e one a t t r i b u t e c a s e .

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I n c a s e o f s e r i a l c o m p a r i s o n s , t h e t r a c e o f one a t t r i b u t e undergoes f u r t h e r changes d u r i n g t h e t i m e when t h e o t h e r a t t r i b u t e i s p r o c e s s e d . Again, u s i n g t h e same t e c h n i q u e s , one c o u l d d e r i v e t h e f o r m u l a f o r t h e p r o b a b i l i t y o f a g i v e n d e c i s i o n . Given t h e e s t i m a t e s of t h e r e l e v a n t p a r a m e t e r s o b t a i n e d from o n e - a t t r i b u t e c a s e , one c o u l d t e s t t h e h y p o t h e s i s o f p a r a l l e l v e r s u s s e r i a l processing

.

It i s w o r t h w h i l e t o p o i n t o u t h e r e t h a t i n t h e system

o f models o f p e r c e p t i o n shown i n t h i s s e c t i o n , f o r t h e f i r s t t i m e i n t h e l i t e r a t u r e t h e r e were i n t r o d u c e d t o t h e model ( v i a mechanisms o f t r a n s p o r t a t i o n ) t h e memor y p a r a m e t e r s . Also, a s i g n i f i c a n t n o v e l t y i s t h e t h e o r e t i c a l a n a l y s i s of p e r c e p t i o n , open t o e x p e r i m e n t a t i o n v i a i n t e r a c t i o n a l s y s t e m o f models. The new t h e o r y a l s o p a s s e s t h e t e s t , c r u c i a l f o r a l l p e r c e p t i o n mod e l s , namely e x p l a n a t i o n o f v i s u a l i l l u s i o n s ( s e e Nowakowska 1 9 8 3 a , b ) . I n a d d i t i o n t o g l a n c e s , one u s e s t h e phenomenon o f f i n e eye movements, which p l a y e s s e n t i a l r o l e i n s h a r p e n i n g and memorizing t h e p e r c e i v e d image. Now, d u r i n g t h e f i n e eye movements, c e n t e r e d a t some p o i n t , t h e image of t h e f i g u r e on t h e r e t i n a p e r f o r m s a movement, c o n s e c u t i v e l y c o v e r i n g and u n c o v e r i n g any p o i n t c l o s e t o t h e boundary o f t h e f i g u r e . I n o t h e r words, t h e f i g u r e " r e t u r n s " t o t h i s p o i n t from t i m e t o t i m e . The f r e q u e n c y of t h i s r e t u r n p r o c e s s depends on t h e l o c a t i o n o f o f t h e p o i n t r e l a t i v e t o t h e f i g u r e . I n f a c t , we have h e r e many i n t e r r e l a t e d p r o c e s s e s , one f o r e v e r y p o i n t c l o s e t o t h e boundary o f t h e f i g u r e . It t u r n s o u t t h a t t h e p r o p e r t i e s o f t h e s e p r o c e s s e s

FORMAL SEMIOTICS

3 69

o f f e r a p o s s i b l e e x p l a n a t i o n o f t h e MUller-Lyer i l l u s i o n ( f o r a r e v i e w o f d i f f e r e n t a p p r o a c h e s and e x p l a n a t i o n s o f v i s u a l i l l u s i o n s , s e e e . g , Eijkman e t a l . (1981), Lee (19811, C a e l l i ( 1 9 8 1 ) o r Ullmann ( 1 9 7 9 ) ) . An a l t e r n a t i v e e x p l a n a t i o n o f v i s u a l i l l u s i o n s based on f i n e eye movements i s g i v e n i n Nowakowska (198313). The i l l u s i o n p r o c e s s e s are proposed t h e r e as s p e c i f i c modeling o c c u r r i n g a t a r e l a t i v e l y low l e v e l o f cognit i o n . The i l l u s i o n i t s e l f i s e x p l a i n e d t h r o u g h t h e n o t i o n o f competing p e r c e p t u a l s y s t e m s -- b a s i c and c o n t e x t u a l -- which y i e l d d i f f e r e n t v a l u e s o f t h e observed a t t r i b u t e s . The MUller-Lyer i l l u s i o n c o n s i s t s o f m i s p e r c e p t i o n of

l e n g t h s o f arrows on F i g . 3 . 4 .

>

/

F i g . 3.4 C o n s i d e r a n enlargement of t h e e n d p o i n t s o f t h e a r r o w s ( s e e F i g . 3 . 5 ) . The movements of t h e eye a r e e q u i v a l e n t

F i g . 3.5

t o movements o f t h e images of t h e s e a r r o w s . Consequentl y , any f i x e d p o i n t on t h e r e t i n a w i l l be c o n s e c u t i v e l y c o v e r e d and n o t c o v e r e d b y t h e image.

3 70

CHAPTER 4

N a t u r a l l y , s u c h changes w i l l be more f r e q u e n t f o r p o i n t s which a r e c l o s e t o t h e edge o f t h e a r r o w , from o u t s i d e or from i n s i d e . On t h e o t h e r hand, f o r p o i n t s f a r away from t h e e d g e , l y i n g o u t s i d e , t h e p e r i o d s o f b e i n g uncovered w i l l t e n d t o be l o n g e r , and conv e r s e l y f o r p o i n t s l y i n g i n s i d e t h e arrow. Each p e r i o d when a p o i n t i s n o t c o v e r e d c o n s i s t s o f two p a r t s : d u r i n g t h e f i r s t p a r t , t h e t r a c e o f t h e stimul u s s t i l l p e r s i s t s i n memory, w h i l e i n t h e second p a r t (which may be z e r o ) , t h e s t i m u l u s has a l r e a d y faded

.

Q u a l i t a t i v e l y s p e a k i n g , t h i s meachanism e x p l a i n s t h e i l l u s i o n . To s e e i t , c o n s i d e r two p o i n t s , a t t h e same d i s t a n c e from t h e end o f t h e a r r o w , l i k e p and q on Fig. 3.5. I n c a s e of outward a r r o w , t h e p o i n t g w i l l be c o v e r e d r a t h e r o f t e n , and f o r l o n g e r p e r i o d s o f t i m e , simply be v e r t i c a l movements o f t h e arrow ( i n a d d i t i o n t o h o r i z o n t a l movements). On t h e o t h e r hand, t h e p o i n t p on t h e inward arrow w i l l b e covered r a t h e r seldom, and f o r s h o r t e r p e r i o d s o f time. C o n s e q u e n t l y , f o r t h e p o i n t q t h e f r a c t i o n o f times when i t i s c o v e r e d , p l u s t h e times when i t i s n o t covered, but t h e t r a c e of t h e s t i m u l u s s t i l l p e r s i s t s , i s q u i t e h i g h . A s opposed t o t h a t , f o r p o i n t p t h e p e r i o d s when t h e r e i s a t r a c e o f t h e s t i m u l u s i n memor y w i l l t e n d t o be r a r e and s h o r t . I n e f f e c t , two p o i n t s l o c a t e d a t t h e same d i s t a n c e from t h e end o f t h e a r r o w w i l l g i v e d i f f e r e n t f r a c t i o n s o f t i m e when t h e s t i m u l u s p e r s i s t s .

FORMAL SEMIOTICS

371

The e x a c t e x p r e s s i o n for t h e f r a c t i o n o f t h e t i m e when t h e image p e r s i s t s depends on t h e a s s u m p t i o n s about t h e f a d i n g p r o c e s s . I n p a r t i c u l a r , if one assumes t h a t t h e f a d i n g p r o c e s s i s s u c h t h a t t h e t r a c e of t h e s t i mulus d i s a p p e a r s a f t e r e x p o n e n t i a l l l y d i s t r i b u t e d t i m e , one may o b t a i n t h e e x p r e s s i o n f o r t h e e x p e c t d d u r a t i o n o f t h e t i m e Y o f p e r s i s t e n c e of t r a c e i n memory. We have namely t h e f o l l o w i n g theorem

If c i s t h e decay r a t e of t h e t r a c e o f t h e s t i m u l u s i n memory, t h e n

THEOREM.

t

where F i s t h e L a p l a c e t r a n s f o r m o f t h e d i s t r i b u t i o n P F o f t h e "pause" f o r p o i n t p ( i . e . d i s t r i b u t i o n o f P t h e l e n g t h s o f t i m e when p &s n o t c o v e r e d ) . F o r t h e p r o o f , s e e Nowakowska ( 1 9 8 3 a ) A s a consequence, t h e a v e r a g e d u r a t i o n of t h e t r a c e of

t h e s t i m u l u s " p e r c y c l e " i s ( d e n o t i n g by U t h e l e n g t h pf p e r i o d o f b e i n g covered)

When c t e n d s t o 0 ( t r a c e p e r s i s t s f o r a l o n g t i m e ) , t h i s r a t i o t e n d s t o 1. On t h e o t h e r hand, when c t e n d s t o i n f i n i t e ( t r a c e fades f a s t ) , t h i s r a t i o tends t o the proportion E(U)/[E(U) t E(Z)].

3 12

CHAPTER 4

3 . 5 . Events r e p r e s e n t a t i o n o f a n o b j e c t : p r e - e v e n t s and memory T h e model here i s based on t h e f o l l o w i n g i n t u i t i o n s . During t h e p r o c e s s of i n s p e c t i o n , t h e eye s e n d s a cont i n u o u s s t r e a m of i n f o r m a t i o n . Somehow o n l y a p a r t of i t i s s e l e c t e d for remembering, and t h e q u e s t i o n i s t o f i n d p l a u s i b l e mechanisms which e x p l a i n how such

a s e l e c t i o n i s accomplished. Such t y p e o f s e l e c t i o n o c c u r s , i n f a c t , on two l e v e l s . On t h e one hand, i n t h e p r o c e s s o f r e c o g n i t i o n and i d e n t i f i c a t i o n o f some o b j e c t , t h e f e a t u r e s a r e anal y s e d , matched, e t c . and e v e n t u a l l y t h e d e c i s i o n about t h e meaning i s r e a c h e d . A s a r u l e , a f t e r some t i m e , t h e p e r s o n remembers more t h a n j u s t t h e meaning o f t h e o b j e c t ; he a l s o remembers some d e t a i l s . To i l l u s t r a t e i t , suppose t h a t we a s k a p e r s o n to

i d e n t i f y t h e make o f a c a r from a photograph. Some t i m e a f t e r t h e t a s k i s a c c o m p l i s h e d , t h e p e r s o n may be a b l e t o t e l l t h e c o l o u r o f t h e c a r , even though t h e c o l o u r p l a y e d no r o l e i n i d e n t i f i c a t i o n of t h e make. A s i m i l a r p r o c e s s o c c u r s a t a somewhat h i g h e r l e v e l , w i t h a n i n t e r v e n t i o n o f c o n s c i o u s components. F o r i n -

stance, certain are the appears

a p e r s o n i n s p e c t s a p a i n t i n g and " n o t i c e s " d e t a i l s : t h e two s h a p e s a t d i f f e r e n t c o r n e r s same, some e l e m e n t s a r e symmetric, l i g h t a t some p l a c e s i n a s p e c i a l way, e t c .

It may be t h o u g h t t h a t t h e b a s i c mechanisms a r e t h e same f o r b o t h c a s e s , and one may t r y t o d e s i g n a model

373

FORMAL SEMIOTICS

which would c o v e r b o t h c a s e s

--

o f c o n s c i o u s and sub-

conscious analysis. When a s t i m u l u s a r r i v e s , i t i s e i t h e r d i s r e g a r d e d , o r s t o r e d i n memory f o r l a t e r c o m p a r i s o n , a n a l y s i s , e t c . The c o n c e p t u a l framework h e r e i s as f o l l o w s . The s t r e a m o f i n f o r m a t i o n i s t r e a t e d as d i s c r e t e , f o r m i n g a s t r e a m

of pre-events,

occurring a t times t = 0 , 1 , 2 ,

... .

Each

pre-event' b e l o n g s t o some t y p e . The c o n c e p t of t y p e i s

a p r i m i t i v e o n e , and n e e d s t o be e x p l i c a t e d i n e a c h c a s e o f a p p l i c a t i o n . I n s p e c i a l c a s e s , i t may be s h a p e , c o l o u r , a b i l i t y t o evoke a s s o c i a t i o n s , e t c . A t any g i v e n t i m e , t h e p e r s o n i s r e c e p t i v e t o some

t y p e s , and n o t r e c e p t i v e t o o t h e r t y p e s , He w i l l s t o r e i n h i s memory o n l y t h o s e p r e - e v e n t s which have t h e t y p e t o which he i s r e c e p t i v e a t t h i s t i m e . F i n a l l y , s t a t e o f r e c e p t i v e n e s s c h a n g e s , and i n p a r t i c u l a r , i t may be m o d i f i e d by p r e - e v e n t s . The main q u e s t i o n which w i l l b e a s k e d i s t o d e t e r m i n e t h e p r o b a b i l i t y t h a t a given pre-event

w i l l become

s t o r e d i n memory, t h a t i s , t h a t i t w i l l become as e v e n t . The

frequency of such c a s e s w i l l t h e r e f o r e

d e t e r m i n e t h e o v e r a l l memory l o a d r e s u l t i n g from t h e inspection. Let e l ,

e 2 , . . . b e t h e p r e - e v e n t s which o c c u r a t t i m e s

t = 1,2,.,.,

and l e t M d e n o t e t h e s e t o f t y p e s , w i t h

g ( e . ) b e i n g t h e type o f pre-event 1

e

i'

L e t p x , x 6 M be t h e p r o b a b i l i t y t h a t a pre-event w i l l be o f t y p e x ; t h e t y p e s a r e assumed t o b e sampled i n d e p e n d e n t l y w i t h t h e same d i s t r i b u t i o n .

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374

N e x t , we make t h e f o l l o w i n g a s s u m p t i o n a b o u t r e c e p t i v i t y : a t any g i v e n t i m e , t h e s t a t e o f r e c e p t i v i t y i s

d e s c r i b e d b y a p a i r ( Q , f t ) , where

-- Q C M i s t h e s e t o f t y p e s o f p r e - e v e n t s t o which t h e person i s receptive at t i m e t ;

--

f t ( x ) , x G &, i s t h e " r e s i d u a l " t i m e o f r e c e p t i v i t y

f o r t y p e x a t time t ( s o t h a t , i n absence o f subsequent changes, t h e person w i l l remain r e c e p t i v e t o t y p e x u n t i l t h e time t t f t ( x ) ) ;

--

w i t h e a c h t y p e t h e r e i s a s s o c i a t e d a s e t B ( x ) of

t y p e s which i t " a c t i v a t e s " ;

-- if y i s s u c h a t y p e , t h e n t h e time o f r e c e p t i v i t y t o y i s Tx P(Tx

,Y

,Y

>

, where t ) = e- k ( X , Y ) t

L e t us d e n o t e now

s o t h a t C(x) i s t h e c l a s s o f a l l t y p e s y which may modify t h e r e c e p t i v i t y f o r t y p e x. F u r t h e r , l e t

be t h e p r o b a b i l i t y t h a t a p r e - e v e n t t h e receptivity t o type x. We have t h e n t h e f o l l o w i n g t h e o r e m .

w i l l n o t cause

375

FORMAL SEMIOTICS

THEOREM. P r o b a b i l i t y P t h a t a p r e - e v e n t

s t o r e d i n memory ( w i l l

w i l l become

become an e v e n t ) e q u a l s

F o r t h e p r o o f , s e e Nowakowska (1983a) To a p p r e c i a t e t h e consequences o f t h i s theorem, i t i s w o r t h w h i l e t o s i m p l i f y t h e s i t u a t i o n , and r e d u c e t h e number o f p a r a m e t e r s . Thus, assume t h a t

--

t h e r e i s o n l y a f i n i t e number m o f t y p e s , a l l w i t h t h e same p r o b a b i l i t y , so t h a t px = l / m . T h i s means t h a t t y p e s a r e sampled i n d e p e n d e n t l y from t h e uniform distribution;

--

t h e a v e r a g e t i m e o f r e c e p t i v i t y i s t h e same f o r

any t y p e , r e g a r d l e s s o f t h e t y p e o f p r e - e v e n t which c a u s e d t h e r e c e p t i v i t y . T h i s means t h a t r ( x , y ) = f o r a l l x,y;

--

e a c h pre-event a c t i v a t e s t h e same number r o f t y p e s , s o t h a t e a c h s e t C(x) c o n s i s t s o f r e l e m e n t s .

Under t h e s e a s s u m p t i o n s , t h e p r o b a b i l i t y t h a t a p r e e v e n t w i l l become s t o r e d i n memory e q u a l s

1

P = 1

+

h m(e r

-

1)

s o t h a t t h e a v e r a g e time between two c o n s e c u t i v e memo-

376

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r y storages i s

The e s s e n c e o f t h i s model may be summarized as f o l l o w s :

a p r e - e v e n t ( a c a n d i d a t e for an e v e n t ) becomes s t o r e d i n memory, i f i t i s p r e c e d e d s u f f i c i e n t l y r e c e n t l y by a n o t h e r p r e - e v e n t of an a p p r o p r i a t e t y p e . The probl e m l i e s i n b u i l d i n g a model which would d e s c r i b e t h e frequency of e v e n t s , t h e i r t y p e s , e t c . , i . e . d e s c r i b e t h e s t a t i s t i c a l f e a t u r e s of what i s b e i n g perceived, i n t h e sense of storage of t h e m a t e r i a l t o be p r o c e s s e d l a t e r . The theorem g i v e s t h e p r o b a b i l i t y d i s t r i b u t i o n o f

s p a c i n g s between e v e n t s . T h i s d i s t r i b u t i o n i s approx i m a t e l y e x p o n e n t i a l , s o t h a t s t o r a g e moments c o n s t i t u t e a Poisson process. The main g o a l of t h e s y s t e m o f p e r c e p t i o n p r e s e n t e d h e r e was t o e x p l i c a t e t h e phenomenon of f i l t e r i n g , or p a r t i a l c o n t r o l , i n t h e p r o c e s s of p e r c e p t i o n o f obj e c t s . It was shown how one c a n , i n a c o n s i s t e n t , psyc h o l o g i c a l l y p l a u s i b l e , and e x p e r i m e n t a l l y t e s t a b l e way, s p e a k o f t h e p r o c e s s of f i l t e r i n g ( c o n t r o l ) i n

p e r c e p t i o n . Moreover, t h e o r i z i n g shown h e r e d e p a r t s s u b s t a n t i a l l y from t h e known u n t i l now m a t h e m a t i c a l a p p r o a c h e s t o p a t t e r n r e c o g n i t i o n i n t h e domain o f m a t h e m a t i c a l psychology and a r t i f i c i a l i n t e l l i g e n c e . One may hope t h a t i t w i l l i n f l u e n c e t h e new s t y l e of r e s e a r c h .

FORMAL SEMIOTICS

377

4 . OBSERVABILITY AND CHANGE

I n t h i s s e c t i o n w e s h a l l c o n s i d e r some problems conn e c t e d w i t h t a k i n g o b s e r v a t i o n s , d e s c r i p t i o n s and i d e n t i f i c a t i o n of o b j e c t s .

4.1. Joint observability We s t a r t from a g e n e r a l framework t o e x p r e s s t h e problems o f s p e c i a l importance f o r s c i e n t i f i c r e s e a r c h , namely j o i n t o b s e r v a b i l i t y (or: m u l t i o b s e r v a b i l i t y ) . Generally, a researcher i s interested i n a c e r t a i n phenomenon. The l a t t e r c o n c e p t i s a p r i m i t i v e o n e , i n t h e s e n s e t h a t we s h a l l n o t g i v e any f o r m a l means of d i s t i n g u i s h i n g a phenomenon from non-phenomenon; i n s t e a d , we s h a l l g i v e a f o r m a l scheme t o d e s c r i b e phenomena. F o r m a l l y , a c o n c e p t u a l framework i s a f a m i l y

(4.1) where e a c h X i s a random ( i n extreme c a s e : d e t e r g,s m i n i s t i c ) v a r i a b l e , depending on t h e i n d i c e s from G and S , and C s i s a c l a s s o f s u b s e t s of G , t o be i n t e r preted l a t e r . The i n t e n d e d i n t e r p r e t a t i o n i s s u c h t h a t e l e m e n t s o f G

a r e l a b e l s o f v a r i a b l e s ( t h e i r names), w h i l e e l e m e n t s o f S a r e l a b e l s o f " c o n t r o l " p a r a m e t e r s . I n extreme c a s e w e may have S c o n s i s t i n g o f one e l e m e n t o n l y , s o

CHAPTER 4

378

t h a t t h e phenomenon i s n o t c o n t r o l l a b l e .

The phenomenon i s now i n t e r p r e t e d as some s o r t o f t h e o r e t i c a l r e l a t i o n s h i p between v a r i a b l e s i i n t h e s y s t e m ( 4 . l ) , t o which one may a s s i g n a name. Such phenomenon may o r may n o t o c c u r i n a p a r t i c u l a r i n s t a n c e , depending on t h e a c t u a l v a l u e s o f t h e v a r i a b l e s . To f i x t h e i d e a s and g u i d e somewhat t h e i n t u i t i o n , i t i s n e c e s s a r y t o g i v e some examples.

Example 1. I n many problems of s c i e n t i f i c r e s e a r c h one i s i n t e r e s t e d i n a r e l a t i o n s h i p between two v a r i a b l e s , s a y x and y . T y p i c a l l y , o b s e r v a t i o n s have t h e form o f pairs (xl,yl), ( x n , y n ) of t h e v a l u e s o f t h e depen-

...,

d e n t v a r i a b l e y a s s o c i a t e d w i t h v a l u e s of t h e independ e n t v a r i a b l e x. One may t h e n t r y t o f i n d t h i s r e l a t i o n s h i p be u s i n g r e g r e s s i o n methods, e t c . I n p r a c t i c e , w e have h e r e two c o n c e p t u a l l y d i s t i n c t s i t u a t i o n s : ( a ) t h e v a l u e s of independent v a r i a b l e x can be c o n t r o l l e d by t h e e x p e r i m e n t e r ; ( b ) t h e v a l u e s of x a r e n o t c o n t r o l l a b l e . The f i r s t c a s e may be exempl i f i e d by t a k i n g x as a d o s e , and y as a r e s p o n s e ; t h e second by t a k i n g a s x t h e number o f c a r a c c i d e n t s , and by y t h e number o f d e a t h s caused by them. The f i r s t c a s e w i l l be r e p r e s e n t e d by t a k i n g p o s s i b l e v a l u e s o f x as l a b e l s s o f c o n t o l l a b l e v a r i a b l e s , and t a k i n g a s G a s i n g l e t o n ( s i n c e o n l y one random v a r i a b l e , namely y , i s o b s e r v e d ) . I n t h e second c a s e , we t a k e S as a s i n g l e t o n ( s i n c e t h e r e i s no c o n t r o l ) , and G = {'1,23 f o r v a r i a b l e x , and 2

--

, where

for variable y.

1 stands

FORMAL SEMIOTICS

Example 2 .

3 79

For a phenomenon s u c h a s " c o n t a g i o n " , t h e

s e t o f v a r i a b l e s which may p o t e n t i a l l y b e o b s e r v e d i s e x t r e m e l y l a r g e . It may c o n t a i n v a r i o u s r e s p o n s e s o f human and a n i m a l organisms t o d i f f e r e n t t y p e s o f cont a c t s w i t h i n f e c t i o n s m a t e r i a l . T h i s s e t may a l s o cont a i n v a r i a b l e s p e r t a i n i n g t o o b s e r v a t i o n o f immunity, e p i d e m i c a l s p r e a d , e t c . The r e s e a r c h e r s t u d y i n g "cont a g i o n " would have t o narrow down t h e scope o f h i s s t u d y t o some s u b s e t s o f S and G . N O W , w i t h o u t any loss of g e n e r a l i t y , we may assume t h a t

e l e m e n t s o f S e x c l u d e one a n o t h e r , s o t h a t i n e a c h i n s t a n c e e x a c t l y one element w i l l be s e l e c t e d ( t h e c a s e o f no c o n t r o l may b e i n t e r p r e t e d a s h a v i n g S cons i s t i n g o f j u s t one e l e m e n t ) . I n c a s e when one may c o n t r o l a number o f p a r a n e t e r s , e l e m e n t s o f S w i l l be v e c t o r s , e a c h r e p r e s e n t i n g a p o s s i b l e c o n f i g u r a t i o n of parameters. A s r e g a r d s C s J i t s e l e m e n t s w i l l be i n t e r p r e t e d as

l a b e l s o f v a r i a b l e s which may be o b s e r v e d j o i n t l y i n t h e same i n s t a n c e ( " r e a l i z a t i o n " o f t h e s e t o f random

v a r i a b l e s ) , when t h e c o n t r o l p a r a m e t e r i s s . The c l a s s e s Cs a r e assumed t o s a t i s f y t h e f o l l o w i n g HYPOTHESIS. F o r e v e r y s d S , t h e c l a s s Cs i s nonempty, and moreover, whenever A C Cs and B C A , t h e n B G Cs. We have t h e n

a l a t t i c e under o p e r a t i o n o f I n t e r s e c t i o n , i . e . whenever A , B E Cs, then A 0 B C Cs.

THEOREM. The c l a s s

If A =

{'g

Cs i s

.,gn3(Cs,

then a l l variables

380

CHAPTER 4

may be o b s e r v e d i n t h e same i n s t a n c e . We s h a l l a l s o s a y t h a t i n t h i s c a s e , t h e s e t A a l l o w s multi-observ-

a t i o n under s ( t h i s i s n o t to be c o n f u s e d w i t h r e p l i c a t i o n o f o b s e r v a t i o n s o f t h e same v a r i a b l e ) . We may now i n t r o d u c e some c o n c e p t s which a r e c o n n e c t ed w i t h t h e above s e t u p i n a n a t u r a l w a y . F i r s t l y , l e t u s s a y t h a t a s e t A C G i s s-maximally m u l t i - o b s e r v a b l e , i f A & Cs and moreover, i f A C B and B e C

*

S'

then A = B.

L e t Cs d e n o t e t h e c l a s s o f a l l s-maximally multi-observable subsets of G . Denote now

R(s) =

UA

(4.2)

A €Cs We s h a l l c a l l R(s) t h e o b s e r v a b i l i t y domain under cont r o l s , and i n c a s e when R(s) = G , w e s a y t h a t s a l l o w s complete i n d i v i d u a l o b s e r v a b i l i t y .

If G E Cs, we s a y t h a t s a l l o w s f u l l m u l t i - o b s e r v a b i l i - , . t y . I n d e e d , from Hypotheses a b o u t Cs i t f o l l o w s t h a t we have

If

then

s a l l o w s complete i n d i v i d u a l = {G\, s o t h a t t h e o n l y maximo b s e r v a b i l i t y , and a l l y multi-observable s e t i s G .

THEOREM.

G € Cs,

* Cs

I f R(s) # G , t h e n we s a y t h a t s p r e v e n t s o b s e r v a b i l i t y

381

FORMAL SEMIOTlCS

of any X

g,s

with g

t'

G

-

R(s).

A s a l r e a d y m e n t i o n e d , a phenomenon i s i d e n t i f i e d w i t h

some t h e o r e t i c a l p r o p e r t y o f c e r t a i n v a r i a b l e s . Its r e a l i z a t i o n may t h e r e f o r e be i d e n t i f i e d w i t h :

-- a s e t o f l a b e l s , gl, g 2 , . . . , gn o f v a r i a b l e s , which d e t e r m i n e whether or n o t t h e phenomenon o c c u r r e d i n a given instance;

--

a c e r t a i n s u b s e t Q o f n-dimensional s p a c e , which n c o r r e s p o n d s t o r e a l i z a t i o n s o f t h e phenomenon.

We s a y namely t h a t t h e phenomenon, s a y Ti , o c c u r r e d under s, or o c c u r r e d i n a g i v e n r e a l i z a t i o n under c o n t r o l s, i f

DEFINITION. We s a y t h a t

i s m u l t i - o b s e r v a b l e under

s, i f It i s q u i t e p o s s i b l e t h a t a g i v e n phenomenon i s m u l t i -

o b s e r v a b l e under some c o n t r o l p a r a m e t e r s, and n o t o b s e r v a b l e under some o t h e r c o n t r o l s s'. I f t h e phenomenon

(A,Qn)

i s c h a r a c t e r i z e d by t h e p a i r

w i t h A = {glJ...,gn?

a s d e s c r i b e d above, t h e n

i s t h e s e t o f a l l c o n t r o l v a l u e s s under which multi-observable.

If s k Q ( R ) , t h e n

li i s

Ti i s n o t m u l t i - o b s e r v a b l e under s,

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and one h a s t o r e s t r i c t t h e o b s e r v a t i o n s t o

some

s u b s e t B C A , w i t h B E Cs. Now, observations of X

f o r g E B u s u a l l y do n o t g,s p r o v i d e complete i n f o r m a t i o n whether i s realized o r not i n a p a r t i c u l a r instance. I f B = j gl,...,gml, then the vector

p r o v i d e s o n l y p a r t i a l i n f o r m a t i o n about t h e r e a l i z a t i o n o f f i . L e t f g m + l , . . . , g n l = A - B be t h e unobserva b l e v a r i a b l e s ( i f one d e c i d e s t o o b s e r v e v a r i a b l e s i n B y i . e . v a r i a b l e s unobservable j o i n t l y w i t h t h o s e from B), and l e t

By d e f i n i t i o n , t h e phenomenon

occurred i f

and d i d n o t o c c u r o t h e r w i s e . Given t h e o b s e r v e d v a l u e s can t h e n d e t e r m i n e t h e p r o b a b i l i t y , or p o s s i -

zB, one

b i l i t y , that

i s realized, that i s

or

The first q u a n t i t y r e q u i r e s t h e knowledge of j o i n t

p r o b a b i l i t y d i s t r i b u t i o n of

ZB

and &A-B,

which may

383

FORMAL SEMIOTICS

be a v a i l a b l e o r n o t . The second depends on t h e d e g r e e with additional o f p o s s i b i l i t y of "complementing" c o o r d i n a t e s , s o t h a t t h e r e s u l t i n g j o i n t v e c t o r would be i n Qn.

zB

If t h e j o i n t d i s t r i b u t i o n i s a v a i l a b l e f o r any s e t B ,

one can p o s e t h e problem o f o p t i m a l , o r most informat i v e , o b s e r v a b i l i t y . If I A ( B ) s t a n d s for t h e amount of i n f o r m a t i o n about -A-B X contained i n X ( i . e . I A ( B ) -B i s t h e conditional entropy of X given X B ) , t h e n A-B t h e o p t i m a l s e t B may be chosen as s u c h t h a t o p t i m i zes I A ( B ) . A s a n example, one can t h i n k h e r e o f t h e problem of

p r e d i c t i o n o f some phenomenon. To d e t e r m i n e whether o c c u r r e d or n o t , one n e e d s t h e o b s e r v a t i o n s of v a r i a b l e s

Suppose t h a t l a b e l gn c o n c e r n s t h e v a r i a b l e i n some f u t u r e , hence not o b s e r v a b l e a t p r e s e n t . Other v a r i a b l e s a r e o b s e r v a b l e a t p r e s e n t , though p e r h a p s n o t all ( d u e t o , s a y , c o s t o f o b s e r v a t i o n s , which may a l l o w t o o b s e r v e o n l y a c e r t a i n number, s a y r , o u t o f n-1 v a r i a b l e s ) . The q u e s t i o n i s t h e n o f s e l e c t i n g t h e most informative set of r v a r i a b l e s out of X , s > * * * Ygx l's i . e . v a r i a b l e s which would p r o v i d e as muih informaf?ion

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384

4.2.

Masks and f i l t e r s

I n t h e p r e c e d i n g a n a l y s i s , no s p e c i f i c a s s u m p t i o n s were made about t h e n a t u r e of e l e m e n t s o f t h e s e t of l a b e l s G. I n p r a c t i c e , l a b e l s g 6 G a r e u s u a l l y m u l t i d i m e n s i o n a l , w i t h one c o o r d i n a t e r e p r e s e n t i n g t i m e . Thus, we may w r i t e G as a C a r t e s i a n p r o d u c t G = H x T

where T i s t h e t i m e a x i s , and H i s a s e t o f l a b e l s which d e s c r i b e t h e n a t u r e o f t h e v a r i a b l e w i t h o u t spec i f y i n g t i m e . I n o t h e r words, g = ( h , t ) i s t o be i n t e r p r e t e d as o b s e r v a t i o n of v a r i a b l e l a b e l e d h a t t i m e t ( e . g . h may b e "blood p r e s s u r e " , t o be o b s e r v e d

a t some f i x e d moments t l , t 2 , . . . a f t e r a n a d m i n i s t r a t i o n of a drug, e t c . ) . Instead of X i t w i l l now be c o n v e n i e n t t o w r i t e g,s X ( t ) f o r t h e v a l u e o f t h e v a r i a b l e l a b e l e d h , under h,s c o n t r o l s , a t t i m e t ( i n t h e example above, s m a y s t a n d f o r t h e dose o f t h e drug, e t c . ) . The c l a s s Cs o f j o i n t l y o b s e r v a b l e v a r i a b l e s a c q u i r e s now a s t r u c t u r e , due t o t h e s t r u c t u r e imposed on G. Thus, a s e t A

Cs i s now a s e t o f p a i r s (4.3)

N a t u r a l l y , some o f t h e v a l u e s h i , as w e l l as some o f t h e v a l u e s t i , may r e p e a t . Let u s t h e r e f o r e d e f i n e , for a fixed h E H

and s e t A g i v e n b y ( 4 . 3 )

385

FORMAL SEMIOTICS

t o be c a l l e d t e m p o r a l t r a c e o f h under A , or h-mask of A . The s e t T A ( h ) i s simply t h e s e t o f a l l moments

a t which t h e v a r i a b l e l a b e l e d h i s t o be o b s e r v e d . I f T A ( h ) = t l , . , . , t , t h e n w e observe (under c o n t r o l k s ) the values X ( t 1 ) , . . . ,X h Y s ( t k ) . h,s

On t h e o t h e r hand, one can t a k e a f i x e d moment t , and consider the s e t MA(t)

= { h C H:

( h , t ) 15 A )

o f a l l v a r i a b l e s which a r e t o be o b s e r v e d a t t i m e t . The s e t M A ( t ) w i l l be c a l l e d a t-mask o f o b s e r v a t i o n , and t h e c l a s s o f a l l t-masks w i l l b e c a l l e d s h o r t l y a mask. T h i s n o t i o n g e n e r a l i z e s t h e c o n c e p t o f mask i n t r o d u c e d b y K l i r (1972). Given t h e above n o t i o n s , one can e x p r e s s v a r i o u s cons t r a i n t s on j o i n t o b s e r v a b i l i t y

.

Thus, f o r g i v e n T A ( h ) , d e f i n e AA(h) =

min

Ilt-t'I

: t,t'

T A ( h ) , t # t f f.(4.4)

T h i s v a l u e r e p r e s e n t s t h e minimal t e m p o r a l s p a c i n g

between o b s e r v a t i o n o f v a r i a b l e h, I f we p u t A(h) =

inf

AA(h)

(4.5)

A 6 Cs

then

A ( h ) d e s c r i b e s t h e minimal t e m p o r a l s p a c i n g

a d m i s s i b l e f o r v a r i a b l e h . I n o t h e r words, any h-mask

i n v o l v i n g two moments c l o s e r t h a n A ( h ) i s n o t admis s i b l e ; t h e r e i s no s e t A o f j o i n t l y o b s e r v a b l e v a r i a b l e s which would a l l o w o b s e r v i n g X ( t ) for t c l o s h,s er than A(h>.

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386

Now, masks d e s c r i b e t h e t e m p o r a l

p a t t e r n s of variab-

l e s t o be o b s e r v e d , and a l s o t h e s e t s o f v a r i a b l e s t o b e o b s e r v e d a t any g i v e n t i m e . I n t h i s s e n s e , t h e y describe the observer's decision (within t h e constrai n t s o f j o i n t o b s e r v a b i l i t y ) about t h e v a r i a b l e s which a r e t o be r e c o r d e d i n a database.

given instance, t o e n t e r t h e

However, even i f t h e o b s e r v e r d e c i d e s t o r e c o r d some v a l u e s , i t may happen t h a t t h e s e v a r i a b l e s w i l l n o t be a v a i l a b l e ; t h e y may be c e n s o r e d , o r f i l t e r e d . T h i s means i n p r a c t i c e t h a t some v a l u e s , i n a g i v e n i n s t a n c e o f t h e phenomenon, may n o t " o c c u r " , o r t h e i r obs e r v a t i o n may be o b s c u r e d i n some o t h e r w a y . A t y p i c a l case occurs i n gathering d a t a f o r medical

r e s e a r c h , where t h e d a t a i n a follow-up s t u d y o f a group o f p a t i e n t s may be u n a v a i l a b l e simply b e c a u s e some o f t h e p a t i e n t s d i e d b e f o r e t h e t i m e o f o b s e r v a tion, left the city, etc. G e n e r a l l y , suppose t h a t a p e r s o n d e c i d e s t o o b s e r v e t h e v a r i a b l e s from some s e t A C where A h a s t h e S'

form ( 4 . 3 ) . T h i s r e q u i r e s o b s e r v i n g t h e v a r i a b l e s

By a f i l t e r w e s h a l l mean a random v a r i a b l e , whose

v a l u e s are s u b s e t s o f A , o r simply -- a random s u b s e t o f A . Thus, a f i l t e r i s d e s c r i b e d b y i t s d i s t r i b u t i o n , i . e . a function

f:

2A

+[O,ll

s u c h t h a t t h e sum o f a l l p r o b a b i l i t i e s e q u a l s o n e , that is

FORMAL SEMIOTICS

z

387

f ( B ) = 1.

BC A

The o p e r a t i o n of a f i l t e r i s t o be i n t e r p r e t e d s o t h a t w i t h p r o b a b i l i t y f ( B ) only t h e v a r i a b l e s

w i l l be o b s e r v a b l e , w h i l e t h e v a r i a b l e s w i t h i n d i c e s (h,t)

k A - B w i l l n o t be o b s e r v a b l e ( i n t h e g i v e n

instance). A f i l t e r with f ( 0 )

> 0 may

be c a l l e d a f i l t e r w i t h

b l a c k o u t p o s s i b i l i t y : i t may happen t h a t none of t h e v a r i a b l e s w i l l be o b s e r v a b l e . On t h e o t h e r e x t r e m e , i f f ( A ) = 1, t h e f i l t e r w i l l n o t d i s t u r b any o b s e r v a t i o n s . V a r i o u s t y p e s of f i l t e r s may now be d e f i n e d t h r o u g h t h e p r o p e r t i e s of p r o b a b i l i t y d i s t r i b u t i o n f(B). I n p a r t i c u l a r , a f i l t e r w i l l be c a l l e d s i m p l e , i f there i s a function

such that

T h i s c o r r e s p o n d s t o t h e c a s e when e a c h v a r i a b l e i s

“ f i l t e r e d “ or n o t f i l t e r e d i n d e p e n d e n t l y , w i t h a p p r o p r i a t e p r o b a b i l i t y . Such f i l t e r s , for t h e c a s e of P o i s s o n p r o c e s s e s , were c o n s i d e r e d i n Nowakowska ( 1 9 8 3 a ) . I n p a r t i c u l a r , if p ( h , t ) = p for a l l ( h , t ) & A , and A c o n t a i n s n e l e m e n t s , t h e n t h e p r o b a b i l i t y of a f i l t e r B w i t h k e l e m e n t s i s g i v e n by t h e b i n o m i a l formula

388

CHAPTER 4

(:)pk(l-p)n-k,

k = O,l,...,n,

so t h a t t h e p r o b a b i l i t y of a blackout ( a l l v a r i a b l e s u n o b s e r v a b l e ) i s (1-p) n

.

The main problem l i e s i n d e t e r m i n i n g t o which e x t e n t t h e e x i s t e n c e of a f i l t e r d i s t o r t s t h e p o s s i b i l i t y of s t a t i s t i c a l i n f e r e n c e about t h e phenomenon. T h i s ext e n t may v a r y ; i n t h e c a s e o f P o i s s o n p r o c e s s e s c i t e d above, f i l t e r i n g m e r e l y slowed down t h e p r o c e s s o f e s t i m a t i o n , w i t h o u t a f f e c t i n g i t i n any o t h e r way. On t h e o t h e r hand, i n c a s e of c e n s o r i n g d a t a e . g . i n med i c a l r e s e a r c h , t h e e f f e c t may be s u b s t a n t i a l , and requiring special s t a t i s t i c a l techniques.

4.3. O p e r a t i o n s on masks and f i l t e r s O f t e n masks, i . e . p a r t i c u l a r c o n s t r a i n t s on j o i n t obs e r v a b i l i t y (and a t t h e same t i m e , p l a n s o f c o l l e c t i n g d a t a , m e e t i n g t h e s e c o n s t r a i n t s ) a r e superimposed on one a n o t h e r . T h i s may happen when t h e c o n s t r a i n t s a r e due t o v a r i o u s f a c t o r s ; f o r i n s t a n c e , one c o n s t r a i n t may be due t o c o s t s o f t a k i n g o b s e r v a t i o n s , o t h e r s may be due t o a v a i l a b i l i t y o f equipment, w h i l e s t i l l

o t h e r s -- t o p h y s i c a l l i m i t a t i o n s . A s a r e s u l t , t h e o b s e r v a t i o n s a r e t a k e n a c c o r d i n g t o a mask which i s an i n t e r s e c t i o n , t o be r e f e r r e d t o as multi-mask.

...,

C s a r e masks, t h e n t h e AN( F o r m a l l y , i f A1, A2, multi-mask g e n e r a t e d by t h e m i s A = A A fi, AN. 1 The theorem o f t h e p r e c e d i n g s e c t i o n e n s u r e s t h a t

.. .

we have A g C s ( i . e . t h e c l a s s Cs i s c l o s e d under t a k ing intersections).

FORMAL SEMIOTICS

389

...

Another problem a r i s e s when a mask A = { ( h l , t l ) , , ( h n , t n ) ] i s t r a n s l a t e d by some amount. Such a t r a n s l a t i o n i s d e f i n e d by

provided t h a t At

Cs.

G e n e r a l l y , we may d e f i n e

s o t h a t t h e s e two t i m e s d e t e r m i n e t h e e a r l i e s t and

l a t e s t t i m e of o b s e r v a t i o n under mask A . These times may be c a l l e d anchor and h o r i z o n of t h e m a s k , r e s p e c t -

ively.

N a t u r a l l y , we have

tmin= t + t min A At

, tmax = t + tmax A, At

i . e . t h e a n c h o r and h o r i z o n o f a t r a n s l a t e d mask a r e a l s o s h i f t e d b y t h e same amount. V a r i o u s r e l a t i o n s between masks may be d e f i n e d by considering t h e temporal equivalences (e.g. i d e n t i t y o f t e m p o r a l t r a c e s up t o a t r a n s l a t i o n , e t c . ) , and i d e n t i t y of v a r i a b l e s o b s e r v e d , r e g a r d l e s s of t h e t i m e s , i . e . i d e n t i t y of t-masks and h-masks a s s o c i a t e d w i t h two masks.

...,

Suppose now t h a t we have t h e masks A1, AN, each accompanied b y a f i l t e r c h a r a c t e r i s t i c f o r t h i s mask. These f i l t e r s g e n e r a t e random s e t s B1,. : ,BN, w i t h Then t h e opaque s u p e r p o s i t i o n of t h e s e Bi C. A i . f i l t e r s i s d e f i n e d as

.

3 90

CHAPTER 4

B

=nB , ,

w h i l e t h e i r t r a n s p a r e n t s u p e r p o s i t i o n i s d e f i n e d as

Thus, an opaque s u p e r p o s i t i o n a l l o w s f o r o b s e r v a t i o n of a v a r i a b l e only i f it i s f i l t e r e d through a l l of t h e f i l t e r s Bi,

while a transparent superposition

allows t h e o b s e r v a t i o n of a v a r i a b l e , i f it i s f i l t e r ed by a t l e a s t one o f t h e f i l t e r s B

i'

Suppose now t h a t t h e f i l t e r s a r e g e n e r a t e d i n d e p e n d e n t -

l y . One c a n t h e n a s k f o r t h e d i s t r i b u t i o n o f t h e r e s u l t i n g opaque and t r a n s p a r e n t s u p e r p o s i t i o n o f t h e filters. T h i s problem may be s o l v e d f o r t h e c a s e o f t h e s i m p l e f i l t e r s , w i t h i d e n t i c a l p r o b a b i l i t i e s . Suppose namely t h a t i n f i l t e r Bi

e a c h o f t h e v a r i a b l e s i n Ai

Bi w i t h p r o b a b i l i t y p i ,

is in

independently of other variab-

l e s . We have t h e n THEOREM.

&&

m be t h e s i z e o f t h e s e t

n

A i . Then th e p r o b a b i l i t y t h a t t h e opaque s u p e r p o s h i o n o f f i l t e r s B1,

...,

BN w i l l c o n s i s t o f e x a c t l y k e l e m e n t s

equals Qqk(l

-

where q = plp 2...pN.

q)m-k , k = 0 , 1 ,

...,m ,

(4.6)

I n t h e c a s e o f t r a n s p a r e n t super-

p o s i t i o n , formula ( 4 . 6 ) z l s o h o l d s , but w i t h

FORMAL SEMIOTICS

391

4.4. O b s e r v a b i l i t y o f change I n t h e p r e c e d i n g s e c t i o n s , t h e a n a l y s i s concerned obs e r v a b i l i t y o f v a l u e s o f some v a r i a b l e s . I f a v a r i a b l e i s o b s e r v e d a t t i m e t l and t 2 , and i t s v a l u e s a t t h o s e t i m e s a r e X ( t ) = x and X ( t , ) = y w i t h x # y , t h e n 1 w e know t h a t a change h a s o c c u r r e d between t l and t 2 . For t h e moment, l e t u s d i s r e g a r d t h e d i s t i n c t i o n b e t ween " e s s e n t i a l " or " s i g n i f i c a n t " c h a n g e s , and " i n e s s e n t i a l " or " i n s i g n i f i c a n t f 1 o n e s , and q u a l i f y any inequa l i t y x # y as a change. Two e s s e n t i a l f a c t s about changes a r e w o r t h s t r e s s i n g here. F i r s t l y , i t may happen t h a t t h e i n f o r m a t i o n whether x = y o r x # y may be o b t a i n e d w i t h o u t a c t u a l l y o b s e r v i n g x and y s e p a r a t e l y . T h i s means t h a t i n o r d e r t o d e t e r m i n e whether or n o t a change has o c c u r r e d w e may n o t r e q u i r e t h e knowledge o f b o t h x and y ( t h e s i m p l e s t example i s i n c a s e when x and y a r e i n t e g e r v a l u e d

s i z e s o f some s e t s : one can t h e n d e t e r m i n e whether

x

y

by p r o c e s s o f matching p a i r s , w i t h o u t l e a r n i n g t h e v a l u e s x and y ) . S e c o n d l y , t h e i n f e r e n c e about change works o n l y i n one d i r e c t i o n : i f x # y , t h e n a change h a s o c c u r r e d ; i f x = y , t h e n a l l we know i s t h a t t h e v a l u e s a t t h e endp o i n t s o f an i n t e r v a l a r e t h e same. I f t h e i n t e r m e d i a t e v a l u e s a r e n o t o b s e r v e d , i t i s p o s s i b l e t h a t a ser i e s o f changes o c c u r r e d i n between, e v e n t u a l l y c a n c e l l i n g o u t t o t h e same v a l u e . The p o i n t h e r e i s t h a t i n o r d e r t o i n f e r from t h e ob-

CHAPTER 4

392

s e r v e d f a c t t h a t x = y t h a t t h e change d i d n o t o c c u r , one n e e d s some a d d i t i o n a l knowledge about t h e n a t u r e o f t h e phenomenon a n a l y s e d , and no f o r m a l t h e o r y can r e p l a c e t h i s knowledge. To g i v e a s i m p l e example, suppose t h a t t l and t 2 a r e 1 0 m i n u t e s a p a r t ( s a y ) , and we know t h a t X ( t l )

equals

X(t2).

i s t h e number o f b i r t h s i n a g i v e n h o s p i t a l , c o u n t i n g from some f i x e d moment ( m i d n i g h t , s a y ) u n t i l t , t h e n X ( t l ) = X ( t , ) means t h a t no b i r t h s o c c u r r e d between t l and t 2 . Here we u s e t h e f a c t t h a t , b y deIf X ( t )

f i n i t i o n , t h e f u n c t i o n X ( t ) may o n l y i n c r e a s e . On t h e o t h e r hand, i f X ( t ) i s t h e number o f p e r s o n s i n a room a t t i m e t , and X ( t , ) = X ( t ) , we c a n n o t i n f e r I 2 t h a t no change o c c u r r e d i n between: some p e r s o n s may have e n t e r e d t h e room, and some o t h e r s may have l e f t it. We s h a l l now g i v e a f o r m a l d e s c r i p t i o n o f s p e c i a l t y p e o f masks, t o be c a l l e d cd-masks ( f o r : change d e t e c t i o n masks). A cd mask i s a s e t o f p a i r s ( h , t ) o f t h e form

(4.7) where t i

ti

f o r i = 1,2,

...,n .

N a t u r a l l y , one c o n s i d e r s o n l y masks which a r e i n C S' i . e . a l l o w j o i n t o b s e r v a b i l i t y of t h e v a r i a b l e s i n q u e s t i o n a t s p e c i f i e d times.

n L e t C s b e t h e c l a s s o f a l l masks o f t h e form ( 4 . 7 )

FORMAL SEMIOTlCS

393

with A & Cs.

S ince i n each consecutive p a i r t h e f i r s t e l e m e n t s c o i n c i d e , i t i s more c o n v e n i e n t t o r e p r e s e n t t h e mask ( 4 . 7 ) i n t h e form

t o be i n t e r p r e t e d : o b s e r v e t h e v a r i a b l e l a b e l e d hi a t t i m e s t i and and d e t e r m i n e whether a change has o c c u r r e d between t h e s e moments. N a t u r a l l y , some o f t h e v a l u e s hi may c o i n c i d e , if a v a r i a b l e i s o b s e r v e d more t h a n two t i m e s .

ti

= 0 i f a change o f v a r i a b l e l a b e l e d hi 1 1 1 o c c u r r e d between t i and ti, and Q ( h i , t i , t i ) = 1 o t h e r wise. One can t h e n d e f i n e t h e c h a n g e - d e t e c t i o n f u n c t i o n a s s o c i a t e d w i t h mask ( 4 . 8 ) as

Let Q(h.,t.,t!)

...

(4.9) We have h e r e G ( A ) = 0 whenever a change o c c u r s i n any o f t h e v a r i a b l e s and between t h e t i m e s s p e c i f i e d b y t h e m ask A , and G ( A ) = 1 i f no change was d e t e c t e d ( w h i c h , as a l r e a d y e x p l a i n e d , d o e s n o t i m p l y t h a t a change d i d n o t o c c u r ) . The s e t

H(A)

= lh:

(h,t,t')

A for some t , t f 3

(4.10)

i s c a l l e d t h e b a s e o f t h e cd-mask, C l e a r l y , mask A i s

c a p a b l e of d e t e c t i n g a change o n l y i f i t c o n c e r n s t h e v a r i a b l e s i n a base; o t h e r changes are u n d e t e c t a b l e under mask A . On t h e o t h e r hand, l e t u s d e f i n e t h e set

CHAPTER 4

394

Then t h e s e t T ( A ) i s t h e s e l e c t i o n s e t , i n t h e s e n s e t h a t t h e changes w i l l be r e c o r d e d o n l y i f t h e y o c c u r i n one o f t h e i n t e r v a l s o f t h e s e l e c t i o n s e t . Masks may be o r d e r e d a c c o r d i n g t o t h e r e l a t i o n o f r e -

f i n e m e n t ; g e n e r a l l y , a r e f i n e m e n t o f a cd mask i s a mask c a p a b l e o f d e t e c t i n g m u l t i p l e c h a n g e s , o c c u r r i n g w i t h i n some i n t e r v a l s c o v e r e d by t h e o r i g i n a l m a s k , or changes o f o t h e r v a r i a b l e s ( e x t e n s i o n of a b a s e ) . F o r m a l l y , we s a y t h a t a mask A ' i s a r e f i n e m e n t o f mask A , i f H(A) C H ( A ' ) , and moreover, whenever a t r i p l e (h,t,t')

6 A , t h e n t h e r e e x i s t s a sequence

with (h,t,t('))

1,2

,..., m - 1 ,

t' A ' ,

(h,t (i),t(itl)) E

and ( h , t ( m ) , t ' ) E A ' .

A 1

for i =

Thus, w h i l e mask

A i s " s e n s i t i v e " t o a change i n t h e i n t e r v a l ( t , t ' )

( o f v a r i a b l e l a b e l e d h ) , mask A ' w i l l r e c o r d a change of t h i s v a r i a b l e i f i t o c c u r s anywhere i n one o f t h e i n t e r v a l s of t h e p a r t i t i o n ( 4 . 1 2 ) . O b v i o u s l y , u n d e r r e l a t i o n of r e f i n e m e n t , masks form a l a t t i c e ; however, t h e r e i s no "most r e f i n e d " mask, s i n c e e v e r y i n t e r v a l may be f u r t h e r s u b d i v i d e d , leadi n g t o a more r e f i n e d mask. We have h e r e t h e f o l l o w i n g theorem.

THEOREM. If A'

is

a refinement of A , t h e n G ( A t ) = 1 i m p l i e s G ( A ) = 1, i . e . whenever a r e f i n e d mask d e t e c t s no change, t h e n t h e o r i g i n a l mask d e t e c t s no change either.

FORMAL SEMIOlTCS

Finally,

395

l e t u s o b s e r v e t h a t a mask may be c r e a t e d

sequentially, i n the sense of observations being taken c o n d i t i o n a l l y on p r e v i o u s o b s e r v a t i o n s ( t h i s n o t i o n o f c o n d i t i o n a l mask a p p l i e s t o cd-masks as w e l l as t o t h e u s u a l m a s k s ) . Thus, a c o n d i t i o n a l mask i s r a n dom, and i s formed as f o l l o w s ( f o r s i m p l i c i t y , t h e d e s c r i p t i o n w i l l be g i v e n f o r one v a r i a b l e o n l y ; a g e n e r a l i z a t i o n for t h e c a s e o f more t h a n one v a r i a b l e i s immediate). F i r s t l y , we have a p a i r ( h , t ) , c a l l e d t h e s t a r t e r o f

random mask. Then, s u b s e q u e n t o b s e r v a t i o n s depend on ( t ) . The d e c i s i o n i s g i v e n i n t h e observed v a l u e X h,s form of a f u n c t i o n f , mapping t h e s e t R o f v a l u e s of X ( t ) i n t o t h e set h,s

t h e symbol * s i g n i f i e s s t o p p i n g t h e p r o c e s s o f t a k i n g o b s e r v a t i o n s ; t h u s , w e s t o p t h e p r o c e s s , i f w e have the condition f [ X h , s ( t ) ] = *. O t h e r w i s e , w e have f[XhJs(t)] = t t w i t h t' '> t , and n e x t o b s e r v a t i o n i s t a k e n a t t i m e t t . The p r o c e s s t h e n c o n t i n u e s i n t h i s fashion u n t i l the value * i s obtained, I n t h i s way, t h e moments t l , t 2 , a t which t h e o b s e r v a t i o n s a r e made ( o r , i n general case, the pairs (hi,ti) of variables t o b e o b s e r v e d and t i m e s o f t h e o b s e r v a t i o n s ) depend i n a random way on t h e p r e c e d i n g o b s e r v a t i o n s .

...

4 . 5 . Semantic a s p e c t s o f r e p r e s e n t a t i o n s We s h a l l now c o n s i d e r t h e problems i n t r o d u c t o r y t o

f o r m a l s e m i o t i c s , t o be c o n s i d e r e d i n n e x t s e c t i o n s ,

CHAPTER 4

396

namely t h e g e n e r a l scheme of r e p r e s e n t a t i o n o f o b j e c t s . We b e g i n w i t h s y s t e m s r e f e r r e d t o i n t h e i n t r o d u c t i o n as " c o n s t r a i n e d " , where we d e a l w i t h a c l a s s o f o b j e c t s which are a l i k e i n many r e s p e c t s , and d i f f e r o n l y by v a l u e s o f some a t t r i b u t e s . T h e s y s t e m h e r e w i l l have t h e form

(4.13) with t h e following intended interpretation. F i r s t l y , X i s a c l a s s o f o b j e c t s under c o n s i d e r a t i o n , and i s a n e q u i v a l e n c e on X . Thus, t h e s e t X i s p a r t i t i o n e d i n t o equivalence c l a s s e s of r e l a t i o n A , , The r e l a t i o n

l~ i s c o n n e c t e d w i t h t h e g o a l f o r which t h e s y s t e m i s u s e d : t h e symbol x m y means t h a t x and y and i n d i s t i n g u i s h a b l e from t h e p o i n t of view of t h e

goal of d e s c r i p t i o n , o r a r e "for p r a c t i c a l purposes will i d e n t i c a l " . Equivalence c l a s s e s of r e l a t i o n be c a l l e d t y p e s , and e l e m e n t s o f t h e same e q u i v a l e n c e

c l a s s w i l l be c a l l e d exemplars of a t y p e . The t y p e s w i l l b e d e n o t e d by T1, T 2 , so t h a t

...,

X = T

1

w T

2

u

...

(4.14)

w i t h XP- y i f f x , y 6 Ti for some i . I n t h i s c a s e , x and y a r e b o t h o f t y p e T or a r e exemplars o f t h a t i'

t4pe. Next, A i s t h e s e t o f a l l a t t r i b u t e v a l u e s , and 2 i s a n e q u i v a l e n c e i n A . Thus, s i m i l a r l y as w i t h t h e s e t X , t h e set A i s p a r t i t i o n e d i n t o equivalence c l a s s e s

of r e l a t i o n

, so

that

397

FORMAL SEMIOTICS

Each s e t A i w i l l be i n t e r p r e t e d as a s e t of a l l p o s s i b l e v a l u e s of some a t t r i b u t e . F o r i n s t a n c e , i f t h e f i r s t a t t r i b u t e s t a n d s for c o l o u r , t h e n e l e m e n t s o f A 1 w i l l be " r e d " , "green", e t c . From t h e c o n s t r u c t i o n of t h e s e t s Ai

a s b iff a,b f A

i

it follows that

for some i .

F i n a l l y , as r e g a r d s r e l a t i o n R , i t w i l l b e i n t e r p r e t e d as a r e l a t i o n o f " h a v i n g a v a l u e on a n a t t r i b u t e " . We p o s t u l a t e t h e r e f o r e ( b e g i n n i n g w i t h nonfuzzy c a s e ) t h a t R i s a binary r e l a t i o n i n X x A s a t i s f y i n g t h e f o l l o w i n g two c o n d i t i o n s :

(vb

c Ai)[xRb + a

=

blf

.

The f i r s t c o n d i t i o n means s i m p l y t h a t e a c h o b j e c t has

a v a l u e on a t l e a s t one a t t r i b u t e . T h e second c o n d i t i o n a s s e r t s t h a t whenever o b j e c t x

has a v a l u e on an a t t r i b u t e , t h i s v a l u e i s u n i q u e . I n

o t h e r words, e i t h e r a n a t t r i b u t e i s i n a p p l i c a b l e f o r a n o b j e c t , or it has e x a c t l y one v a l u e on t h i s a t t r i bute. Consequently, i t i s

I(x)

=

ti:

n a t u r a l t o denote

l a E Ai w i t h xRa]

(4.16)

CHAPTER 4

398

and from a s s u m p t i o n ( a ) i t f o l l o w s t h a t I ( x ) # 0 f o r e v e r y x. We may now d e f i n e v a r i o u s k i n d s o f e q u i v a l e n c e s on X , e x p r e s s i n g them t h r o u g h a t t r i b u t e s . F i r s t l y , we s h a l l d e f i n e

x

%

A

y i f (+a

E A) xRa i f f Y R ~ .

Thus, two o b j e c t s a r e i n r e l a t i o n -A,

i f t h e y have

t h e same v a l u e s o f a l l a t t r i b u t e s . S e c o n d l y , one may d e f i n e x -I

x

y, i f I(x) = I(y). If

y , t h e o b j e c t s w i l l be s a i d t o belong t o t h e

same a t t r i b u t e t y p e s . I n p r a c t i c e , i t means t h a t t h e s e t s o f a t t r i b u t e s a p p l i c a b l e t o them c o i n c i d e , a l t h o u g h t h e a t t r i b u t e v a l u e s may be d i f f e r e n t . L e t U1,

U2, ... b e t h e e q u i v a l e n c e c l a s s e s o f

-

l e t V1,

V2,

-I.

c e x A,,,

...

be t h e e q u i v a l e n c e c l a s s e s o f

y i m p l i e s t h a t x -I

u n i o n o f some U ' s . j

y , each

Thus, t h e p a r t i t i o n

finement o f p a r t i t i o n

...,

Sin-

Vi must b e t h e

tUi

i s a re-

yVij.

S i n c e our o b j e c t i v e i s t o d i s t i n g u i s h o b j e c t s t h e i r equivalence

A' and

2 , i.e.

up t o

up t o t h e i r t y p e s T1, T 2 ,

t h e f o l l o w i n g d e f i n i t i o n seems a p p r o p r i a t e .

D E F I N I T I O N . The s y s t e m ( 4 . 1 3 ) i s s a i d t o be i n f o r m a -

t i o n a l l y c o m p l e t e , i f f o r e v e r y Ui

there exists a type

.

T . w i t h Ui c T Otherwise, t h e system i s i n f o r m a t i o n J j a l l y incomplete. T h i s means t h a t t h e s y s t e m i s i n f o r m a t i o n a l l y c o m p l e t e ,

399

FORMAL SEMIOZTCS

i f t h e knowledge o f a l l a t t r i b u t e s o f t h e o b j e c t

a l l o w s t o i d e n t i f y i t , up t o t h e c l a s s o f e q u i v a l e n c e i n t h e degree s u f f i c i e n t f o r t h e g o a l . On t h e r o t h e r hand, i f t h e system i s n o t complete i n f o r m a t i o n a l l y , t h e n t h e r e e x i s t o b j e c t s which have t h e same v a l u e s o f a t t r i b u t e s ( h e n c e t h e y cannot be d i s t i n g u i s h e d u s i n g t h e a t t r i b u t e s from A ) , b u t a r e of d i f f e r e n t t y p e s , h e n c e p l a y v a r i o u s r o l e s from t h e p o i n t o f view o f t h e g o a l . of t h e r e l a t i o n v, i . e .

4.5.1. Algebra o f d e s c r i p t i o n s .

For e v e r y x , t h e s e t

I ( x ) describes t h e c l a s s of attributes applicable t o x , and for e v e r y i < I ( x ) t h e r e i s e x a c t l y one element a i ( x ) k A i such t h a t x R a i ( x ) . T h i s means t h a t f o r e v e r y x we have a f u n c t i o n , w i t h domain I ( x ) and values i n s e t s Ai, i & I ( x ) . This function provides a s i m p l e s t d e s c r i p t i o n of o b j e c t x. To d e v e l o p a conveni e n t f o r m a l i s m for i n t r o d u c i n g a n a l g e b r a o f d e s c r i p = {1,2,..,,NI be t h e s e t o f a l l a t t r i b u t e s ions, l e t ( o r , more p r e c i s e l y , a t t r i b u t e names). We s h a l l now d e f i n e a d e s c r i p t i o n as a v e c t o r

(w,

B ~ i, 6

W)

(4.17)

c o n s i s t i n g of a s e t W C i s a nonempty s e t o f a t t r i b u t e s , and Bi C A i i s a s u b s e t o f t h e s e t Ai o f v a l u e s of a t t r i b u t e i . Obviously, s p e c i f i c a t i o n of t h e v a l u e s of a t t r i b u t e s o f a n o b j e c t i s a d e s c r i p t i o n of t h i s o b j e c t i n t h e s e n s e o f d e f i n i t i o n ( 4 . 1 7 ) , w i t h I ( x ) = W , and e a c h Bi b e i n g a s i n g l e t o n Bi = 1 ai ( x ) ? . However, one c a n

CHAPTER 4

400

t h i n k a l s o o f d e s c r i p t i o n s which do n o t s p e c i f y e x a c t l y t h e v a l u e s o f a t t r i b u t e s , b u t g i v e o n l y some s e t s o f v a l u e s , presumed t o " c o v e r " t h e v a l u e s a i ( x ) , One can now d e f i n e some r e l a t i o n s and o p e r a t i o n s on d e s c r i p t i o n s o f t h e form ( 4 . 1 7 ) . F i r s t l y , i f

a r e two d e s c r i p t i o n s , we s a y t h a t do i s c o n t a i n e d i n d ' , o r : d ' i s an e x t e n s i o n o f d , i f

Bi

(b)

B;

f o r each i

C

W.

I n t h i s case w e shall write d C d ' .

Thus, an e x t e n s i o n c o n t a i n s a t l e a s t a3 many a t t r i b u t e s , and whenever a n a t t r i b u t e i s used i n b o t h d e s c r i p t i o n s , then the set o f values i n extension d l i s a t l e a s t as l a r g e as t h e s e t o f v a l u e s i n d . N e x t , l e t d and d 1 b e two d e s c r i p t i o n s w i t h i d e n t i c a l bases, W = W'. Then t h e i r i n t e r s e c t i o n d n d ' i s def i n e d as t h e d e s c r i p t i o n

while t h e i r disjunction d

(w,

B

~

i~

d ' i s defined as

B U' i C W). iy

F i n a l l y , a n o t h e r o p e r a t i o n w i l l be t h e j o i n o f d e s c r i p t ions, applicable t o descriptions d,dl with W D W f = @

.

401

FORMAL SEMIOTICS

Then t h e j o i n (W

\I

d

W',

+

d ' i s d e f i n e d as

Bi,

i

E W , B;, i < W').

It s h o u l d be o b s e r v e d t h a t w h i l e i n t e r s e c t i o n and d i s j u n c t i o n c o n c e r n t h e same s e t s o f a t t r i b u t e s , t h e o p e r a t i o n o f j o i n a l l o w s u s t o add new a t t r i b u t e s . A description with singleton base, i . e . with W =

CiT,

c o n s i s t i n g t h e r e f o r e o f a s i n g l e s e t Bi w i l l be c a l l e d e l e m e n t a r y . I n a d d i t i o n , i f Bi = ? a i l , w i t h a i E Ai, t h e d e s c r i p t i o n w i l l be c a l l e d simple. It f o l l o w s t h a t e v e r y d e s c r i p t i o n d = (W, Bi, i E W) of t h e form ( 4 . 1 7 ) i s a j o i n o f e l e m e n t a r y d e s c r i p t o n s d(Bi)

= ( f i J Bi), ,

so that

d(Bi).

d =

i E W T h i s form w i l l be c a l l e d a c a n o n i c a l r e p r e s e n t a t i o n o f

a description. Moreover, e v e r y e l e m e n t a r y d e s c r i p t i o n i s a d i s j u n c t i o n o f s i m p l e d e s c r i p t i o n s , namely d(Bi)

=

u

d(ia)).

a e Bi

C o n s e q u e n t l y , e v e r y d e s c r i p t i o n d has a c a n o n i c a l rep r e s e n t a t i o n as a j o i n o f d i s j u n c t i o n s o f s i m p l e descriptions d = i G W

u

d({aJ).

a c Bi

The o p e r a t i o n s d e f i n e d above impose a s t r u c t u r e on

t h e s e t o f a l l d e s c r i p t i o n s . A s w i l l b e shown i n n e x t

CHAPTER 4

402

s e c t i o n t h i s s t r u c t u r e imposes s t r u c t u r a l c o n s t r a i n t s on s e m a n t i c s o f d e s c r i p t i o n s .

4.5.2. A s s e r t i o n s and q u e r i e s . L e t now D d e n o t e t h e c l a s s o f a l l d e s c r i p t i o n s ( 4 . 1 7 ) . Such d e s c r i p t i o n s may be used i n two w a y s . One way may be d e s c r i b e d as an a s s e r t i o n " o b j e c t x i s d" ( i . e . " d e s c r i p t i o n d f i t s o b j e c t x " , o r " c o v e r s " o b j e c t x , e t c . ) . Here t h e s t a t e m e n t s p e c i f i e s a p a i r ( x , d ) , where x i s a n o b j e c t and d i s a d e s c r i p t i o n , presumed t o f i t t o t h i s o b j e c t . Such an a s s e r t i o n w i l l be d e n o t e d by x!d. I n a n o t h e r way, one may a s k f o r o b j e c t o r o b j e c t s which s a t i s f y a given d e s c r i p t i o n d . I n t h i s c a s e , only d i s g i v e n ; we s h a l l i n t e r p r e t i t as a q u e r y , t o b e d e n o t ed by ? d . I n b o t h c a s e s , w e s h a l l need a r e l a t i o n c o n n e c t i n g objects with descriptions , representing the t r u t h about t h e s e o b j e c t s . We s a y namely t h a t x i s d e s c r i b e d by d = ( W ,

i t W)

Bi,

( o r : d i s a t r u t h about x , d i s

t r u e f o r x ) , if I ( x ) = W,

ai(x)

$r

Bi

for all i t I(x).

N O W , as r e g a r d s a s s e r t i o n s x ! d , we have

x!d i s t r u e i f f

d i s a t r u t h f o r x.

2

Next,let d = d ( B . ) be t h e c a n o n i c a l r e p r e s e n t a t i o n 1 * 4 W of d as a j o i h o f e l e m e n t a r y d e s c r i p t i o n s . We have t h e n THEOREM.

x!d i s t r u e i f f d(Bi) i s t r u e f o r x f o r a l l i .

403

FORMAL SEMIOTlCS

Now, as r e g a r d s q u e r i e s , t h e s i t u a t i o n i s a s f o l l o w s . F i r s t l y , for e a c h d = (W, Bi, i 6 W) d e n o t e S(d) = q x € X : I(x) = W , a i ( x ) t Bi,

i t

W]

(4.18)

t h a t i s , t h e s e t of a l l o b j e c t s f o r which t h e d e s c r i p -

tion d i s true. The o p e r a t i o n s o f d e s c r i p t i o n s may now be a s s o c i a t e d w i t h o p e r a t i o n s ' o n s e t s S ( d ) as f o l l o w s . L e t d and d ' be d e s c r i p t i o n s w i t h t h e same b a s e W ,

so

B;, i € W ) , and l e t S ( d ) and S ( d ' ) be t h e c o r r e s p o n d i n g s e t s o f o b j e c t s f o r which d and d ' a r e t r u e . We have t h e n

that d = (W,

Bi,

i € W ) and d' = ( W ,

S(d u d ' ) =

and S(d n d ' ) = S(d)

#A

(4.20)

S(d').

A s r e g a r d s t h e o p e r a t i o n of j o i n , l e t d = W , Bi,

and d ' = ( W ' , S(d

+

BI, i € W 1 ) w i t h W d ' ) = S(d)

r\

S(d').

fi

W1 =

i € W)

0. Then (4.21)

C o n s e q u e n t l y , f o r m u l a s ( 4 . 1 9 ) - ( 4 . 2 1 ) e s t a b l i s h an isomorphism between t h e s t r u c t u r e o f t h e c l a s s of s e t s S ( d ) under o p e r a t i o n s of u n i o n s and i n t e r s e c t i o n s , and t h e c l a s s o f a l l d e s c r i p t i o n s , w i t h o p e r a t i o n s of i n t e r s e c t i o n , d i s j u n c t i o n and j o i n . We s h a l l now i n t e r p r e t S ( d ) as a f u l l r e p l y t o t h e query ?d. If S ( d ) = 0, i.e. t h e r e a r e no o b j e c t s x

404

CHAPTER 4

f o r w h i c h x ! d , t h e n t h e q u e r y ?d i s a n s w e r e d n e g a t i v e l y . If S ( d ) # @,

t h e answer i s p o s i t i v e . I f , i n a d d i t -

i o n , there e x i s t s an index j such t h a t S ( d ) C T

j’

where j i s a c l a s s of a b s t r a c t i o n of t h e e q u i v a l e n c e yv , w e s a y t h a t t h e q u e r y ? d h a s s a t i s f a c t o r y a n s w e r , o r t h a t d e s c r i p t i o n d i s s a t i s f a c t o r y . The j u s t i f i c a t i o n of t h e t e r m l i e s h e r e i n t h e f a c t t h a t t h e s e t S ( d ) c o n t a i n s o b j e c t s which are i d e n t i c a l from t h e p o i n t o f view o f t h e g o a l , i . e . i f x,y 6 S ( d ) , t h e n X

N

y.

We may now g i v e

a n i n t e r p r e t a t i o n o f t h e a b o v e con-

s i d e r a t i o n s w h i c h w i l l b e u s e f u l i n t h e s e q u e l , when we p a s s t o fuzzy a t t r i b u t e s , l i n g u i s t i c d e s c r i p t i o n s i n form o f v e r b a l c o p i e s , and g e n e r a l l y , l i n g u i s t i c measurements

and s e m i o t i c s .

E a c h d e s c r i p t i o n may b e r e g a r d e d a s a scheme b u i l t o u t o f s e t s o f a t t r i b u t e v a l u e s by l o g i c a l o p e r a t i o n s o f d i s j u n c t i o n , c o n j u n c t i o n and j o i n ( r e d u c i b l e a l w a y s t o c a n o n i c a l f o r m ) . To e a c h scheme t h e r e c o r r e s p o n d s a s e t o f o b j e c t s , whose s t r u c t u r e i s i s o m o r p h i c t o t h e l o g i c a l s t r u c t u r e o f t h e scheme of d e s c r i p t i o n .

4 . 5 . 3 . F u z z y a t t r i b u t e s a n d v e r b a l c o p i e s . L e t us now e x t e n d t h e t h e o r y o f t h e p r e c e d i n g s e c t i o n s s o as t o i n c l u d e t h e p o s s i b i l i t y o f some d e s c r i p t o r s b e i n g fuzzy. T h i s s i t u a t i o n i s o f s p e c i a l importance i n t h e c a s e when w e d e a l w i t h v e r b a l d e s c r i p t i o n s . The m o d i f i c a t i o n w i l l c o n c e r n t h e s e t s B

allowed i n t h e d e s c r i p t i o n s d = (W, Bi, i 6 W ) ; t h e y may now b e f u z z y ; t h e s e t W , h o w e v e r , w i l l n o t be assumed f u z z y . i

405

FORMAL SEMIOnCS

T e c h n i c a l l y s p e a k i n g , B . i s now a fuzzy s u b s e t o f A i , 1 hence c h a r a c t e r i z e d by membership f u n c t i o n mg ( a ) ,

a G A i . Moreover, we s h a l l r e s t r i c t t h e d e s c r i p t i o n s t o f u z z y s e t s o f some s p e c i a l form o n l y ; i n t u i t i v e l y s p e a k i n g , o n l y t o t h o s e fuzzy s e t s which may be "named". T o e x p r e s s t h e s e r e s t r i c t i o n s , we i n t r o d u c e two a d d i -

t i o n a l n o t i o n s t o t h e system (4.13), namely V and m , s o t h a t now we d e a l w i t h t h e s y s t e m

(x,

cu

, A,

3

,

R,

V, m )

(4.22)

where t h e f i r s t f i v e symbols have t h e same i n t e r p r e t a t i o n as b e f o r e , w h i l e V and m a r e i n t e r p r e t e d as follows. F i r s t l y , V i s a s e t o f a d m i s s i b l e names f o r s e t s o f

a t t r i b u t e v a l u e s . We assume h e r e t h a t t h e r e l a t i o n i s d e f i n e d a l s o on V , and t h a t V i s p a r t i t i o n e d i n t o

t h e same number o f d i s j o i n t c l a s s e s as A , i . e .

v = v1 u v*

u

...

LJ

vN'

The e l e m e n t s o f Vi w i l l be i n t e r p r e t e d as a d m i s s i b l e names o f v a l u e s o f a t t r i b u t e s i n A i . S e c o n d l y , m i s a f u n c t i o n which maps V r A i n t o t h e i n t e r v a l 0,1], i . e .

m: V x A 4[0,11, i t s v a l u e s t o be d e n o t e d by m v ( a ) . It w i l l be assumed t h a t the function m satisfies t h e following condition: if v c Vi, t h e n m v ( a > = 0 f o r a l l a 9? A i .

CHAPTER 4

406

The meaning of t h e above c o n d i t i o n i s as f o l l o w s . For

e a c h v t V, t h e f u n c t i o n m v ( a ) , t r e a t e d a s a f u n c t i o n of a , d e t e r m i n e s a f u z z y s u b s e t o f A . I f v V then i' t h e c o n d i t i o n s t a t e s t h a t t h e s u b s e t c h a r a c t e r i z e d by

( a ) has s u p p o r t i n A i J i . e . it i s a fuzzy s u b s e t of

m

vi A . only. 1

O o n s i d e r now t h e c l a s s o f a l l d e s c r i p t i o n s o f t h e form d =

(W, B i ,

i € W)

such t h a t t h e f o l l o w i n g c o n d i t i o n

ho I d s : f o r e a c h i E W t h e r e e x i s t vil, such t h a t

Bi

vi2,.

. . ,v i n

i

CVi

has membership f u n c t i o n

.

mvilr\ vi2 n . . n v i n i

( a ) = min j

mV

(a). i j

Such d e s c r i p t i o n s w i l l b e simply d e n o t e d b y d = (vij,

i

,..., n i ) .

W, j = 1

Now, a s b e f o r e , a d e s c r i p t i o n d may be used i n two w a y s . If d i s used i n c o n j u n c t i o n w i t h some o b j e c t x , as an a s s e r t i o n "x i s d " , i t w i l l be c a l l e d a v e r b a l copy o f x . I f used by i t s e l f , i t c o n s t i t u t e s e i t h e r a q u e r y ? d , as i n t h e p r e c e d i n g s e c t i o n , o r as a g e n e r a t i n g p r o j e c t o r of an imaginary o b j e c t x . To s e e t h e d i f f e r ence between v e r b a l copy and g e n e r a t i n g p r o j e c t o r , it i s enough t o n o t e a d i f f e r e n c e between d e s c r i p t i o n o f , s a y , White House, and a d e s c r i p t i o n o f t h e house of a f i c t i t i o u s c h a r a c t e r i n a novel. I n t h e f i r s t c a s e , i t i s p o s s i b l e t o d e t e r m i n e how w e l l t h e v e r b a l copy d e s c r i b e s t h e o b j e c t , whether some d e t a i l s a r e

407

FORMAL SEMIOTICS

o m i t t e d or m i s r e p r e s e n t e d , e t c ( t e c h n i c a l l y , one can d e t e r m i n e how f a i t h f u l and how e x a c t i s t h e v e r b a l c o p y ) . I n t h e second c a s e , when d i s used as a genera t i n g p r o j e c t o r , i t makes no s e n s e t o c o n s i d e r t h e “ t r u t h ” of t h e d e s c r i p t i o n , hence n e i t h e r i t s f a i t h fulness o r exactness. I n C h a p t e r 6 , t h e problem of v e r b a l c o p i e s w i l l be considered again, i n connection w i t h s c i e n t i f i c descripti o n s and t h e o r y o f knowledge ( h e n c e t h e y w i l l be used i n t h e s e n s e o f d e s c r i p t i o n s o f an o b j e c t x , p e r h a p s of a consi der able complexity) . I n connection with t h e form “x i s d ” , it i s necessary t o analogous t o t h o s e

verbal copies, i . e . assertions of w i t h d being a fuzzy d e s c r i p t i o n , determine t h e t r u t h conditions, i n t h e nonfuzzy c a s e .

N a t u r a l l y , t h e t r u t h i n t h i s c a s e w i l l b e of f u z z y c h a r a c t e r . D e n o t i n g by mg ( a ) t h e membership f u n c t i o n ( a ) , cbrresponding t o a l l fuzzy m V i l 4 Vi2O.. ’ini d e s c r i p t o r s concerning i - t h a t t r i b u t e , t h e t r u t h value a s s o c i a t e d w i t h i - t h a t t r i b u t e w i l l be d e f i n e d as

.

(4.23) where a s b e f o r e , a 1. ( x ) i s t h e unique t r u e v a l u e of t h e a t t r i b u t e i o f o b j e c t x. The s e t pi = f a 6 A ~ :m

(a) Bi

>

0’)

(4.24)

w i l l be c a l l e d t h e e x a c t n e s s s e t . Now, f a i t h f u l n e s s

l e v e l o f t h e v e r b a l copy o f x may be d e f i n e d a s t h e v e c t o r of numbers

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CHAPTER 4

and i t s e x a c t n e s s i s c h a r a c t e r i z e d by t h e v e c t o r o f sets

(4.26) I n t u i t i v e l y , the highest l e v e l of f a i t h f u l n e s s occurs i f a l l c o o r d i n a t e s o f v e c t o r ( 4 . 2 5 ) are e q u a l 1, w h i l e exactness increases with the decrease of s e t s F so i' t h a t t h e most e x a c t copy has e v e r y F b e i n g a s i n g l e i

ton. I n t h e nonfuzzy c a s e , when e a c h of t h e s e t s B

i

is

" c r i s p " , a f a i t h f u l v e r b a l copy o f x i s a d e s c r i p t i o n

of (some o r a l l ) a t t r i b u t e s of x , which happens t o be t r u e . The t r u t h may be broad o r q u i t e s p e c i f i c ; e . g . i n d e s c r i b i n g an a t t r i b u t e s u c h as h e i g h t o f a p e r s o n , a f a i t h f u l v e r b a l copy might g i v e t h e s e t B o f v a l u e s i o f a t t r i b u t e " h e i g h t " such as "more t h a n one f o o t , and l e s s t h a n 8 f e e t " , o r i t may g i v e more narrow and i n f o r m a t i v e bounds. I n t h e c a s e o f f u z z y a t t r i b u t e s , t h e problem o f t r u t h i s more c o m p l i c a t e d . T o u s e examples t y p i c a l i n f u z z y s e t t h e o r y , i t may be h a r d t o d e t e r m i n e t h e t r u t h o f d e s c r i p t i o n o f someone as " r a t h e r young" and " f a i r l y t a l l " . Here t h e c o n c e p t o f f a i t h f u l n e s s r e p l a c e s t h a t o f t r u t h , and i t s l e v e l i s measured by t h e members h i p v a l u e s of t h e p e r s o n ' s t r u e age and h e i g h t i n t h e f u z z y s e t s " r a t h e r young" and " f a i r l y t a l l " . Similar s i t u a t i o n e x i s t s with t h e notion o f exactness, which i n t h e n i n f u z z y c a s e c o r r e s p o n d s t o p r e c i s i o n .

FORMAL SEMIOTICS

409

A d e s c r i p t i o n o f someone's h e i g h t as 5'11'' i s p r e c i s e up t o t h e n e a r e s t i n c h ; a d e s c r i p t i o n "Over 6 f e e t t a l l " i s much l e s s p r e c i s e ( t h o u g h b o t h may be f a l s e ) I n a s i m i l a r w a y , i n t h e f u z z y c a s e , one may have more e x a c t and l e s s e x a c t d e s c r i p t i o n s , depending on t h e s u p p o r t s of t h e fuzzy s e t s g i v e n as d e s c r i p t o r s . The e s s e n t i a l f a c t here i s t h a t b o t h f a i t h f u l n e s s and

e x a c t n e s s a r e m u l t i d i m e n s i o n a l c o n c e p t s : a v e r b a l copy may b e f a i t h f u l w i t h r e g a r d s t o some a t t r i b u t e s

o n l y , and i t s e x a c t n e s s may a l s o v a r y from a t t r i b u t e to attribute. To sum u p , t h e c o n c e p t o f f a i t h f u l n e s s , though based on t h e underlying concept of t r u t h , i s not e q u i v a l e n t t o

i t . I t r e p r e s e n t s a m u l t i d i m e n s i o n a l composite t r u t h and i s c l o s e r t o t h e n o t i o n o f v a l i d i t y o f d e s c r i p t i o n . I n t h i s s e n s e , t h e n o t i o n o f f a i t h f u l n e s s may be o f i n t e r e s t not only f o r psychologists, but also t o logici a n s , p h i l o s o p h e r s and s p e c i a l i s t s i n a r t i f i c i a l i n t e l l i g e n c e , as a n example o f forming composite t r u t h valuations. A d d i t i o n a l l y , an i m p o r t a n t consequence i s o b t a i n e d t h r o u g h a n a l y s i s o f v a r i a b i l i t y o f s u c h judgments i n t i m e . A s shown i n C h a p t e r 3, i f one s t u d i e s s u c h v a r i a b i l i t y a c r o s s p o p u l a t i o n , one o b t a i n s i n f o r m a t i o n about a m b i g u i t y and u n c e r t a i n t y / v a g u e n e s s o f t r u t h or f a i t h f u l n e s s . This a d d i t i o n a l information, concerning s t a b i l i t y o f j u d g m e n t s , leads t o some p a r a d o x i c a l c o n c l u s i o n s about some a r e a s o f vagueness o f t r u t h -p a r a d o x i c a l b e c a u s e t r u t h i s u s u a l l y e x p e c t e d t o be accompanied by c l a r i t y . The most i m p o r t a n t , however, i s t h e f a c t t h a t vagueness o f t r u t h t e n d s t o i n c r e e a s e

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t h e v a l i d i t y of vague s t a t e m e n t s . I n d e e d , t h i s was t h e r e a s o n why P y t h i a s i n a n c i e n t Greece were s o successful i n t h e i r predictions. The above c o n s i d e r a t i o n s show a l s o some d i f f i c u l t i e s i n t h e f o u n d a t i o n s of f u z z y s e t t h e o r y of Zadeh. T h i s t h e o r y was i n t r o d u c e d i n o r d e r t o s p e a k p r e c i s e l y about i m p r e c i s i o n , but i t i s e a s y t o show t h a t even more s o p h i s t i c a t e d and c l o s e r t o i n t u i t i o n m a t h e m a t i c a l t o o l s d e s i g n e d f o r d e a l i n g w i t h vagueness c a n n o t a v o i d t h e i r own u n c e r t a i n t y and v a g u e n e s s a r e a , due t o spec i fi c psycho l o g i c a1 mechani sms o f j udgment format i o n and change. T h i s has some p h i l o s o p h i c a l consequences c o n c e r n i n g c o g n i t i v e bounds i n removing u n c e r t a i n t y and v a g u e n e s s from judgments. The l a t t e r phenomenon was h a r d l y n o t i c e d t h u s f a r , s i n c e l i t t l e i n t e r e s t was d i r e c t e d a t s t a b i l i t y and v a r i a b i l i t y o f a s s i g n m e n t s o f v a l u e s o f membership f u n c t i o n s ( s e e , however, Zimmermann and Zysno, 1 9 8 0 ) . The r e s u l t s p r e s e n t e d h e r e p o i n t o u t t o t h e n e c e s s i t y o f c o n c e n t r a t i n g t h e r e s e a r c h n o t o n l y on d e v e l o p i n g fuzzy s e t t h e o r y a s a mathematical d i s c i p l i n e , but a l s o on i t s p s y c h o l o g i c a l f o u n d a t i o n s , and on some immanent p r o p e r t i e s of t h e l a t t e r . O m i t t i n g t h e s e a s p e c t s may i n consequence lower t h e v a l u e o f t h e r e s u l t s o b t a i n e d t h u s f a r . Theory o f f u z z y s e t s i s h e r e a good example o f i n t e r v e n t i o n of p s y c h o l o g i c a l p r o c e s s e s i n t o mathem a t i c a l s t r u c t u r e s b u i l t on some a s s u m p t i o n s about these processes. Now, g i v e n t h e above n o t i o n s o f f a i t h f u l n e s s and e x a c t ness,

i t i s e a s y t o e x p l a i n why t h e vagueness o f

P y t h i a ' s predictions increased t h e i r v a l i d i t y . F i r s t

41 1

FORMAL SEMIOTICS

o f a l l , by g i v i n g t o o f e w , d i v e r g e n t , and f u z z y a t t r i b u t e s , i t s was p o s s i b l e t o f i n d many o b j e c t s t h a t somehow s a t i s f i e d t h e p r e d i c t i o n s . The a d d i t i o n a l e f f e c t o f i n c r e a s e o f v a l i d i t y was due t o vagueness By g i v i n g t o o f e w ( o r none) f u z z y v a l u e s o f a t t r i b u t e s , i t

.

was p o s s i b l e t o i n c r e a s e i t s t i l l f u r t h e r . The f u n c t i o n s o f f a i t h f u l n e s s and e x a c t n e s s a l l o w a l s o a d e f o r m a t i o n o f v e r b a l copy o f t h e o b j e c t , o r a l t e r n a t i v e l y -- some imaginary deforming o p e r a t i o n s on objects. Being e x a c t on some some o t h e r open f o r or c o n v e n t i o n , i s a res (deformation of

a t t r i b u t e s , and l e a v i n g v a l u e s o f complementation due t o knowledge f o u n d a t i o n f o r comics o r c a r i c a t u objects).

A copy may be e x a c t b u t n o t f a i t h f u l , i . e . i t may be

"shifted" with respect t o the f u l s e t o f a t t r i b u t e s does n o t s e t ) . T h i s allows t o introduce t h e s o u r c e o f comic e f f e c t s i n

t r u e state ( i t s faithintersect the basic d e f o r m a t i o n s which a r e p l a y s or n o v e l s .

Another d i s t o r t i o n may be due t o o v e r l y e x t e n s i o n o f f a i t h f u l n e s s s e t , beyond t h e p a r s i m o n i o u s optimum for t h e t r u e s t a t e of a given o b j e c t . T h i s c o n c e n t r a t i o n o f f a i t h f u l n e s s f a c t o r may c a u s e p a r t i a l loss o f e x a c t ness. The i m p o r t a n c e o f e x a c t n e s s l i e s i n t h e f a c t t h a t i t

a l l o w s t o i d e n t i f y ( p o s s i b l y w i t h some e r r o r s ) t h e objects. One c a n speak o f i n e x a c t n e s s o f e x a c t n e s s , i n t h e s e n s e o f v a r i a b i l i t y o f judgments i n t i m e ( u n d e r a s s u m p t i o n of i n v a r i a n c e o f t h e t r u e s t a t e of t h e o b j e c t ) .

412

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F a i t h f u l n e s s has a l s o i t s v a g u e n e s s , s i n c e f o r a g i v e n f i x e d t r u e s t a t e , t h e f a i t h f u l n e s s s e t may b e t o o

small o r t o o l a r g e w i t h r e s p e c t t o i t s o p t i m a l s i z e , e . g . b e c a u s e o f changes i n l o c a t i o n o f t h e s e s e t s , due t o change i n judgment from o c c a s i o n t o o c c a s i o n . T h i s may l e a d t o a d e c r e a s e o f t h e p o s s i b i l i t y of i d e n t i f i c a t i o n of the object. The n o t i o n o f v e r b a l copy shows how c o m p l i c a t e d c o g n i t i v e l y may be even t h e s i m p l e s t r e l a t i o n s between o b j e c t

and i t s r e p r e s e n t a t i o n , and i n d i c a t e s t h e n e c e s s i t y o f t a k i n g i n t o a c c o u n t ( i n a model o f a g i v e n phenomenon) t h e c o g n i t i v e bounds i n h e r e n t f o r t h i s phenomenon.

5 . FORMAL SEMIOTIC SYSTEMS

5.1. Eepresentations A t t h e e n d , w e s h a l l c o n s i d e r t h e problem o f r e p r e s e n t -

a t i o n of o b j e c t s i n t h e g e n e r a l c a s e , n o t n e c e s s a r i l y b y v e r b a l u t t e r a n c e s ; t h e r e p r e s e n t a t i o n s w i l l be

g e n e r a l l y r e f e r r e d t o as s i g n s . A s opposed t o t h e a n a l y s i s of t h e p r e c e d i n g s e c t i o n s ,

we s h a l l d e a l h e r e w i t h " f r e e " or " u n c o n s t r a i n e d " des c r i p t i o n s . I n o t h e r words, we no l o n g e r have a s i t u a t i o n o f a s e t o f o b j e c t s which a r e a l l " a l i k e " , exc e p t t h a t t h e y may d i f f e r o n l y on some s e t o f w e l l d e f i n e d a t t r i b u t e s . C o n s e q u e n t l y , w e do n o t assume any s p e c i f i c s i m i l a r i t i e s between o b j e c t s which may be described o r represented.

413

FORMAL SEMIOTICS

The o b j e c t s c o n s i d e r e d now w i l l g e n e r a l l y be r e f e r r e d t o a s s i g n s , which c a r r y ( o r convey, r e p r e s e n t ) some meanings. We b e g i n w i t h t h e s i m p l e s t f o r m a l s y s t e m , and t h e n e n r i c h it i n imposing a d d i t i n n a l s t r u c t u r a l constraints

.

Thus, we c o n s i d e r t h e system, c a l l e d a r e p r e s e n t a t i o n system (5.1) where M and Z a r e s e t s of meanings and s i g n s , and R i s a (non-fuzzy) b i n a r y r e l a t i o n which c o n n e c t s s i g n s and meanings. The symbol zRm s i g n i f i e s t h a t " z r e p r e s e n t s m t r , o r "meaning o f s i g n z i s m f ' . The meaning o f a g i v e n s i g n may be n o t u n i q u e , and

also a g i v e n meaning may be r e p r e s e n t e d by v a r i o u s s i g n s . It i s t h e r e f o r e n a t u r a l t o i n t r o d u c e t h e s e t s R ( z ) = ( m E: M : zRmf

(5.2)

and R-l(m) = ( z E Z : zRm')

.

(5.3)

If R ( z ) # PI, t h e s i g n z i s c a l l e d m e a n i n g f u l , o r i n t e r p r e t a b l e : t h e r e e x i s t s a t l e a s t one meaning m which t h i s s i g n r e p r e s e n t s . N e x t , two s i g n s , z and z ' a r e ( p a r t i a l l y ) synonymous, i f R ( z ) n R ( z f ) # 0, i . e . if t h e y have some meanings i n common. We have f u l l synonymy, i f R ( z ) = R ( z ' ) , and hyponymy, i f R ( z ) C R ( z t ) . F i n a l l y , an i m p o r t a n t c a s e o f p a r t i a l

414

CHAPTER 4

synonymy i s e q u i p o l l e n c e , R(z')

-

R ( z ) and R ( z )

0

when a l l s e t s R ( z )

-

R(z'),

R(z') a r e nonempty.

We have h e r e t h e f o l l o w i n g t h e o r e m , which c h a r a c t e r i z e s t h e p r o p e r t i e s o f t h e r e l a t i o n s i n t r o d u c e d above. THEOREM. The r e l a t i o n o f f u l l synonymy i s an e q u i v a l -

e n c e , w h i l e p a r t i a l synonymy i s r e f l e x i v e and symmetri c , b u t n o t n e c e s s a r i l y t r a n s i t i v e . Hyponymy i s r e f l e x i v e and t r a n s i t i v e , b u t n o t symmetric, w h i l e e q u i p o l l e n c e i s symmetric, b u t n e i t h e r t r a n s i t i v e n o r r e f l e x - ive

.

L e t u s now c o n s i d e r t h e s e t s R - l ( m )

defined by ( 5 . 3 ) ,

i . e . s e t s o f a l l r e p r e s e n t a t i o n s o f meaning m

.

I n g e n e r a l , t h e meaning m i s c a l l e d r e p r e s e n t a b l e , i f # PI. The r e p r e s e n t a t i o n i s u n i q u e , i f R- 1(m)

R '(m)

i s a singleton.

On t h e o t h e r hand, i f R-'(m) n R - l ( m f ) # B , t h e meanings m and m ' have some common r e p r e s e n t a t i o n s . I n s u c h a c a s e , any s i g n z E R-'(m) r\ R - ' ( m ' ) i s s a i d to be ambiguous, a s i t r e p r e s e n t s b o t h m and m ' . 1

( m ) p R-'(m') = 0, t h e n m and m ' a r e o r t h o g o n a l : any s i g n which r e p r e s e n t s one o f t h e s e meanings, does n o t r e p r e s e n t t h e o t h e r . F i n a l l y , i f R-

5.2. Multiple representations The f i r s t e x t e n s i o n o f t h e s y s t e m ( 5 . 1 ) w i l l i n v o l v e simultaneous a n a l y s i s of several representation sys-

415

FORMAL SEMIOTICS

tems, or e q u i v a l e n t l y , s e v e r a l t y p e s o f s i g n s . Without loss o f g e n e r a l i t y , we r e s t r i c t t h e a t t e n t i o n t o t h e case of n = 2 representations; t h e generalization t o t h e c a s e n > 2 i s immediate. Thus, t h e s y s t e m now have t h e form (5.4) where Z1 and Z 2 a r e two c l a s s e s o f s i g n s , assumed t o be d i s j o i n t , i . e . Z1o Z 2 = 0 . A s b e f o r e , R i < M % Z i , i = 1,2, a r e r e l a t i o n s which c o n n e c t meanings w i t h t h e corresponding s i g n s of a given t y p e . The s e t s R 1 ( z ) , R 2 ( z ) , RT1(m) and R;'(m) are d e f i n e d a s i n ( 5 . 2 ) and ( 5 . 3 ) , and t h e c o n c e p t s o f synonymy, p a r t i a l and f u l l , e t c . c a r r y o v e r t o e a c h o f t h e s e t s

Z1 and 2 2 s e p a r a t e l y . However, t h e synonymy may a l s o c o n c e r n s i g n s o f v a r i o u s t y p e s , i . e . p a i r s of s i g n s z E Z1 and z ' C Z2. O f p a r t i c u l a r importance h e r e i s e q u i p o l l e n c e : t h e u s e of s i g n s z Z1 and z' Z2 which a r e e q u i p o l l e n t , leads t o s h a r p e n i n g o f meaning, n a r r o w i n g i t down t o t h e i n t e r s e c t i o n R 1 ( z )

n R2(z1).

C o n s i d e r now t h e s e t s R;l(m) and R;'(m), contained r e s p e c t i v e l y i n Z and Z2. L e t u s n o t e t h e f o l l o w i n g 1 theorem.

and z 2 C R ; l ( m ) , z1 C R;'(m) and z a r e ( p a r t i a l l y ) synonymous. 12

THEOREM. z

then the signs

An i m p o r t a n t s p e c i a l c a s e i s c o v e r e d h e r e by t h e folowi n g d e f i n i t i o n , which c o n c e r n s t h e e x p r e s s i n v e power of s i g n s .

CHAPTER 4

416

DEFINITION. Suppose t h a t t h e s e t s R;l(m)

and R ; l ( m )

a r e nonempty, and s u c h t h a t t h e c o n d i t i o n

h o l d s . Then t h e r e p r e s e n t a t i o n system CM, Z1, Rl> i s s a i d t o have more e x p r e s s i v e power f o r m t h a n t h e system (M,

Z2,

R2).

The term i s j u s t i f i e d by t h e f a c t t h a t i f ( 5 . 5 ) h o l d s , t h e n e a c h s i g n z1 which r e p r e s e n t s m i n t h e f i r s t system does s o w i t h l e s s a m b i g u i t y (more p r e c i s i o n ) t h a n a s i g n z 2 i n t h e second s y s t e m .

...,

I n t h e g e n e r a l c a s e , we have n d i s j o i n t s e t s , Z1, Z o f t a x o n o m i c a l t y p e s o f s i g n s , and t h e c o r r e s p o n d n R n . We may t h e n r e g a r d Z1, ..., i n g r e l a t i o n s R1, Z as a p a r i t i t i o n o f t h e s e t n

...,

z

=

z 1 i) z 2

u

,..

zn,

and c o n s i d e r one r e l a t i o n R , b e i n g t h e union o f t h e r e l a t i o n s Ri. Given two p a r t i t i o n s , s a y F and F ' , o f t h e same s e t Z we may d e f i n e t h e i r prods a y , i n t o s e t s Z i and Z;, u c t F n F' as F 0 F' = f Z i

A

T o d e f i n e t h e union

Z'

j'

i = 1

,..., n ,

j = 1

,..., m].

F y F' o f two p a r t i t i o n s , i t i s

most c o n v e n i e n t t o u s e t h e e q u i v a l e n c e r e l a t i o n s c o r r e s p o n d i n g t o F and F ' . Any p a r t i t i o n d e f i n e s as e q u i v a l e n c e r e l a t i o n , two e l e m e n t s b e i n g e q u i v a l e n t ,

41 7

FORMAL SEMIOTICS

i f t h e y a r e i n t h e same s e t o f p a r t i t i o n . L e t t h e s e

- ' . Then t h e r e l a t i o n - F u F t , r e l a t i o n s be -and to be d e n o t e d s i m p l y by .-", i s d e f i n e d as t h e t r a n s i t i v e e x t e n s i o n o f t h e sum o f r e l a t i o n s and - ' , i . e . z rv" z' i f

-

3 zl,...,z t : z z1

-'

-

z

1

z2 and..,

or z and z

-'

-

z

1

and z1

z or z t

N

z2 or z'.

We have t h e n t h e f o l l o w i n g theorem. THEOREM. For a l l p a r t i t i o n s F ,

N e e d l e s s to s a y , b o t h

\J

and n

F' we have

may be d e f i n e d for

more t h a n two t e r m s . While i n t e r s e c t i o n of two p a r t i t i o n s i s a familiar operation, leading t o a f i n e r p a r t i t i o n , obtained by subdividing t h e c a t e g o r i e s , t h e o p e r a t i o n o f union o f p a r t i t i o n s a p p e a r s , a t f i r s t , somewhat a r t i f i c i a l and o f l i t t l e p r a c t i c a l u s e . Howe v e r , t h i s o p e r a t i o n i s of primary i m p o r t a n c e for des c r i b i n g t h e c o g n i t i v e p r o c e s s e s which i n v o l v e a s s o c i a t i o n s . One can namely a r g u e t h a t i f two o b j e c t s ( s i g n s , e t c . ) a r e a s s o c i a t e d i n o n e ' s mind, t h e n t h e y must b e l o n g t o t h e same c l a s s i n a p a r t i t i o n F u F 1 u F". for s u i t a b l e p a r t i t i o n s F , F', F",.. .

.

5.3. Fuzzy meanings L e t u s now i n t r o d u c e a n e x t e n s i o n o f t h e s y s t e m , con-

s i s t i n g o f a f u z z i f i c a t i o n o f t h e r e l a t i o n R , and o f

..

CHAPTER 4

418

a d d i t i o n o f temporal a s p e c t s . For s i m p l i c i t y , we r e t u r n h e r e t o t h e c a s e o f one r e presentation set Z . Thus, we c o n s i d e r t h e system

where now T i s t h e s e t of c o n s i d e r e d t i m e moments, and e a c h Rt i s a f u z z y r e l a t i o n i n M x Z , i . e . a function Rt:

M

A

Z

-+

[O,ll

.

(5.7)

Here t h e v a l u e R ( m , z ) i s t h e d e g r e e t o which z i s a t r e p r e s e n t a t i o n o f m a t t i m e t. It i s now p o s s i b l e t o c o n s i d e r t h r e e f a m i l i e s o f f u z z y s e t s , o b t a i n e d by t r e a t i n g t h e f u n c t i o n ( 5 . 7 ) as de-

p e n d i n g on one v a r i a b l e o n l y . Thus, for any f i x e d t E T and m t M y t h e f u n c t i o n R t ( m , - ) may be i n t e r p r e t e d a s a membership f u n c t i o n o f a s u b s e t o f Z , namely o f o f a l l r e p r e s e n t a t i o n s o f m a t t. We s h a l l s a y t h a t m i s f u l l y r e p r e s e n t a b l e a t t , i f

sup R t ( m , z ) ztz

(5.8)

= 1

i . e . if t h e s e t o f r e p r e s e n t a t i o n s o f m a t t i s norma l . I n g e n e r a l , t h e number d (m) = 1

t

-

sup

Rt(m,z)

(5.9)

Z E Z

may b e c a l l e d t h e r e p r e s e n t a b i l i t y d e f i c i t o f m a t t .

FORMAL SEMIOTICS

419

N e x t , one may c o n s i d e r R (m,z) as a f u n c t i o n o f m , for t f i x e d t and z . I t t h e n r e p r e s e n t s a fuzzy s e t o f meani n g s a s s o c i a t e d w i t h s i g n z a t time t , i . e . t h e s e t of a l l meanings f o r which z s e r v e s as r e p r e s e n t a t i o n . Again, i f sup

Rt(m,z)

= 1,

m t M

t h e n z may b e c a l l e d normal: t h e r e i s a t l e a s t one meaning which z r e p r e s e n t s i n f u l l d e g r e e . The most i n t e r e s t i n g c o n s t r u c t i o n i s o b t a i n e d when m and z a r e f i x e d , and one c o n s i d e r s R t ( m , z ) as a funct i o n o f t . T h i s g i v e s a f u z z y s e t o f moments, or fuzzy p e r i o d , when z r e p r e s e n t s m, o r a t e m p o r a l t r a c e o f r e p r e s e n t a t i o n o f m by z . We s h a l l s a y t h a t meaning m o f s i g n z e v o l v e s r e g u l a r l y , i f f o r a l l t l c t 2 < t we have

3

T h i s means t h a t t h e f u n c t i o n Rt(m,z) has a t most one

peak. A t y p i c a l behaviour i s t h e r e f o r e such t h a t t h e

d e g r e e t o which z r e p r e s e n t s m f i r s t i n c r e a s e s , and s t a r t s t o d e c r e a s e a f t e r r e a c h i n g t h e maximum. T h i s may be e x e m p l i f i e d by v a r i o u s s i g n s which g r a d u a l l y a c q u i r e r e c o g n i t i o n and e n t e r t h e u s a g e , and subseque n t l y " f a d e " ( e . g . become l e s s f a s h i o n a b l e ) . A c a s e i n p o i n t h e r e may be v a r i o u s s l a n g e x p r e s s i o n s . We have h e r e t h e f o l l o w i n g theorem THEOREM. The c l a s s o f a l l meanings which e v o l v e regul a r l y i s c l o s e d under c o n j u n c t i o n o f meanings.

420

CHAPTER 4

For t h e p r o o f , s e e Nowakowska ( 1 9 8 3 a ) .

5.4. R e l a t i o n s h i p s between s t r u c t u r e s I n t h i s and p r e c e d i n g s e c t i o n , t h e b a s i c c o n c e p t u a l scheme c o n s i s t e d of two s e t s o f o b j e c t s o f some k i n d ( o b j e c t s b e i n g r e p r e s e n t e d , and o b j e c t s r e p r e s e n t i n g t h e m ) , and t h e r e l a t i o n o f r e p r e s e n t a t i o n . T h i s sche-

m e i s i n t e r p r e t a b l e i n many w a y s , and t h e c h o i c e of assumptions about t h e o b j e c t s of b o t h s e t s was c o n s t r a i n e d by t h e intended i n t e r p r e t a t i o n . Thus, i n S e c t i o n

4, w e c o n s i d e r e d o b j e c t s which are

r e p r e s e n t e d , as b e l o n g i n g t o some common c l a s s ( e . g . b o o k s ) , and o b j e c t s r e p r e s e n t i n g them were d e s c r i p t i o n s , v e r b a l , n u m e r i c a l , e t c . I n t h i s c a s e , i t was n a t u r a l t o c o n s i d e r t h e s t r u c t u r e o f t h e s e t o f o b j e c t s as induced b y t h e g o a l , and by t h e v a l u e s of a t t r i b u t e s . Another c a s e o c c u r s i n s e m i o t i c s , where o b j e c t s r e p r e s e n t e d a r e meanings, and o b j e c t s which r e p r e s e n t them are signs. S t i l l a n o t h e r p o s s i b i l i t y i s when o b j e c t s b e i n g r e p r e s e n t e d a r e phenomena, and o b j e c t s r e p r e s e n t i n g them a r e t h e i r models or t h e o r i e s . I n p a r t i c u l a r , r e l a t i v e l y s i m p l e s i t u a t i o n s of t h i s k i n d a r e measurement prob-

lems, t o be d i s c u s s e d i n d e t a i l i n n e x t c h a p t e r .

It ought t o b e c l e a r t h a t t h e s t r u c t u r e o f o b j e c t s r e p r e s e n t e d and s t r u c t u r e o f t h e i r r e p r e s e n t a t i o n s , may d i f f e r w i d e l y , depending on t h e i n t e n d e d i n t e r -

pretation.

FORMAL SEMIOTICS

42 1

We s h a l l now d e s c r i b e t h e s i t u a t i o n i n g e n e r a l c a s e ,

r e t a i n i n g t h e symbols M and 2 f o r t h e c l a s s o f o b j e c t s r e p r e s e n t e d and f o r t h e c l a s s o f t h e i r r e p r e s e n t a t i o n s , s i n c e o u r main i n t e n t i o n i s t o e x p l o r e a p p l i c a t i o n s to s e m i o t i c s , where e l e m e n t s o f M a r e meanings, and e l e ments o f Z a r e s i g n s . It ought t o be k e p t i n mind, however, t h a t t h i s i s n o t t h e o n l y p o s s i b l e i n t e r p r e t a tion.

5.4.1. S t r u c t u r e o f r e p r e s e n t e d o b j e c t s . G e n e r a l l y , i t i s n e c e s s a r y t o t a k e i n t o account t h e p o s s i b i l i t y t h a t :

-- t h e s e t M h a s some s t r u c t u r e , as d e s c r i b e d

by var-

i o u s r e l a t i o n s between o b j e c t s ;

--

t h e e l e m e n t s of t h e s e t m may p o s s e s s some i n t e r n -

al structure. A s reagards .the first p o s s i b i l i t y , a c e r t a i n s t r u c t u r e

of t h e s e t o f o b j e c t s was i n t r o d u c e d i n S e c t i o n 4 , namely a n e q u i v a l e n c e ., d e s c r i b i n g " i d e n t i t y from t h e p o i n t o f view o f t h e g o a l " , and r e l a t i o n s i n d u c e d by a t t r i b u t e v a l u e s . However, i n t h i s c a s e , no i n t e r n a l s t r u c t u r e o f o b j e c t s was t a k e n i n t o c o n s i d e r a t i o n . A g e n e r a l d e s c r i p t i o n would i n v o l v e a r e l a t i o n a l s t r u c -

t u r e o f t h e form

where i n t u r n e a c h U(m) i s a system

.,

Qmj>

(5.12)

CHAPTER 4

422

A s b e f o r e , M i s a s e t o f o b j e c t s under c o n s i d e r a t i o n , S . a r e some r e l a t i o n s on M ( b i n a r y , t e r n a r y , e t c . ) , 1

and U ( m ) i s a system which d e s c r i b e s t h e i n t e r n a l s t r u c t u r e of o b j e c t m. Here Km i s a s e t o f " c o n s t i t u e n t p a r t s " o f m, w h i l e Q m l , , Q m j a r e some r e l a t i o n s between t h e p a r t s o f m, t h e i r a t t r i b u t e s , e t c .

.. .

The c h o i c e of s p e c i f i c r e l a t i o n s , t h e i r c h a r a c t e r ,

--

e t c . , depends

as a l r e a d y s t r e s s e d

--

on t h e i n t e n d -

ed i n t e r p r e t a t i o n . We g i v e below one example o f such r e l a t i o n s , c o n c e r n i n g t h e c a s e when e l e m e n t s o f m are meanings o r , more g e n e r a l l y , c o n c e p t s , composed o u t of e l e m e n t a r y c o n c e p t s (sememes). I n t u i t i v e l y , we have h e r e t h r e e k i n d s o f r e l a t i o n s : one of them a l l o w s t o o r d e r ( p a r t i a l l y ) t h e meanings

as t o t h e i r

l e v e l s of g e n e r a l i t y ; a n o t h e r c o n c e r n s

t h e meanings on t h e same g e n e r a l i t y l e v e l , and d e s c r i b e s t h e s i t u a t i o n t h a t t h e s e meanings may b e " c l o s e r " o r " f a r t h e r a p a r t " . F i n a l l y , t h e t h i r d concerns t h e meanings ( o r : c o n c e p t s ) on d i f f e r e n t l e v e l s of gener a l i t y , and r e f l e c t s t h e f a c t t h a t t h e l e s s g e n e r a l one may be a c o n s t i t u e n t p a r t o f t h e more g e n e r a l o n e . A c c o r d i n g l y , we c o n s i d e r a t r i p l e t of r e l a t i o n s

&Y

t , P)'

(5.13)

Here g i s a b i n a r y r e l a t i o n on M , assumed t o b e r e f -

l e x i v e and t r a n s i t i v e ; t h e symbol mgm' means t h a t m i s l e s s g e n e r a l ( i n t h e s e n s e o f weak i n e q u a l i t y ) t h a n m'. N a t u r a l l y , some meanings may remain incomparable according t o g.

423

FORMAL SEMIOTICS

I f mgm' and m ' g m ,

we s h a l l w r i t e m = m ' , and s a y g t h a t t h e s e meanings a r e on t h e same l e v e l o f g e n e r a l ity.

N e x t , t i s a q u a r t e r n a r y r e l a t i o n on M , h o l d i n g b e t ween q u a d r u p l e t s which a r e a l l on t h e same l e v e l o f (ml,m2)t(m3,rn4) i s t o be i n t e r g e n e r a l i t y . Formally, p r e t e d t h a t ml and m 2 a r e c l o s e r one t o a n o t h e r t h a n rn3 and m 4 . T h i s r e l a t i o n i s assumed t o be t r a n s i t i v e among t h e p a i r s o f e l e m e n t s o f m , and a l s o s a t i s f y t h e following conditions :

and f o r a l l m,ml,m2

F i n a l l y , t h e r e l a t i o n t i s t o concern only t h e pairs on t h e same g e n e r a l i t y l e v e l , which may be e x p r e s s e d as

The f i r s t two c o n d i t i o n s a s s e r t t h e

symmetry o f

r e l a t i o n t w i t h r e s p e c t t o each p a i r , w h i l e t h e t h i r d a s s e r t s t h a t t h a t t h e " c l o s e n e s s " o f m and m i t s e l f i s t h e highest closeness possible. F i n a l l y , p i s t h e r e l a t i o n " b e i n g a p a r t of'", which h o l d s between some e l e m e n t s o f M which a r e on d i f f e r e n t g e n e r a l i t y l e v e l s . A c c o r d i n g l s , i t i s assumed t h a t

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p i s r e f l e x i v e and t r a n s i t i v e on M , and t h a t whenever

mpm' ( m i s a p a r t o f m f ) , t h e n a l s o rngm': mpm'

+

mgm'.

T h i s means simply t h a t t h e r e , l a t i o n s p and g s a t i s f y

the condition p

c

g.

Thus, under t h e above s t r u c t u r e , e l e m e n t s o f M have h i e r a r c h i c a l c h a r a c t e r , r e p r e s e n t a b l e as a t r e e . Given t h e n o t i o n of c l o s e n e s s , as r e p r e s e n t e d by r e l a t i o n t , one can d e s c r i b e s u c h t r e e s of c o n c e p t s (or meanings) i n t e r m s o f v a r i o u s p a r a m e t e r s , s u c h as w i d t h or depth. Semantic w i d t h may be d e f i n e d as t h e d i a m e t e r o f a floor o f t h e t r e e o f h i e r a r c h y , w h i l e s e m a n t i c d e p t h i s t h e number of floors. It may be p o s t u l a t e d t h a t t h e c o n c e p t s on h i g h e r g e n e r a l i t y l e v e l c o r r e s p o n d t o c o n s t r u c t s , while o t h e r s correspond t o o b j e c t s t h a t are d i r e c t l y observable. If t h e s e c o n s t r u c t s and n o t i o n s a r e of modal t y p e , and

t h e i r i n t e n s i t i e s may be r e p r e s e n t e d i n n u m e r i c a l domain, t h e n t h e f i r s t c a t e g o r y i s a n o b j e c t o f i n t e r e s t o f t e s t t h e o r y , w h i l e t h e second -- o f measurement t h e o r y . A s rega rds numerical r e p r e s e n t a t i o n ,

i t i s merely one

o f many p o s s i b l e i n s e m i o t i c s .

Finally, s t i l l another possible s t r u c t u r e occurs i f we have some d e p e n d e n c i e s between e l e m e n t s of M . Such d e p e n d e n c i e s may be d e s c r i b e d i n v a r i o u s w a y s . Thus,

425

FORMAL SEMIOTICS

one may have a mapping, s a y T : M 4 M y w i t h m 1 = T ( m ) b e i n g i n t e r p r e t e d t h a t m 1 depends on m . The f u n c t i o n T may depend on s e v e r a l a r g u m e n t s , s o t h a t m 1 = T(ml,...,mk),

m 1 on m 1 >

'* *

which means a m u l t i p l e dependence o f >mk'

The dependence may a l s o be o f a s t o c h a s t i c c h a r a c t e r ,

as b e s t e x e m p l i f i e d by some n a t u r a l phenomena. E l e ments o f M a r e t h e n o b s e r v a b l e v a r i a b l e s ( o r , more p r e c i s e l y , t h e i r l a b e l s ) , and a r e l a t i o n s h i p between a p a i r ml = X and m2 = Y i s r e p r e s e n t a b l e as a j o i n t d i s t r i b u t i o n o f t h e s e v a r i a b l e s . Thus, t h i s r e l a t i o n s h i p h a s t h e form of t h e j o i n t d i s t r i b u t i o n f u n c t i o n F ( x , y ) = P(X 4 x , Y 4 y ) . O c c a s i o n a l l y , i t makes s e n s e t o r e p r e s e n t such a dependence

numerically, e.g

as a c o r r e l a t i o n c o e f f i c i e n t between X and Y , e t c .

Homomorphisms. G e n e r a l l y , t h e e l e m e n t s o f t h e s e t Z (which a r e used t o r e p r e s e n t o b j e c t s i n M ) , a r e a l s o r e l a t i o n a l s t r u c t u r e s o f some k i n d . Moreover, t h e r e a r e a l s o r e l a t i o n s on t h e s e t Z . The n a t u r e o f

5.4.2.

t h e s e r e l a t i o n s i s , t o a l a r g e e x t e n t , similar t o t h a t of t h e c o r r e s p o n d i n g r e l a t i o n s on M . The above s e n t e n c e r e q u i r e s " s i m i l a r i t y " of M and 2 on " c o r r e s ponding" r e l a t i o n s . When p u t i n a p r e c i s e form, t h i s means e x i s t e n c e o f a homomorphism between t h e s t r u c t u r es. Suppose t h a t t h e s t r u c t u r e o f 2 i s r e p r e s e n t a b l e a s

where

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w i t h s i m i l a r i n t e r p r e t a t i o n as i n t h e c a s e ( 5 . 1 1 ) and

( 5 . 1 2 ) for t h e s t r u c t u r e o f M.

For s i m p l i c i t y o f p r e s e n t a t i o n , suppose t h a t a l l r e l a t i o n s S i , S!1 a r e b i n a r y .

DEFINITION. If t h e r e e x i s t s a f u n c t i o n Z -?M

h:

s u c h t h a t for p a i r s lations , z1S;.z2

implies

J

(Si ,Si ) , 1

1

...,

( S i k y S i k ) of re-

h(zl)Si h(z2) j

for j = l , . . . , k , t h e n w e s a y t h a t h e s t a b l i s h e s a homomorphism o f Z and M w i t h r e s p e c t t o r e l a t i o n s Si . j

T h i s homomorphism c o n c e r n s m u l t i o b j e c t s and m u l t i s i g n s ,

i n t h e s e n s e o f t h e r e l a t i o n s between s i g n s and t h e c o r r e s p o n d i n g r e l a t i o n s between meanings o f t h e s e signs , without e n t e r i n g i n t o t h e i n t e r n a l s t r u c t u r e s o f s i g n s and meanings. A s a n example, one may t h i n k h e r e o f meanings of s i g n s

composed o u t of two or more s i g n s .

,..

1

.,Si r e p r e s e n t s t h e t y p e , or Here t h e s e t {Si t r a c e o f hornomordhism, akd i s c a l l e d t h e p r o j e c t i o n s e t . I n o t h e r words, t h i s s e t c o n t a i n s t h e i n f o r m a t i o n about t h e r e l a t i o n s which a r e p r e s e r v e d under homomorphism.

FORMAL SEMIOTICS

427

DEFINITION. Let h b e a f u n c t i o n which e s t a b l i s h e s a homomorphism between Z and M . I f t h e r e e x i s t s a f u n c t i o n

XlQA(Z)

,i.x2 J

implies

u(x1)QZyi ~ ( x , ) , j

t h e n we s a y t h a t homomorphism h i s s t r u c t u r a l l y i c o n i c a t element z . Thus, a s t r u c t u r a l l y i c o n i c homomorphism i s such t h a t i t n o t o n l y r e p r e s e n t s t h e r e l a t i o n s between o b j e c t s , but also represents t h e i r i n t e r n a l structure, o r at l e a s t some o f i t s f e a t u r e s . The s e t Q i l , . . . , Q Z i c o n t a i n s t h e i n f o r m a t i o n about t h e r e l a t i o n s o f t h e o b j e c t which a r e r e p r e s e n t e d , and i s c a l l e d t h e s e l e c t ion s e t . I n p a r t i c u l a r , suppose t h a t t h e r e a r e f u n c t i o n a l dep e n d e n c i e s between e l e m e n t s o f M , and t h a t t h e y a r e p r e s e r v e d under homomorphism h. T h i s means t h a t t h e r e such t h a t i f T 1 ( z ) = e x i s t s a f u n c t i o n T 1 : Z -+Z, z ’ , then also T[h(z)l = h ( z l ) . I f f u n c t i o n T corresponds t o a c a u s a l r e l a t i o n connect-

i n g p a i r s ( m , T ( m ) ) , t h e n homomorphism h i s s a i d t o be i n d e x i n g , and t h e s i g n s z a r e i n d e x e s o f t h e m’s c o r r e s p o n d i n g t o them under f u n c t i o n h . On t h e o t h e r hand, i f t h e f u n c t i o n h p r e s e r v e s no s t r u c t u r a l a s p e c t s o f e l e m e n t s o f M , and no m u t u a l r e l a t i o n s between them, we have a p u r e l y symbolic re-

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presentation. To g i v e some examples, a p h o t o g r a p h p r e s e r v e s a t l e a s t

p a r t o f t h e s t r u c t u r e o f t h e o b j e c t , and a l s o m u t u a l r e l a t i o n s between them. On t h e o t h e r hand, a map shows mutual r e l a t i o n s , e . g . between towns, b u t g i v e s no i n f o r m a t i o n about t h e s t r u c t u r a l p r o p e r t i e s o f t h e towns. These examples show t h a t s i g n s may o r may n o t be i c o n i c

, and

y e t c o n s t i t u t e a good r e p r e s e n t a t i o n .

S i m i l a r l y , r e p r e s e n t a t i o n of meanings t h r o u g h words, i s p u r e l y s y m b o l i c : e . g . t h e words " s h o r t " and " l o n g " do n o t r e f l e c t t h e r e l a t i o n s between meanings by any r e c o g n i z a b l e f e a t u r e s , s u c h as number o f l e t t e r s , e t c .

5 . 5 . S i g n s and i n f o r m a t i o n L e t u s o b s e r v e f i r s t t h a t t h e above c o n s i d e r a t i o n s can immediately e x t e n d e d t o t h e c a s e o f s e v e r a l t y p e s o f s i g n s , hence t o m u l t i - t y p e r e p r e s e n t a t i o n s , as it was done i n S e c t i o n 4 , by c o n s i d e r i n g p a r t i t i o n s Z = Z1 U U Zn i n t o sign-types. A r e p r e s e n t a t i o n

...

o f a meaning i n form o f a s i g n may be u n i - t y p e , o r homogeneous, i f i t h a s t h e form o f a s i n g l e element o f one o f t h e s e t i n t h e p a r t i t i o n . I n g e n e r a l , i t may be a v e c t o r o f t h e form ( z ~ ,z 2 , . . , z n ) , where e a c h z i i s e i t h e r a n element o f Z i , o r z = 6 , t o i symbolize t h e f a c t t h a t s i g n s o f i - t h t y p e a r e n o t

.

used. G e n e r a l l y , s u c h composite s i g n s are some s t r u c t u r e s , i n which one s i g n a p p e a r s i n a c o n t e x t of other signs, of d i f f e r e n t types. I n a sense, t h e t h e o r y here i s p a r a l l e l t o t h e t h e o r y of multimedia1 communication, and t h e n o t i o n s of s u p p o r t i n g , i n h i b i t -

429

FORMAL SEMIOTICS

ion, cancelling, e t c . carry over t o the present case. It i s n o t w o r t h w h i l e t o r e p e a t t h e d e f i n i t i o n s o f t h e s e n o t i o n s ; it i s s u f f i c i e n t t o r e c a l l t h a t t h e s e d e f i n i t i o n s were o b t a i n e d by c o n s i d e r i n g t h e meanings of 2 s i g n , s a y z Zi, i n context of s i g n s z i 1'"" Z Z , z n , by comparing i t w i t h t h e meaning i-1' of zi = 6 i n t h e same c o n t e x t . I n s h o r t , one c o n s i d e r s t h e e f f e c t o f r e p l a c i n g t h e s i g n z by "no-sign" 6 . i

..

Now, s i g n s p r o v i d e i n f o r m a t i o n about meanings, a l t h o u g h t h e i n f o r m a t i o n e l i c i t e d by t h e o b s e r v e r ( u n d e r t h e o b s e r v a b i l i t y c o n s t r a i n t s ) need n o t c o i n c i d e w i t h t h e i n f o r m a t i o n i n t e n d e d by t h e p e r s o n who produced t h e sign. Consider f o r s i m p l i c i t y a single- type s y s t e m , w i t h nonfuzzy r e l a t i o n R , i . e . a s y s t e m ( M , Z , R ) . F o r any s i g n z w e c o n s i d e r t h e n t h e set R ( z ) o f a l l i t s meani n g s , i . e . t h e s e t o f a l l meanings which z r e p r e s e n t s . The s e t R ( z ) may be c a l l e d t h e s e t o f a l l ttSharpll meanings o f z . Suppose t h a t t h e i n t e n d e d meaning was m 0 (2 M , V a r i o u s c o n c e p t s may now be d e f i n e d t h r o u g h and t h e t h e p r o p e r t i e s o f s e t s R(z), l o c a t i o n o f m 0' e q u i v a l e n c e r e l a t i o n IV i n M , which p a r t i t i o n s M i n t o e q u i v a l e n c e c l a s s e s , t o b e d e n o t e d by [ m ] . F i r s t l y , i t i s c l e a r t h a t s i g n z c o n t a i n s more informR(zl). a t i o n , o r i s more i n f o r m a t i v e t h a n z ' , i f R(z) T h i s means s i m p l y t h a t z has s m a l l e r s e t o f p o s s i b l e meanings which may be a t t a c h e d t o i t .

c

If

R(z) c [m] for some meaning m , t h e n we may s a y

t h a t sign z i s s u f f i c i e n t for meaning m s T h i s condit i o n means simply t h a t z c a r r i e s i n f o r m a t i o n which

CHAPTER 4

430

may n o t a l l o w t o i d e n t i f y t h e meaning u n i q u e l y , but

a l l meanings m which are i n R ( z ) s a t i s f y t h e c o n d i t i o n

t h a t t h e y a r e e q u i v a l e n t from t h e p o i n t of view o f the goal. A s i g n may be s u f f i c i e n t f o r meaning m , b u t t h i s does

n o t imply t h a t i t s u s e for conveying i n f o r m a t i o n m 0 i s s e n s i b l e . G e n e r a l l y , w e may s a y t h a t z i s f e a s i b l e f o r m o , i f mo G R ( z ) . If R ( z ) C [m,], then z i s adequate f o r m 0' I n most c a s e s , t h e s e t R ( z ) i n t e r s e c t s w i t h a number

[ m ] . Given a s i g n z , i t i s therefore n a t u r a l t o distinguish three classes of meanings: t h o s e which a r e c o n t a i n e d i n R ( z ) t o g e t h e r w i t h t h e i r whole e q u i v a l e n c e c l a s s , t h o s e which a r e e x c l u d e d from R ( z ) , and t h o s e which a r e n e i t h e r cont a i n e d , n o r e x c l u d e d . Thus, w e may p u t of equivalence c l a s s e s

We may to the ion t o t o any

t h e n s a y ' t h a t A ( z ) i s t h e lower a p p r o x i m a t i o n meaning z , w h i l e B ( z ) i s t h e upper approximati.t. The i n t e n d e d meaning, s a y m , may b e l o n g 0 of t h e s e t h r e e c l a s s e s .

I n f u z z y c a s e , when t h e r e l a t i o n

R i s described i n

terms of i t s c h a r a c t e r i s t i c f u n c t i o n Rt ( p o s s i b l y depending on time t , as i n s e c t i o n 5 . 3 ) , we o b t a i n t h e membership f u n c t i o n R t ( z , m ) , e x p r e s s i n g t h e d e g r e e

43 1

FORMAL SEMIOTICS

t o which t h e s i g n z r e p r e s e n t s meaning m . I t i s n o t p o s s i b l e t o d e f i n e s e t s A ( z ) , B ( z ) and C ( z ) , b u t t h e upper and lower a p p r o x i m a t i o n of meaning o f z may be d e f i n e d ( f o r time t ) , as t h e k e r n e l and c a r r i e r o f t h e membership f u n c t i o n R t , s o t h a t we may now p u t A(z) =

im:

Rt(m,z)

= 17

Thus, i n t h i s c a s e , t h e s e t C ( z ) of e x c l u d e d meanings w i l l be t h e s e t C ( z ) = {m: R t ( m , z ) = O f

.

G e n e r a l l y , a meaning m may be c a l l e d e x p r e s s i b l e , or d e s c r i b a b l e , i f t h e r e e x i s t s a s i g n z ( p e r h a p s compos i t e ) , such t h a t A(z)

= B ( z ) = [m]

( h e r e t h e d e f i n i t i o n i s t h e same i n t h e fuzzy and nonf u z z y c a s e , s o t h a t t h e s e t s A ( z ) and B ( z ) may be i n t e r p r e t e d a c c o r d i n g t o e i t h e r o f t h e two f o r m u l a s above). Somewhat more g e n e r a l l y , we s a y t h a t meaning m i s k-describable, (or: k-expressible, k-definable), i f t h e r e e x i s t s a s i g n z such t h a t [ m l R ( z ) and moreov e r , R ( z ) i s a union of e x a c t l y k e q u i v a l e n c e c l a s s e s o f some meanings. We have h e r e t h e f o l l o w i n g theorem. THEOREM. If meaning m i s k - d e f i n a b l e ,

a s i g n z and meanings ml,m2,

... 'mk-l

then t h e r e e x i s t s such t h a t

CHAPTER 4

432

m i d mj,

i , j = 1,.

..,k-1,

i # j,

and k- 1

u

~ ( z = ) C ~ I Y

[mil.

i=l

We may s a y t h a t ( u n d e r z ) t h e meaning m i s confounded w i t h k-1 o t h e r non-equivalent

meanings. T h u s , k - d e f i n a b i l i t y means simply confounding w i t h k - 1 o t h e r meanings. I f meaning rn i s k - d e f i n a b l e ,

and i s n o t ( k = l ) - d e f i n a b le, t h e n we s a y t h a t m i s s t r i c t l y k - d e f i n a b l e .

L e t D d e n o t e t h e c l a s s of a l l meanings rn which a r e k - d e f i n a b l e f o r some k . D E F I N I T I O N . The s i g n s z 1 , z 2

a r e s a i d t o obey t h e mean-

i n g t r u n c a t i o n law, if

Thus, under meaning t r u n c a t i o n l a w , a composite s i g n c a r r i e s o n l y t h o s e meanings which a r e common t o b o t h s i g n s ( n a t u r a l l y , t h i s i s a r a r e and r a t h e r s p e c i a l c a s e , s i n c e i n most i n s t a n c e s when two s i g n s a r e composed, a new meaning a r i s e s ) . However, we have

THEOREM. g n d e r meaning t r u n c a t i o n l a w , t h e c l a s s D

is

c l o s e d under a l l s e t - t h e o r e t i c a l o p e r a t i o n s , i . e . u n i o n s ; i n t e r s e c t i o n s and c o m p l e t e n t a t i o n . The assumption o f t h i s theorem, namely t h e t r u n c a t i c n law, may be e x p e c t e d t o h o l d i n c a s e when meanings

433

FORMAL SEMIOTICS

a r e r e p r e s e n t e d as d e s c r i p t i o n s i n t e r m s o f a t t r i b u t e s , i . e . f o r s p e c i a l category of s i g n s . I n such a c a s e , e a c h new i t e m o f i n f o r m a t i o n ( d e s c r i p t o r ) narrows down t h e s e t o f o b j e c t s which a r e r e p r e s e n t e d b y t h e composite sign. On t h e o t h e r hand, i n a r t , t h e assumption o f t r u n c a t i o n o f meaning does n o t o c c u r -- on t h e c o n t r a r y , we have a p p e a r e n c e s of new meanings, due t o c o m b i n a t i o n s of s i g n s . We s a y t h a t s e m i o t i c system (or: i n f o r m a t i o n s y s t e m ) i s k - s e l e c t i v e , i f any meaning i s k - d e f i n a b l e . A s i g n z which meets t h e r e q u i r e m e n t s o f d e f i n a b i l i t y f o r meani n g rn w i l l b e c a l l e d a c h a r a c t e r i s t i c s i g n f o r m. I n c a s e when meaning m i s n o t k - d e f i n a b l e f o r any k , we have t o c o n s i d e r t h e upper and lower a p p r o x i m a t i o n s , as d e f i n e d above. The i n f o r m a t i o n c o n t a i n e d i n any s i g n w i l l t h e n b e ambiguous, i n t h e s e n s e t h a t i t a l w a y s g i v e s a t l e a s t two e q u i v a l e n t meanings, o f which

one i s e x c l u d e d and t h e o t h e r i s n o t . Imagine now t h a t t h e d e s c r i p t i o n , i n form o f a t t r i b u t e v a l u e s , is g i v e n s e q u e n t i a l l y , and l e t R1 '> R2 3

...

be t h e s u c c e s s i v e i n f o r m a t i o n s e t s , a f t e r s p e c i f y i n g t h e f i r s t , s e c o n d , e t c . a t t r i b u t e v a l u e . Thus, if dk

i s t h e d e s c r i p t i o n i n v o l v i n g t h e f i r s t k s e t s B1,

...,

o f a t t r i b u t e v a l u e s ( f u z z y or n o t ) , t h e n we have k Rk = R(d ) . We may t h e n s a y t h a t a t t r i b u t e k i s rek dundant i n t h e c o n t e x t B1,.. ,Bk-l i f Fik =

B

.

Rk-l'

An a t t r i b u t e which i s r e d u n d a n t i n any c o n t e x t , i t i s t o t a l l y r e d u n d a n t . The most common t y p e o f redundancy o c c u r s , i f t h e a t t r i b u t e i s dependent on t h e p r e c e d i n g

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...,

o n e s . Thus, B

i s dependent on B1, B k k-1 i f t h e i n f o r m a t i o n s e t for any o b j e c t x a f t e r dk - i s t h e same as a f t e r d k'

5.6. S t a t i s t i c a l semiotics L e t us now c o n n e c t t h e t h e o r y o f s i g n s and t h e i r meani n g s w i t h t h e t h e o r y o f random o b j e c t s and t h e i r obs e r v a b i l i t y , introduced a t t h e beginning of t h i s c h a p t e r . Thus, s i g n s and meanings a r e now b o t h random o b j e c t s , a p p e a r i n g w i t h some f r e q u e n c i e s . These freq u e n c i e s w i l l , i n g e n e r a l , depend on t h e u s e r o f t h e s y s t e m , i . e . on t h e p e r s o n who p r o d u c e s t h e s i g n ( w e s h a l l d e n o t e t h i s p e r s o n by x). On t h e o t h e r hand, t h e p e r s o n who p e r c e i v e s t h e s i g n , t o b e d e n o t e d b y y , u s e s i n g e n e r a l some mask, and t h e r e f o r e o b t a i n s o n l y some i n f o r m a t i o n about t h e s i g n , p e r h a p s i n s u f f i c i e n t t o i d e n t i f y t h e s i g n completely. The system w i l l now c o n s i s t s o f t h e f o l l o w i n g e l e m e n t s

where M , Z and R are as b e f o r e , IT i s a p r o b a b i l i t y d i s t r i b u t i o n on M , s o t h a t X(m) i s t h e f r e q u e n c y w i t h which t h e s e n d e r x u s e s meaning m , p ( z l m ) i s a f a m i l y o f p r o b a b i l i t y d i s t r i b u t i o n s on Z , where p ( z m ) r e p r e s e n t s t h e p r o b a b i l i t y t h a t sender w i l l choose s i g n z t o r e p r e s e n t meaning m . Next, I i s t h e c l a s s of a l l p o s s i b l e d a t a ( i n f o r m a t i o n ) about z which i s a t t a i n a b l e under a g i v e n mask (assumed t o be f i x e d

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by t h e p e r c e i v e r y ) , and q i s t h e f a m i l y o f p r o b a b i i -

l i t y d i s t r i b u t i o n s q ( z ) I ) , t h a t t h e s i g n was z , g i v e n observation I. Clearly ,

i s t h e u n c o n d i t i o n a l p r o b a b i l i t y t h a t s i g n z w i l l be

produced, s o t h a t p ( z ) i s t h e o v e r a l l f r e q u e n c y o f s i g n z ( i . e . f r e q u e n c y o f random o b j e c t s z). If we d i s r e g a r d t h e c o n t e n t o f t h e s i g n , t h e n one can

u s e t h e Shannon f o r m u l a

as t h e a v e r a g e amount o f i n f o r m a t i o n c a r r i e d by a s i g n . Here t h e amount o f i n f o r m a t i o n c a r r i e d by an i n d i v i d u a l sign i s log l / p ( z ) . N a t u r a l l y , t h i s approach i s of l i m i t e d u s e , s i n c e t h e f r e q u e n c y o f s i g n s has l i t t l e t o do w i t h t h e s e m a n t i c i n f o r m a t i o n which t h e y c a r r y . S i m i l a r l y , one can u s e t h e j o i n t d i s t r i b u t i o n

o f s i g n and meaning, t o d e f i n e t h e j o i n t e n t r o p y , and hence d e f i n e ( a s u s u a l i n i n f o r m a t i o n t h e o r y ) t h e amount o f i n f o r m a t i o n c a r r i e d by one v a r i a b l e about t h e o t h e r ( i n t h i s c a s e one v a r i a b l e i s t h e s i g n , and t h e o t h e r i s t h e meaning).

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I n t h e c a s e under c o n s i d e r a t i o n , t h e assignment o f meaning, or i n t e r p r e t a t i o n s , may b e o b t a i n e d from Bayes' f o r m u l a . Here t h e problem i s t o d e t e r m i n e t h e c o n d i t i o n a l p r o b a b i l i t y of meaning, s a y m , g i v e n t h e 0 o b s e r v e d d a t a I . L e t u s s t a r t from t h e s i m p l e s t c a s e when I = z , s o t h a t z i s o b s e r v a b l e , We have h e r e

Given z , one can a s s i g n t o i t t h a t meaning m

which 0 maximizes t h e above e x p r e s s i o n . T h i s a p p r o a c h has l o n g t r a d i t i o n i n a n a l y s i s o f i n f o r m a t i o n , and we s h a l l omit the details. T o summarize, and t o i n c l u d e t h e i n f o r m a t i o n I , our

random s e m i o t i c s y s t e m c o n s i s t s o f t h e e l e m e n t s o f t h e nonrandom s y s t e m , and t h e a p p r o p r i a t e p r o b a b i l i t y d i s t r i b u t i o n s . The f o l l o w i n g c h a i n of e v e n t s d e s c r i b e s the situation:

-- s e n d e r x wants t o convey meaning m o . T h i s o c c u r s w i t h p r o b a b i l i t y $ ( m o ) , which e x p r e s s e s t h e f a c t t h a t some meanings a r e i n t e n d e d more o f t e n t h a n o t h e r s . --

s e n d e r x t h e n c h o o s e s s i g n z 0' w i t h p r o b a b i l i t y p ( z I m ) . S i g n z o i s a r e p r e s e n t a t i o n of m o , i . e . we 0 0 u s u a l l y w i l l have

mo 6. R ( z o ) .

-- t h e s i g n z o i s p e r c e i v e d by t h e o b s e r v e r y t h r o u g h a mask, which may g i v e ambiguous i n f o r m a t i o n about zo. T h i s a m b i g u i t y i s d e s c r i b e d by t h e p r o b a b i l i t y d i s t r i b u t i o n q(zolI), where I i s t h e i n f o r m a t i o n o b t a i n e d .

FORMAL SEMIOTICS

43 I

The problem i s t o d e t e r m i n e t h e p r o b a b i l i t y t h a t meani n g m was i n t e n d e d , g i v e n t h e i n f o r m a t i o n I . We may 0 write

Thus, w e need t h e p r o b a b i l i t y t h a t s i g n z i s r e a l l y z, g i v e n t h e o b s e r v a t i o n I , and t h e p r o b a b i l i t i e s concerni n g t h e c h o i c e s o f meanings and s i g n s b y t h e s e n d e r . The f i r s t f a c t o r i n t h e l a s t f o r m u l a depends on t h e c h a r a c t e r i s t i c s o f t h e s e n d e r , and t h e second f a c t o r on t h e c h a r a c t e r i s t i c s o f t h e p e r c e i v e r .

--

5 . 7 . Comments I n t h e formal s e m i o t i c s of t h i s s e c t i o n , o b s e r v a t i o n o f an o b j e c t i s r e l a t i v e t o t h e s i t u a t i o n , t o measurement p r o c e d u r e s , and t o mappings i n t h e n u m e r i c a l o r l i n g u i s t i c domains, which a r e t h e p r e r e q u i s i t e s o f f o r m u l a t i n g t h e o b s e r v a t i o n a l judgments. The l a t t e r mapping a l l o w s u s t o r e p l a c e t h e o p e r a t i o n s on o b j e c t s by o p e r a t i o n s on symbols -- e i t h e r i n m a t h e m a t i c a l domain or i n l i n g u i s t i c s p a c e -- o f c l a s s i f i c a t i o n s

o r d e s c r i p t i o n s . T h i s mapping, and i t s c h a r a c t e r , i s d e t e r m i n e d by t h e c l a s s o f t r a n s f o r m a t i o n s o f t h e o b s e r v a t i o n a l s y s t e m , which impose c o n s t r a i n t s on

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on i t s adequacy and i n v a r i a n c e . I n o t h e r words, one may d e a l w i t h t h e whole c l a s s of d e s c r i p t i o n s o r num e r i c a l a s s i g n m e n t s , from which one chooses o n l y t h o s e which a r e c h a r a c t e r i z e d b y s u f f i c i e n t l e v e l o f i n v a r i a n c e . An i m p o r t a n t problem h e r e i s t h e c h o i c e of adeq u a t e r e p r e s e n t a t i o n s o f o b j e c t s and r e l a t i o n s between them. The system o f c a u s a l r e l a t i o n s between o b j e c t s , i n t r o duced p r i m a r i l y f o r o b j e c t s , must be e x t e n d e d t o s e t s o f o b j e c t s which c o n s t i t u t e s i t u a t i o n s , t h e c a u s a l r e l a t i o n s between networks o f o b j e c t s , s e t s o f s e n t e n c e s d e s c r i b i n g f a c t s o r laws, c l a s s e s o f p r o c e d u r e s , n u m e r i c a l or l i n g u i s t i c mappings, and c o n d i t i o n s f o r invariance. T h i s i s a dynamical s y s t e m , n o t o n l y on t h e s e n s e o f

changes of o b j e c t s , and t h e c o r r e s p o n d i n g changes o f descriptions, p o s s i b i l i t i e s of continuous c o l l e c t i o n o f i n f o r m a t i o n , e t c . , b u t a l s o because some m o d i f i c a t i o n s o f o b s e r v a t i o n a l systems i n d u c e t h e improved and e n r i c h e d d e s c r i p t i o n s o f r e l a t i o n s between o b j e c t s . I n consequence, i t also l e a d s t o e l i m i n a t i o n o f e r r o r s , e x t e n s i o n of o l d , or i n t r o d u c t i o n o f new t h e o r y f o r t h e e x t e n d e d s e t of o b s e r v a t i o n a l s e n t e n c e s . T o sum up, t h e dynamics o f r e l a t i o n s between o b s e r v a -

t i o n s and c o l l e c t e d knowledge a b o u t o b j e c t s , on t h e one hand, and p r o d u c t i o n o f t h e i r c o p i e s on a g i v e n l e v e l o f r e p r e s e n t a t i o n , shows t h e m e t h o d o l o g i c a l and i n t e g r a t i v e r o l e of s e m i o t i c c o n c e p t i o n s o f t h e f o u n d a t i o n s of c o g n i t i o n . THera i s a n e s p e c i a l l y i m p o r t a n t problem c o n n e c t e d w i t h t h e above s k e t c h e d s e m i o t i c a l a p p r o a c h , namely t h a t

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439

of o p t i m i z a t i o n o f o p e r a t i o n s on r e p r e s e n t a t i o n s . The main q u e s t i o n h e r e i s which c o g n i t i v e p r o c e s s e s i n t e r vene and f a c i l i t a t e t h e o p t i m i z a t i o n p r o c e s s e s . Depending on t h e medium and v a r i o u s g o a l s o f o p t i m i z a t i o n , i t i s n e c e s s a r y t o l o o k f o r r e l a t i o n s between c o g n i t i v e p r o c e s s e s and r e p r e s e n t a t i o n s . For t h e soci a l s c i e n c e s , t h e s e m i o t i c a l a p p r o a c h o f f e r s c l e a r and e s t a b l i s h e d l o g i c a l l y r e s e a r c h on s t r u c t u r e o f knowledge and i t s r e l a t i o n t o r e a l i t y . For p h i l o s o p h y , i t o f f e r s u n i f i c a t i o n o f o n t o l o g y and e p i s t e m o l o g y . For t h e l a t t e r , one g e t s a n e x t e n s i o n o f i t s c l a s s i c a l s c o p e . For l o g i c -- one g e t s p r o b a b l y a s i m p l i f i c a t i o n and improvement o f l o g i c a l c a l c u l i . For l i n g u i s t i c s -- i t a l l o w s t o c o n s i d e r u n i f i e d s y s tems l y i n g a t t h e b a s e o f a l l n a t u r a l and f o r m a l l a n guage s

.

Also, f o r t h e f o u n d a t i o n s of i n f o r m a t i o n s c i e n c e , t h e i n t r o d u c t i o n o f s t r u c t u r a l e l e m e n t s i n t o what i s des c r i b e d b o t h i n memory and l a n g u a g e , may l e a d t o new more economical p r i n c i p l e s o f computer o p e r a t i o n s , and new t h e o r e t i c a l s o l u t i o n s .

CHAPTER 4

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4

BIBLIOGRAPHY FOR CHAPTER

and B I R C H , D .

ATKINSON, J . W . 1970

The Dynamics o f A c t i o n . N e w Y o r k . W i l e y .

CAELLI, T .

1 9 8 1 V i s u a l P e r c e p t i o n . London. Pergamon P e r e s s . Eijkman, E . G . J . ,

1981

Jongsma, H . J .

and V i n c e p t , J .

"Two D i m e n s i o n a l F i l t e r i n g , O r i e n t e d L i n e D e t e c t o r s a n d F i g u r a l A s p e c t s as D e t e r m i n a n t s o f V i s u a l I l l u s i o n s " . P e r c e p t i o n a n d Psychop h y s i c s . 29; 4 .

352-358.

FELLER. W .

1957

An I n t r o d u c t i o n t o P r o b a b i l i t y T h e o r y a n d I t s Qplications.

N e w York. Wiley

.

K l i r , G.J.

1972

Trends i n G e n e r a l Systems Theory. N e w York. Wiley

LEE, S.H.

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(ed.)

1981 O p t i c a l Information Processing. Fundamentals. Berlin. Springer. NOWAKOWSKA , M .

1967

" Q u a n t i t a t i v e A p p r o a c h t o t h e Dynamics o f P e r cept ion". E n e r a 1 Systems, X I I .

81-95.

1973 Language of M o t i v a t i o n a n d Language o f A c t i o n s . The Hague. Mouton.

1 9 7 8 " V e r b a l a n d N o n v e r b a l Communication as a Mult i d i m e n s i o n a l Language". ( i n P o l i s h ) . Vol. I X ,

S t u d i a Semiotyczne.

181-196.

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1979

"FUZZY Concepts: Their Structure and Problems of Measurement". In M.M. Gupta, R.K. Ragade and R.R. Yager (eds.) Advances in Fuzzy Set Theory and Application. Amsterdam. North Holland. 361-387. 1979a "Foundations of Formal Semiotics: Objects and Their Verbal Copies". Ars Semeiotica. Vol. 11, nr. 2. 133-1118. 1980 "Semiotic Systems and Knowledge Representation". Intern. Journal of Man-Machine Studies. 13. 223-257. 1980a "Empirical Semiotics". Ars Semeiotica. Vol. 111, nr. 2. 1981 "Formal Semiotics and Multidimensional Semiotic Systems". Cybernetics and Systems: An International Journal. Vol. 12. 83-100. 1983a Cognitive Sciences. New Perspectives. New York. Academic Press. 1983b "Dynamics of Perception: Some New Models". Intern. Journal of Man-Machine Studies. 18.

175-197. SHAPLEY, L . S . and SHUBIK, M. "A Method of Evaluating the Distribution of 1954 Power in a Committee System". The American Political Science Review. 48. 787-792. ULLMANN, S. 1979 The Interpretation of Visual Motion. Cambridge, Mass. MIT Press. ZIMMERMANN, H.J. and ZYSNO, P. 1980 "Latent Connectives in Human Decision Making". Fuzzy Sets and Systems. 4, 37-51.

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SELECTED TOPICS I N

5

MEASUREMENT THEORY

The measurement o f a t r a i t o f o b j e c t s o f a c e r t a i n s e t

c o n s i s t s of a s s i g n i n g numbers t o t h e s e o b j e c t s , so as t o r e f l e c t a d e q u a t e l y t h e measured t r a i t . Measurement t h e o r y i s concerned w i t h f o r m a l c o n d i t i o n s which must be met i n o r d e r f o r such a n assignment o f numbers t o e x i s t . I n t h i s r e s p e c t , measurement t h e o r y ought t o be d i s t i n g u i s h e d from s c a l i n g t h e o r y , which deals w i t h t h e t e c h n i q u e s o f a s s i g n i n g numbers t o o b j e c t s s o as t o r e f l e c t t h e measured t r a i t . The e m p i r i c a l s i t u a t i o n s whose a n a l y s i s became simpli f i e d , made more p r e c i s e ( o r even i n some c a s e s made p o s s i b l e ) by q u a n t i f i c a t i o n , a p p e a r i n numerous d i s c i p l i n e s . T h i s r e s u l t e d i n t h e s t u d i e s of e x i s t e n c e and p r o p e r t i e s o f measurement, conducted t o a l a r g e e x t e n t i n d e p e n d e n t l y i n d i f f e r e n t domains. An a t t e m p t o f s y n t h e s i s o f t h e r e s u l t s c o n c e r n i n g measurement i n v a r i o u s d i s c i p l i n e s , such as p h y s i c s , economics, psychology, e t c . , l e a d s t o t h e c o n c l u s i o n t h a t a l l known problems o f t h e s o - c a l l e d fundamental measurement ( i . e . measurement which i s n o t based on r e s u l t s o f p r e v i o u s measurements) can be reduced t o

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one o f t h r e e problems: o r d i n a l measurement, measurement b y u s i n g s t a n d a r d s e q u e n c e s , and s o l v i n g i n e q u a l i t i e s ( s e e Krantz e t a l . 19-71). T h i s c h a p t e r w i l l c o n t a i n a p r e s e n t a t i o n o f a number

o f measurement axiom s y s t e m s , s o as t o c o v e r t h e most t y p i c a l c a s e s o c c u r r i n g i n p r a c t i c e . I n each c a s e , t h e system w i l l concern some a b s t r a c t r e p r e s e n t a t i o n o f emp i r i c a l r e l a t i o n s . The most i m p o r t a n t o f t h e s e r e l a t i o n s are connected w i t h comparison of o b j e c t s w i t h r e s p e c t t o some p r o p e r t y ( i . e . a s s e r t i n g t h a t one o b j e c t i s "smaller" than t h e o t h e r ) . O t h e r r e l a t i o n s c o n c e r n "composite" o b j e c t s . Here t h e p r o t o t y p e s a r e c o n c a t e n a t i o n s , i n which t h e f e a t u r e s o f t h e o b j e c t s behave i n an a d d i t i v e way ( a s t y p i f i e d b y p u t t i n g r o d s end t o end s o t h a t t h e i r l e n g t h s comb i n e a d d i t i v e l y ) Another k i n d on "composite" o b j e c t s a r e t h o s e c h a r a c t e r i z e d by two or more i n d e p e n d e n t f e a t u r e s , each c o n t r i b u t i n g t o t h e o v e r a l l c h a r a c t e r i s t i c s ( a t e x e m p l i f i e d by u t i l i t i e s o f " b a s k e t s " cont a i n i n g v a r i o u s goods.)

.

T h e r e a r e i m p o r t a n t s p e c i a l c a s e s , a p a r t from p h y s i c s ,

when t h e f i r s t t y p e o f c o n c a t e n a t i o n ( w i t h a d d i t i v i t y ) h o l d s , namely c o n c e r n i n g s u b j e c t i v e p r o b a b i l i t y and u t i l i t y , and c o n c e r n i n g r i s k p e r c e p t i o n . The c o r r e s p o n d i n g axiom systems w i l l be p r e s e n t e d ; as r e g a r d s r i s k , a new model w i l l a l s o be shown, which a c c o u n t s f o r some seemingly i r r a t i o n a l b e h a v i o u r (from t h e p o i n t o f view o f t h e model o f s u b j e c t i v e e x p e c t e d u t i l i t y ) . A s a n i l l u s t r a t i o n of measurement s y s t e m s , a t h e o r y of

s u b j e c t i v e t i m e w i l l be shown, t o g e t h e r w i t h h y p o t h e s e s

SELECTED TOPICS INMEASUREMENT THEORY

445

which e x p l a i n t h e d i s t o r t i o n s o f p e r c e p t i o n o f t i m e . F i n a l l y , t h e c h a p t e r w i l l c o n t a i n some r e s u l t s concerni n g a n e m p i r i c a l c o u n t e r p a r t of measurement t h e o r y , namely s c a l i n g theory. 1. BASIC PROCEDURES OF MEASUREMENT

1.1. O r d i n a l measurement

L e t A d e n o t e t h e s e t o f o b j e c t s under c o n s i d e r a t i o n s . It w i l l be assumed t h a t t h e s e o b j e c t s may be compared from t h e p o i n t o f view o f a g i v e n a t t r i b u t e . F o r any a,b E A , l e t a ). b s t a n d f o r t h e f a c t t h a t t h e o b j e c t a i s e v a l u a t e d h i g h e r (from t h e p o i n t o f view o f t h e a t t r i b u t e under c o n s i d e r a t i o n ) t h a n o b j e c t b. S i m i l a r l y , a - w b w i l l mean t h a t a and b are i n d i s t i n g u i s h a b l e from t h e p o i n t of view o f t h e a t t r i b u t e i n q u e s t i o n .

The e l e m e n t s o f t h e s e t A may be r o d s , o r p i e c e s o f a s t r i n g , compared from t h e p o i n t o f view o f t h e i r l e n g t h ; d i n n e r s , compared a c c o r d i n g t o p r e f e r e n c e ; p a i n t i n g s , compared a c c o r d i n g t o t h e i r a e s t h e t i c v a l u e , and so f o r t h . The problem o f measurement o f t h e a t t r i b u t e which un-

d e r l i e s t h e comparisons, c o n s i s t s o f c o n s t r u c t i o n o f a f u n c t i o n f, d e f i n e d on A , s u c h t h a t f ( a ) > f ( b ) i f and o n l y i f a ’ > b (hence a l s o f ( a ) = f ( b ) i f and only if a b).

It ought t o be c l e a r t h a t f o r s u c h a measurement t o be p o s s i b l e , t h e r e s u l t s o f comparisons o f e l e m e n t s o f A , e x p r e s s e d by t h e r e l a t i o n s h a n d & must s a t i s f y some c o n s i s t e n c y c o n d i t i o n s . For i n s t a n c e , t h e r e s u l t s o f

446

CHAPTER 5

i n d i f f e r e n c e a s s e r t i o n s a - b must be t r a n s i t i v e : i f a b and b rv c , t h e n a l s o a c ( s i n c e i n t h e opposit e c a s e , we would have f ( a ) = f ( b ) and f ( b ) = f ( c ) , bu t f ( a ) # f ( c ) , which i s i m p o s s i b l e ) . N

L e t us o b s e r v e , however, t h a t i n c a s e s when t h e asserti o n t h a t a . u b i s o b t a i n e d by a n a p p l i c a t i o n o f some e m p i r i c a l p r o c e d u re , t h e t r a n s i t i v i t y of t h e r e l a t i o n i s by no means a u t o m a t i c : t h e r e may e x i s t t r i p l e t s a , b , c such t h a t a and b , as w e l l as b and c, are i n d i s t i n g u i s h a b l e , w h i l e a and c are a l r e a d y d i s t i n g u i s h a b l e . As a n example, one may take t h r e e shades o f some colour.

-

The theorem below p r o v i d e s c o n d i t i o n s imposed on t h e

r e s u l t s o f comparisons which imply t h e e x i s t e n c e o f the function f. Observe f i r s t t h a t I n s t e a d o f c o n s i d e r i n g two r e l a t i o n s 4 and C J ) one can t a k e as a s t a r t i n g p o i n t one r e l a t ion , such t h a t a d b d e n o t e s t h e f a c t t h a t a i s no

4

l a t e r t h a n b ( w i t h r e s p e c t t o t h e a t t r i b u t e under consideration).

4

i s c a l l e d a weak o r d e r i n t h e s e t A , i f i t s a t i s f i e s t h e f o l l o w i n g two c o n d i t i o n s : t r a n s i t i v ity, i . e . The r e l a t i o n

and co n n e cte d n e ss, i.e. either a

4

b,

or

b

47 I


o n l y p r e s e r v e d , b u t i n a se n se , t h e d i f f e r e n c e o f i t s two sides i n c r e a s e s i n d e f i n i t e l y . Indeed, i t i s asserted t h a t f o r some n we s h a l l have f n o h gn o h ' , reg a r d l e s s o f h and h ' . It i s i n t e r e s t i n g t h a t i f one r e s t r i c t s t h e c o n s i d e r -

a t i o n s t o d i s t r i b u t i o n s w i t h f i n i t e v a r i a n c e s , and i m p o s e s some a d d i t i o n a l c o n s t r a i n t s on t h e r e l a t i o n a l s t r u c t u r e Y1 (i.e. i f t h e l o t t e r y X1 i s s t r i c t l y r i s k i e r t h a n Y1), t h e r e e s i s t s an n such t h a t t h e l o t t e r y X1 t t Xn t Z i s more r i s k y t h a n t h e l o t t e r y Y1 t + Yn t W. I n o t h e r words, as n i n c r e a s e s , t h e d i f f e r e n c e between r i s k i n e s s o f X1 + t Xn and Y1 + t Yn ( i . e . n -fo l d p a r t i c i p a t i o n i n l o t t e r y X1 and n- fo l d p a r t i c i p a t i o n i n l o t t e r y Y1) becomes so l a r g e t h a t i t cannot be compensated by p a r t i c i p a t i o n i n any l o t t e r i e s Z and W.

...

...

...

...

Axiom 4 s t a t e s t h a t one a l s o c o n s i d e r s d e g e n e r a t e l o t t e r i e s , i . e . p r i z e s o r l o s s e s which are c e r t a i n t o o ccu r ( w i t h o u t t h e randomness a s s o c i a t e d w i t h a l o t t e r y ) .

CHAPTER 5

414

Axiom 5 asserts t h a t one may change t h e s c a l e i n l o t t e r i e s : i f a random v a r i a b l e X b e l o n g s t o t h e c l a s s o f l o t t e r i e s under c o n s i d e r a t i o n , t h e same i s a l s o t r u e f o r t h e random v a r i a b l e ax. Axioms 6 and 7 a r e c r u c i a l f o r t h e a s s e r t i o n o f t h e theorem. Axiom 6 a s s e r t s t h a t a d d i t i o n o f a p o s i t i v e c o n s t a n t does n o t i n c r e a s e t h e r i s k : X i s a t l e a s t as r i s k y as X t c , i f o n l y c > 0. I n o t h e r words: t h e p a r t i c i p a t i o n i n a l o t t e r y X i s more r i s k y t h a n p a r t i c i p a t i o n i n t h e l o t t e r y X, f o r which one i s b e i n g p a i d e. Axiom 7 a s s e r t s t h a t if t h e l o t t e r y X i s f a i r ( i . e . E ( X ) = O), and a 7 1, t h e n t h e l o t t e r y aX i s more r i s k y t h a n l o t t e r y X. Thus, a game w i t h a l l stakes i n c r e a s e d i n t h e same p r o p o r t i o n i s more r i s k y . The second p a r t o f axiom 7 s t a t e s t h a t whenever E ( X ) = E ( Y ) = 0 , and X i s more r i s k y t h a n Y , t h e n t h e same i s t r u e under any p r o p o r t i o n a l change o f stakes. F i n a l l y , axiom 8 i s t h e o n l y one w i t h no d i r e c t and n a t u r a l i n t e r p r e t a t i o n i n terms o f random v a r i a b l e s , and n o t t h e i r d i s t r i b u t i o n s . It a s s e r t s t h a t i f t h e i s such t h a t t h e d i s t r i b u t i o n s sequence X1, X,, o f t h e terms t e n d t o a c e r t a i n d i s t r i b u t i o n o f a random v a r i a b l e X, and a l s o E(X1) = E ( X 2 ) = and Gc(X1) = 2 d (X,) = t h e n t h e sequence of r i s k s a s s o c i a t e d w i t h l o t t e r i e s X1, X 2 , tends t o the r i s k of t h e l o t -

...

...,

...

n

...

t e r y X. The a s s e r t i o n o f t h e theorem is t h a t t h e r i s k o f t h e

l o t t e r y X depends s o l e l y on t h e v a r i a n c e and mean of X, and t h a t t h i s dependence is l i n e a r o f t h e form R ( X ) = qCi'(X)

-

(1

-

q)E(X),

SELECTED TOPICS IN MEASUREMENT THEORY

415

where q i s some c o n s t a n t between 0 and 1. Thus, f o r " f a i r " l o t t e r i e s , 1.e. when E(X) = 0 , t h e r i s k i s prop o r t i o n a l t o t h e v a r i a n c e . On t h e o t h e r hand, f o r a f i x 2 ed v a r i a n c e G (XI, t h e r i s k becomes smaller when t h e l o t t e r y becomes more f a v o u r a b l e , i . e . when t h e mean

E(X) i n c r e a s e s . 4.2.

Q u a l i t a t i v e scale of probability

The second n o n t r i v i a l example o f c o n c a t e n a t i o n o u t s i d e

t h e scope o f p h y s i c a l s c i e n c e s i s t h e c o n s t r u c t i o n o f the s c a l e of probability. The problem may be f o r m u l a t e d as f o l l o w s . Suppose we

c o n s i d e r a f i x e d c l a s s o f e v e n t s A, B, C,..., f o r which there are d e f i n e d p r o b a b i l i t i e s P ( A ) , P ( B ) , P(C),... Then w e may d e f i n e t h e r e l a t i o n i n t h e c l a s s o f eve n t s by p u t t i n g A B i f P ( A ) 2 P ( B ) . Thus, & i s t h e r e l a t i o n "at l e a s t as p r o b a b l e as".

8

Moreover, t h e p r o b a b i l i t y f u n c t i o n s a t i s f i e s t h e a d d i t i v i t y p r o p e r t y : f o r m u t u a l l y e x c l u s i v e e v e n t s A and B w e have P ( A u B) = P ( A )

+

P(B).

Formally, t h e o p e r a t i o n of a d d i t i o n o f m u t u a l l y e x c l u s i ve ( d i s j o i n t ) e v e n t s may be regarded a s a c o n c a t e n a t i o n ; t h e above c o n d i t i o n s t a t e s simply t h a t t h e p r o b a b i l i t y f u n c t i o n must be a d d i t i v e w i t h r e s p e c t t o c o n c a t e n a t i o n . Consequently, t h e f u n c t i o n P i s a measurement f o r t h e rel a t i o n a l s y s t e m c o n s i s t i n g o f a c e r t a i n number o f e v e n t s , relation and t h e o p e r a t i o n o f c o n c a t e n a t i o n . Here

476

CHAPTER 5

>

the relation was d e f i n e d i n terms of P. One can t h e r e f o r e pose a q u e s t i o n , which i s , i n a s e n s e , oppos i t e : suppose t h a t t h e r e l a t i o n ( " i s a t l e a s t a s prob a b l e a s " ) i s a p r i m i t i v e n o t i o n , and ask f o r t h e cond i t i o n s which i t must meet i n o r d e r for t h e e x i s t e n c e o f t h e p r o b a b i l i t y f u n c t i o n P c o n s t i t u t i n g a measurement for this relation. To f o r m a l l y p r e s e n t t h e r e s u l t s , w e s t a r t w i t h i n t r o -

duc in g some d e f i n i t i o n s . L e t E s t a n d f o r a nonempty set. I t s el emen t s w i l l be c a l l e d ele m en t a ry e v e n t s , and t h e s e t E w i l l b e c a l l e d t h e sample s p a c e , o r space o f e l e m en t ary e v e n t s . A nonempty c l a s s y o f s u b s e t s o f E w i l l be c a l l e d an algebra, i f i t s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s :

1.

If A F Y , t h e n

2.

If

A,Bfy,

E

-A

= -A€Y;

then A u B E Y ;

If, i n a d d i t i o n , t h e c l a s s

f

satisfies the condition 00

then

i s c a l l e d a $ -a l g e b ra .

One can p r o v e t h a t e v e ry a l g e b r a i s c l o s e d a l s o w i t h r e s p e c t t o t h e o p e r a t i o n o f i n t e r s e c t i o n and d i f f e r e n c e o f e v e n t s ( t h i s f o l l o w s a t once from de Morgan laws); on t h e o t h e r hand, e a c h 6 - a l g e b r a i s c l o s e d w i t h resp e c t t o t h e o p e r a t i o n o f c o u n t a b l e i n t e r s e c t i o n s of events. L e t u s o b s e r v e t h a t if

7 i s an

algebra of events, then

417


E

ef

7,

and B E

( s i n c e if A E then -AEyby condition = E G Y by c o n d i t i o n 2. Moreover, 0 = by c o n d i t i o n 1).

1, and A u (-A)

-E

€7

The s y s t e m (E,

y,

P> w i l l be c a l l e d a f i n i t e l y a d d i t i v e

p r o b a b i l i t y space, i f

(a)

3is

a n algebra o f s u b s e t s o f E;

( b ) P i s a n u m e ri c a l f u n c t i o n , d e f i n e d on

9,

such

that -

( b . 2 ) P ( E ) = 1;

P(A u B ) = P ( A )

+

P(B).

On t h e o t h e r hand, i f (c)

i s a 6-alaebra;

( d ) whenever

A1,

A*,

...E'f

then M

and

Ai n A

j

= 0

for i # j ,

00

2

P ( u An> = P(An), n =1 n= 1 t h e n P i s c a l l e d a countably a d d i t i v e p r o b a b i l i t y . The ele m en t s o f t h e a l g e b r a are c a l l e d e v e n t s f 'un c ti o n P i s c a l l e d p r o b a b i l i t y .

, and

L e t u s c o n s i d e r now a s y s t e m o f t h e form (E, ,&) ; f i s a n algebra o f s u b s e t s

here E i s some nonempty s e t ,

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CHAPTER 5

y.

i s a b i n a r y r e l a t i o n on A s already meno f E , and t i o n e d , t h e problem l i e s i n f i n d i n g c o n d i t i o n s on t h e s y s t e m (E, , under which t h e r e e x i s t s a f u n c t i o n , P, b e i n g a ( f i n i t e l y a d d i t i v e ) p r o b a b i l i t y on and such t h a t

2.)

P(A)

2

P(B)

7,

if and o n l y i f , A

B.

L e t u s assume f i r s t t h a t such a f u n c t i o n e x i s t s , and t r y t o f i n d the necessary conditions.

>

y,

must o r d e r weakly t h e s e t i.e. it must be t r a n s i t i v e and connected ( t h i s i s a n obvious necessary requirement). F i r s t of all,

Next, s i n c e P ( E ) = 1 and P ( 0 ) = 0 , we must have E & 0. Moreover, s i n c e 0 P ( A ) 5 1 for e v e r y A € , we must have 0 $ A 3 E. Thus, E i s t h e f i r s t , and 0 i s t h e l a s t element i n t h e o r d e r i n g

2

.

L e t u s assume now t h a t A i s d i s j o i n t w i t h B and a l s o C. Then P ( A U B ) = P ( A ) d i s j o i n t w i t h C , and t h a t B t P ( B ) > P ( A ) + P ( C ) = P ( A u C ) . Conversely, i f A u B 2 A u C, t h e n P(A) t P ( B ) = P(A u B ) P(A U C ) = P ( A ) t P ( C ) , hence P ( B ) P ( C ) , or B & C . Consequently, w e must have t h e m o n o t o n i c i t y c o n d i t i o n : i f A i s d i s j o i n t w i t h B and c, t h e n B )r C i f , and o n l y i f , A u B

>

>

3

> A u C .

L e t u s now c o n s i d e r t h e Archimedean axiom. L e t w d e n o t e t h e e q u i v a l e n c e r e l a t i o n induced b y , i.e. A N B i f A ) B and B > A . Thus, Am B means t h a t A and B are

&

equiprobable.

419


7.

L e t now A d e n o t e a n a r b i t r a r y e v e n t i n We s h a l l s a y i s stant h a t t h e sequence A1, A 2 , o f elements o f d a r d w i t h r e s p e c t t o A, i f f o r e v e r y k t h e e v e n t Ak may b e p a r t i t i o n e d i n t o t h e union Ak = B1 u B2 u c, Bk /V Bk o f d i s j o i n t e v e n t s such t h a t B1m B2 .cv A.

...

-

)i

...

H f A1,

A2,

... i s a s t a n d a r d sequence

...

with respect t o A,

t h e n w e have P ( A k ) = P(B1 u

= P(B1)

+

... u Bk)

...

t P(Bk) = kP(A).

>

It f o l l o w s t h a t i f P ( A ) 0, i.e. A 0, t h e n t h e standard sequence must b e f i n i t e ( s i n c e kP(A) = P ( A k ) 1, hence k l / P ( A ) : any s t a n d a r d sequence based on A may have a t most l / P ( A ) e l e m e n t s ) .

) DEFINITION. A s y s t e m (E, a t i v e probability scale, i f 1. 2.

> E

i s a weak o r d e r i n

>0

and A

>0

w i l l be c a l l e d a q u a l i t -

7;

f o r e v e r y A,

3. If A p\ B = 0 and A n C = 0, t h e n B i f A u B b A U C ;

&C

i f and o n l y

>

4. If A 0, t h e n any sequence which i s s t a n d a r d w i t h r e s p e c t t o A must have o n l y a f i n i t e number o f element s. The f i r s t q u e s t i o n t o a s k i s whether t h e c o n d i t i o n s

CHAPTER 5

480

1

-

4 a r e s u f f i c i e n t t o imply t h e e x i s t e n c e o f probabi-

l i t y f u n c t i o n P on t h e a l g e b r a

3,

which would c o n s t i t u t e a measurement f o r t h e r e l a t i o n 2 . It t u r n s o u t ( s e e Krantz e t a l . , 1971) t h a t t h e answer i s n e g a t i v e : t h e r e e x i s t s y s t e m s o f q u a l i t a t i v e p r o b a b i l i t y (E, , i.e. s y s t e m s s a t i s f y i n g c o n d i t i o n s 1-4, f o r which t h e p r o b a b i l i t y P d o e s n o t e x i s t . We have t h e r e f o r e a probl e m of f i n d i n g a d d i t i o n a l a s s u m p t i o n s , which would s u f f i c e f o r e x i s t e n c e o f P. We p r e s e n t below one s u c h c o n d i t i o n , due t o Savage (1954).

9 ,h )

5. Assume t h a t (E,

9 ,$>

i s a system o f q u a l i t a t i v e

p r o b a b i l i t y . If A , B E 3 and A B, t h e n t h e r e e x i s t s a o f t h e s e t E , such t h a t f o r e v e r y p a r t i t i o n {C1,. ,Cn) i = 1,2,...,n w e have A B u ci.

..

>

B ( b y m o n o t o n i c i t y ) , t h e c o n d i t i o n above S i n c e B U Ci means t h a t whenever A )- B, one can p a r t i t i o n t h e whole s p a c e i n t o e v e n t s C1, Cn, e a c h h a v i n g so small probab i l i t y t h a t e v e r y e v e n t B u Ci i s "between" e v e n t s A and B.

...,

To e x p l a i n b e t t e r t h e meaning o f axiom 5, i t i s w o r t h w h i l e t o p r e s e n t some of i t s consequences.

The q u a l i t a t i v e p r o b a b i l i t y s t r u c t u r e (E, ,&> w i l l be c a l l e d f i n e , if f o r e v e r y A ZS t h e r e e x i s t s a p a r t i t i o n o f t h e set E i n t o C1, , C n such t h a t A Ci n. f o r i = 1,2

,...,

...

+

I n t u i t i v e l y , t h i s p r o p e r t y means t h a t t h e sample s p a c e may be p a r t i t i o n e d i n t o e v e n t s w i t h a r b i t r a r i l y small probabilities. Next, f o r a g i v e n q u a l i t a t i v e p r o b a b i l i t y s t r u c t u r e ,


let A N

*

B, i f f o r e v e r y C , D G

B n D = 0, w e have A U C

r e (E,

7 ,>>

8B

9 , such t h a t

and B u D

&

A.

481

A

A

C = 0,

The s t r u c t u -

i s c a l l e d t i g h t , i f t h e c o n d i t i o n A*-

*

B

i m p l i e s A N B.

To u n d e r s t a n d t h i s c o n d i t i o n , i t i s b e s t t o t r y t o imagi n e a s t r u c t u r e which i s n o t t i g h t . Then t h e r e e x i s t two * e v e n t s , A and B, such t h a t A N B and A 4 B. S i n c e * B, f o r e v e r y C € such t h a t A n C = 0, t h e union A A rl C i s a t l e a s t as p r o b a b l e as B, i . e . A U C $ B. T h i s means t h a t A i s l e s s p r o b a b l e t h a n B, b u t a f t e r a d d i n g t o A any e v e n t C which i s d i s j o i n t w i t h i t , one o b t a i n s t h e e v e n t A u C which i s a t l e a s t as p r o b a b l e as B. I n s t i l l o t h e r words, it means t h a t t h e s t r u c t u r e does n o t c o n t a i n any e v e n t s which would be more p r o b a b l e t h a n A, b u t l e s s p r o b a b l e t h a n B.

-

3

It t u r n s o u t t h a t t h e two l a s t n o t i o n s completely char a c t e r i z e t h e s t r u c t u r e s a t i s f y i n g axiom 5, namely w e

have THEOREM 4.3. The s t r u c t u r e o f q u a l i t a t i v e p r o b a b i l i t y (E, $ s a t i s f i e s axiom 5 i f . and o n l y i f , i t i s

,b)

f i n e and t i g h t . Moreover, axioms 1-5 are s u f f i c i e n t f o r t h e e x i s t e n c e o f p r o b a b i l i t y s c a l e . We have namely TFEOREM 4.4.

If a q u a l i t a t i v e p r o b a b i l i t y s t r u c t u r e

(E, $ , 2, s a t i s f i e s axiom 5, t h e n t h e r e e x i s t s a ( f i n i t e l y a d d i t i v e ) p r o b a b i l i t y f u n c t i o n P on such t h a t P(A)

2

P(B)

i f - and o n l y i f . A

8 B.

3,

For some o t h e r c o n d i t i o n s implying t h e same a s s e r t i o n , as w e l l a s f o r an e x t e n s i v e l i t e r a t u r e on t h e s u b j e c t ,

482

CHAPTER 5

s e e R o b e r t s (1979).

5. DIFFERENCE MEASUREMENT One o f t h e most commonly e n c o u n t e r e d e m p i r i c a l s i t u a t i o n i n which t h e r e i s no o p e r a t i o n o f c o n c a t e n a t i o n o f o b j e c t s , i s t h e s i t u a t i o n of comparison o f d i f f e r e n c e s (intervals). F o r m a l l y , a n i n t e r v a l i s d e f i n e d a s a p a i r of o b j e c t s , and t h e comparisons c o n c e r n t h e r a t i o o f t h e l e n g t h s of i n t e r v a l s (or: the differences of values of a t t r i b u t es i n p a i r s of o b j e c t s ) . The s i m p l e s t e m p i r i c a l s i t u a t i o n may b e d e s c r i b e d as

f o l l o w s . The s u b j e c t i s p r e s e n t e d w i t h a p a i r o f s t i m u l i (e.g. two l i g h t s w i t h d i f f e r e n t i n t e n s i t i e s , two sounds w i t h d i f f e r e n t p i t c h o r i n t e n s i t y , e t c . ) , and i s asked t o i n d i c a t e ( b y means o f a s p e c i a l equipment) a s t i m u l u s which l i e s " i n t h e middle" between t h e two s t i m u l i p r e s e n t e d , i . e . a s t i m u l u s which l i e s "at t h e same d i s t a n ce" from e a c h o f . t h e m . T h i s s c a l i n g t e c h n i q u e i s c a l l e d b i s e c t ion. Let u s d e n o t e t h e i n i t i a l s t i m u l i b y a and b. Then t h e o b j e c t i s t o f i n d a s t i m u l u s c , such t h a t t h e d i f f e r e n -

c e s between c and a , and between b and c , a r e t h e same. T h i s t y p e o f e x p e r i m e n t assumes i m p l i c i t l y t h a t t h e subj e c t i s c a p a b l e o f comparing t h e d i f f e r e n c e s between p a i r s o f s t i m u l i ( a t l e a s t between p a i r s w i t h one element i n common, i . e . he c a n compare i n t e r v a l s w i t h a common e n d p o i n t ) . The comparison o f i n t e r v a l s need n o t be d i r e c t . Thus,


483

i n c a s e of c h o i c e from a p a i r ( a , b ) , t h e p r o b a b i l i t y p ( a , b ) of choosing a i s a c e r t a i n measure of " d i f f e r e n ce" between a and b. I f p ( a , b ) = 4, t h e elements a and 4, we may t h i n k t h a t a i s b are equivalent; i f p(a,b) " t o t h e r i g h t " of b, w i t h larger d e v i a t i o n s p ( a , b ) 4 i n d i c a t i n g l a r g e r d i s t a n c e s . Consequently, t h e knowledge o f p r o b a b i l i t i e s such as p ( a , b ) , p ( c , d ) , . . . may s e r v e as a basis f o r i n f e r e n c e about t h e l e n g t h s o f i n t e r v a l s a b , cd,

>

-

... .

Such a t e c h n i q u e based on p r o b a b i l i t i e s p ( a , b ) , l e a d i n g t o a n assignment o f s c a l e v a l u e s t o s t i m u l i , is c a l l e d t h e t e c h n i q u e of j u s t n o t i c e a b l e d i f f e r e n c e s ( j n d ) . A s a r u l e , a u n i t o f l e n g t h ( d i s t a n c e ) i s t a k e n t o be t h e d i s t a n c e between p a i r a,b such t h a t p ( a , b ) = 3/4. Thus, i n t h e t e c h n i q u e of u n f o l d i n g s c a l e s o f Coombs, t h e subject o r d e r s t h e s t i m u l i according t o h i s preference; i t i s assumed t h a t t h e s u b j e c t has a l s o an " i d e a l p o i n t " i n t h e space under c o n s i d e r a t i o n , and t h a t t h e s t i m u l i are l o c a t e d i n t h e same space. The c l o s e r is a s t i m u l u s t o t h e i d e a l p o i n t , t h e more p r e f e r r e d i t is. Thus, t h e model assumes comparisons o f i n t e r v a l s , counted from the ideal point t o the stimuli. The above examples i n d i c a t e t h a t t h e s i t u a t i o n s i n which t h e e m p i r i c a l data a l l o w us t o i n f e r about t h e d i f f e r -

e n c e s i n p a i r s o f s t i m u l i , i s r e l a t i v e l y common. A t t h e same t i m e , t h e y i n d i c a t e t h e formal v a r i e t y o f such sit u a t i o n s (e.g. comparisons may be p o s s i b l e f o r a l l int e r v a l s , o r o n l y f o r i n t e r v a l s w i t h a common e n d p o i n t ; t h e y may concern a l g e b r a i c d i f f e r e n c e s , o r d i s t a n c e s , e t c . ) . Accordingly, t h e r e e x i s t s a v a r i e t y o f axiom s y s tems, implying measurement f o r such s i t u a t i o n s , each of

484

CHAD772R 5

them c o n s t r u c t e d f o r t h e d e s c r i p t i o n o f some e m p i r i c a l setup. I n t h i s s e c t i o n , some o f such axiom systems w i l l b e p r e s e n t e d . The f i r s t w i l l c o n c e rn s i t u a t i o n s i n which a l l comparisons a r e p o s s i b l e ( n o t o n l y i n t e r v a l s w i t h common e n d p o i n t s ) , and t h e d i f f e r e n c e s are t r e a t e d a l g e b r a i c a l l y ( i . e . a b and ba are of o p p o s i t e s i g n s ) .

5.1.

The s t r u c t u r e of a l g e b r a i c d i f f e r e n c e s

The b a s i c r o l e here i s p l a y e d by a b i n a r y r e l a t i o n on t h e s e t o f p a i r s o f d i f f e r e n c e s ( o r p a i r s o f e l e m e n t s of t h e s e t A ) , hence a q u a r t e r n a r y r e l a t i o n on A. The symbol a b >, cd w i l l be i n t e r p r e t e d a s " t h e d i f f e r e n ce between a and b i s a t l e a s t as l a r g e as t h e d i f f e r e nce between c and d". The problem l i e s i n f i n d i n g c o n d i t i o n s imposed on t h e r e l a t i o n 2 , which would i m p l y t h e e x i s t e n c e o f a func-

t i o n f on A, such t h a t ab

cd

i f , and o n l y i f f ( a )

-

f(b)

>, f(c)

-

f(d).

These c o n d i t i o n s are e x p r e s s e d by t h e f o l l o w i n g theorem (see Krantz e t a l . , 1971).

THEOREM 5.1. Let A be a nonempty s e t , and l e t b i n a r y r e l a t i o n i n A X A, hence a q u a r t e r n a r y r e l a t i o n in A Assume t h a t t h e r e l a t i o n a l s y s t e m < A X A , &> s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s . For any a,b ,c,d , C A: a l , b l , c l C A and a r b i t r a r y sequences al,a2,.

.

..

1.

the relation

)r i s a weak o r d e r i n

A x A;

485


2.

If

ab

3. If ab

4.

cd, t h e n

8

be

a'b'

g ab ) cd & aa,

that ad' -

cd

N

dc

)r

ba;

b'c',

t h e n ac

theR there exist d',

ale'; d" t A such

d"b;

.

5. g al, a2,.. i s a s t r i c t l y bounded s t a n d a r d sequence, i.e. a i + l a i d a 2 a 1' and i t i s n o t t r u e t h a t a 2 a 1 ,U a 1a 1' and t h e r e e x i s t d ' , d " such t h a t d ' d " a i a l b d"d' f o r e v e r y i, t h e n t h e sequence al,a2, finite.

>

...

Then t h e r e e x i s t s a f u n c t i o n f d e f i n e d on A , such t h a t f o r a l l a , b , c , d E A we have (1)

ab

& cd

i f . and o n l y i f

f(a)-f(b)

2

f(c)-f(d).

Moreover, i f a f u n c t i o n g s a t i s f i e s t h e c o n d i t i o n

-

( i ) ,t h e n g = d f + f f o r some oC> 0

r.

The s e n s e o f t h e s e axioms i s as fo l l o ws . The f i r s t axiom

a s s e r t s t h a t t h e r e l a t i o n & i s connected and t r a n s i t i v e , i . e . t h a t e v e ry two d i f f e r e n c e s may be compared, and t h e r e s u l t s o f t h e s e comparisons are c o n s i s t e n t . The second axiom s t a t e s t h a t t h e d i r e c t i o n o f i n e q u a l i t y changes under t h e change o f s i g n . The t h i r d axiom s t a t e s t h a t i f two i n t e r v a l s are c o n s t r u c t e d by j o i n i n g t h e i n t e r v a l s w i t h common end p o i n t s , and i f t h e components o f one i n t e r v a l are l o n g e r t h a n t h e components o f t h e second i n t e r v a l , t h e n t h e r e s u l b i n g first i n t e r v a l i s l o n g e r t h a n t h e second. The f o u r t h axiom asserts t h a t g i v e n two i n t e r v b l s , one can

c u t a p a r t o f t h e l o n g e r one, e q u a l i n l e n g t h t o t h e

CHAPTER 5

486

s h o r t e r , and t h a t t h i s may be done c o u n t i n g from each o f t h e two ends o f t h e l o n g e r i n t e r v a l . F i n a l l y , t h e l a s t axiom i s one more v e r s i o n o f t h e A r chimedean axiom. The a s s e r t i o n o f t h e theorem i s t h a t t h e r e e x i s t s a sca-

l e f f o r t h e a t t r i b u t e , whose d i f f e r e n c e s are compared

>r.

thro u g h r e l a t i o n Moreover, f i s a n i n t e r v a l s c a l e , i.e. a s c a l e d e t e rm i n e d u n i q u e l y up t o t h e c h o i c e o f zer o and u n i t . 5.2.

Bisections

The axiom s y s t e m i n t h e p r e c e d i n g s e c t i o n concerned t h e

relation which o r d e r e d t h e d i f f e r e n c e s ( i n t e r v a l s ) . I n c a s e o f e x p e ri m e n t s c o n c e rn i n g b i s e c t i o n s , o u t l i n e d i n t h e i n i t i a l p a r t o f t h e p r e s e n t s e c t i o n , i t i s more con v e n i en t t o have axioms i n which t h e p r i m i t i v e n o t i o n i s t h e midpoint. Such axioms are e a s i e r t o v e r i f y empirically. The p r i m i t i v e c o n c e p t s here are a weak o r d e r

& in

the

s e t A , and a f u n c t i o n B, which maps A x A i n t o A, i . e . a s s i g n i n g t o e a c h p a i r a , b G A t h e p o i n t B ( a , b ) t' A , i n t e r p r b t e d as a midpoint between a and b. W e s h a l l n o t p r e s e n t here t h e whole s y s t e m of axioms con cer n in g b i s e c t i o n s (see P f a n z a g l 1968, Krantz e t a l , 1971 o r R o b e r t s 1979). Roughly, t h e s e axioms a s s e r t t h a t B(a,a) = a ( t h e midpoint o f a d e g e n e r a t e i n t e r v a l coincides with t h e endpoints of t h i s i n t e r v a l ) ; i f a b b , t h e n B( 9 , c) >/ B ( b , c ) , which means t h a t when one en d p o i n t i s changed from a t o b , t h e midpoint changes i n t h e same

SELZCTED TOPICS IN MEASUREMENT THEORY

487

d i r e c t i o n ( s e e Fig. 5 . 1 ) .

F i g . 5.1.

Monotonicity o f t h e midpoint.

F i n a l l y , t h e s o - c a l l e d bi-symmetry axiom, a s s e r t s t h a t

The sense of t h i s axiom may be i l l u s t r a t e d g e o m e t r i c a l l y ( s e e Fig. 5.2).

F i g . 5.2.

I l l u s t r a t i o n o f bisymmetry axiom.

I n a d d i t i o n , one assumes t h e c o n t i n u i t y o f t h e m i d p o i n t B(a,b). w i t h r e s p e c t t o a and b. The a s s e r t i o n o f t h e theorem i s t h a t t h e r e e x i s t s a f u n c t i o n f , d e f i n e d on A , and such t h a t

a

2b

i f , and o n l y i f , f ( a )

2

f(b);

CHAPTER 5

488

Moreover, i f some f u n c t i o n g s a t i s f i e s t h e above two c o n d i t i o n s , t h e n g = df' f o r some a)O and

tp

P.

Thus, f i s a measurement on a n i n t e r v a l t y p e s c a l e . I n case o f p e r f e c t symmetry i n evaluation o f the midpoint, we have f [ B ( a , b ) ] = $ f ( a ) t $ f ( b ) , i . e . p = $. The t h e orem t a k e s , however, i n t o a c c o u n t t h e p o s s i b i l i t y o f s y s t e m a t i c d e v i a t i o n s ; f o r i n s t a n c e , t h e tendency t o p l a c e t h e m i d p o i n t somewhat c l o s e r t o t h e l e f t e n d p o i n t a, o f t h e i n t e r v a l w i l l b e e x p r e s s e d as t h e p r o p e r t y p etc.

>

5.3. The s c a l e of j u s t n o t i c a b l e d i f f e r e n c e s ( j n d ) A t t h e end, w e s h a l l p r e s e n t two s e t s o f axioms concer-

n i n g d i s c r i m i n a t i o n . I n t h e f i r s t c a s e , t h e problem l i e s i n f i n d a s e t o f axioms f o r c h o i c e p r o b a b i l i t i e s p ( a , b ) , which would i m p l y t h e e x i s t e n c e o f a f u n c t i o n f such t h a t p(a,b)

>

Hepe p ( a , b ) of the p a i r a as h a v i n g (e.g. a may as b r i g h t e r

p(c,d)

i f , and d n l y i f ,

i s t h e p r o b a b i l i t y t h a t upon p r e s e n t a t i o n ( a , b ) , t h e s u b j e c t w i l l i n d i c a t e t h e element more o f t h e a t t r i b u t e under c o n s i d e r a t i o n be p r e f e r r e d t o b , lamp a may b e e v a l u a t e d t h a n lamp b , e t c . ) .

A s b e f o r e , we s h a l l p r e s e n t o n l y t h e b a s i c ideas o f t h e c o n s t r u c t i o n u n d e r l y i n g v a r i o u s axiom s y s t e m s .

SELEC7ED TOPICS IN MEASUREMENT THEORY

489

F i r s t o f a l l , t h e p r o b a b i l i t i e s p ( a , b ) a l l o w u s t o define the relation as f o l l o w s :

3

a

>

b if

and o n l y i f

p(a,b)

3

$.

Thus, we have a m b i f , and o n l y i f p ( a , b ) = $, Generally, if 0 p(a,b) 1 w e speak o f i m p e r f e c t d i s c r i m i n a t i o n between a and b , w h i l e i n c a s e p ( a , b ) = 0 o r 1 we speak o f p e r f e c t d i s c r i m i n a t i o n .

(the so-called strong stochastic t r a n s i t i v i k y ) , or

(moderate s t o c h a s t i c t r a n s i t i v i t y ) , o r f i n a l l y p(a,c> (weak

>4

stochastic transitivity).

CHAPTER 5

490

One of t h e axioms which a r e necessary f o r t h e e x i s t e n ce of s c a l e f i s t h e so-called guadruple c o n d i t i o n . If t h e s c a l e f e x i s t s , i t must s a t i s f y f(a)

-

f(b)

2

f(c)

-

f(d) i f

and only i f

It f o l l o w s t h a t t h e choice p r o b a b i l i t i e s must s a t i s f y t h e condition

T h i s i s t h e quadruple c o n d i t i o n mentioned above. A s a l r e a d y s t a t e d , t h e r e e x i s t many theorems about t h e

e x i s t e n c e of measurement induced by choice p r o b a b i l i t i e s p ( a , b ) . I n g e n e r a l , t h e a s s e r t i o n s of t h e s e theorems s t a t e t h a t t h e r e e x i s t s f u n c t i o n f such t h a t

Moreover, t h e measurement i s on i n t e r v a l s c a l e , i.e. every o t h e r fun c t i o n g s a t i s f y i n g t h e above c o n d i t i o n i s r e l a t e d t o f u n c t i o n f by a p o s i t i v e l i n e a r transforma t i o n g +f + (3 f o r some A 7 0 and f

.

5.4.

Semiorder s

F i n a l l y , i t i s worth while t o mention here one more axiom system, i n t r o d u c e d by Luce (19561, concerning t h e so-called semiorders.


491

The a i m o f t h e o r i g i n a l c o n s i d e r a t i o n s o f Luce was t o

c r e a t e a model f o r s i t u a t i o n s when t h e i n d i f f e r e n c e r e l a t i o n need n o t be t r a n s i t i v e . More p r e c i s e l y , one c o n s i d e r s a c e r t a i n set A , and a b i n a r y r e l a t i o n , say D, where t h e symbol aDb i s i n t e r p r e t e d a s ''a i s d i s t i n c t l y dominating o v e r b"; domination means s i m p l y a h i g h e r v a l u e of t h e a t t r i b u t e under study. For example, t h e e l e m e n t s o f A may b e sounds, c o n s i d e r e d from t h e p o i n t of view o f p i t c h . Then aDb means t h a t a i s d i s t i n c e l y h i g h e r i n p i t c h t h a n b. One assumes t h a t t h e r e l a t i o n D s a t i s f i e s t h e followi n g conditions: 1. It i s n o t t r u e t h a t aDa; 2.

If aDb and cDd, t h e n e i t h e r aDd o r cDb;

3. If aDb and bDc, t h e n f o r e v e r y d e i t h e r aDd o r dDc.

A w e have

The i n t e r p r e t a t i o n of t h e s e axioms i s t h e following.

The f i r s t axiom s t a t e s t h a t no o b j e c t dominates i t s e l f . To i l l u s t r a t e t h e meaning o f t h e remaining axioms, l e t u s imagine t h a t t h e e l e m e n t s o f A are p o i n t s on t h e a x i s , and t h a t aDb if t h e p o i n t a i s t o t h e l e f t o f

p o i n t b , a t a d i s t a n c e exgeeding a c e r t a i n t h r e s h o l d ( s a y , 1 cm). Then axiom 2 may be i l l u s t r a t e d as on F i g . 5.3. T h i s f i g u r e g i v e s two p o s s i b l e l o c a t i o n s o f pairs a,b and c,d, w i t h d i s t a n c e i n each p a i r exceeding 1 c e n t i m e t e r . On t h e upper f i g u r e we have aDd, w h i l e on t h e lower f i g u r e we have cDb. The meaning o f axiom 3 may be i l l u s t r a t e d i n a similar way (see Fig. 5.4). Here w e mark t h o s e p a r t s o f t h e axis

492

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where we have r e s p e c t i v e l y aDd

v

V

"

a

c

b

and

dDc.

"

d

aDd

v

v

v

v

c

a

b

d

cDb Fig. 5.3.

I l l u s t r a t i o n for axiom 2

"

"

\r

a

b

C

Fig.

5.4.

-

I l l u s t r a t i o n for axiom 3.

Let u s now d e f i n e a I b by requirement t h a t i t i s n o t t r u e t h a t aDb, and i t i s n o t t r u e t h a t bDa. The r e l a t i o n I

i s a n i n d i f f e r e n c e : i f a I b , t h e n t h e r e i s no p e r c e p t u a l d i f f e r e n c e between a and b. The axioms 1-3 do n o t i m p l y t r a n s i t i v i t y of t h e r e l a t i o n I: t h e s t i m u l i a , b , c may b e such t h a t a I b , b I c , b u t aDc.


493

One may, however, d e f i n e t h e e q u i v a l e n c e r e l a t i o n as f o l l o w s : we d e f i n e aEb, i f f o r e v e r y c G A t h e c o n d i t i o n a I c h o l d s i f and o n l y i f b I c . The r e l a t i o n E i s e a s i l y s e e n t o be r e f l e x i v e (aEa),

symmetric ( a E b i f and o n l y i f b E a ) , and t r a n s i t i v e ( a E b and bEc imply aEc). The theorem on e x i s t e n c e o f measurement may b e formul-

ated a s follows: If t h e set A i s f i n i t e , and r e l a t i o n D st i s f i e s axioms 1 , 2 and 3, t h e n t h e r e e x i s t s a f u n c t i o n f d e f i n e d on A such t h a t f o r some p o s i t i v e number 6 , t h e c o n d i t i o n aDb h o l d s i f - and o n l y i f f ( a ) 2 f ( b ) + 6

THEOREM 5.2.

I n t h i s c a s e t h e a s s e r t i o n o f t h e theorem does n o t spec i f y t h e t y p e o f measurement. However, f o r some f u r t h e r r e s u l t s c o n ce rn i n g se m i o rd e rs, see R o b e r t s (1979).

6. C O N J O I N T MEASUREMENT

One o f t h e more i n t e r e s t i n g problems o f measurement t h e o r y i s t h e problem o f c o n j o i n t measurement. We d e a l w i t h such measurement i n s i t u a t i o n s , when t h e o b j e c t s under c o n s i d e r a t i o n may b e decomposed i n t o two o r more components, e a c h i n f l u e n c i n g i n a s p e c i f i c way t h e a t t r i b u t e of the object. A t y p i c a l example may be p ro v i d e d by baskets o f goods, e a c h a p p e a r i n g i n some q u a n t i t y . These b a s k e t s are com-

p a r e d from t h e p o i n t o f view o f p r e f e r e n c e . The l a t t e r i s assumed t o depend i n some way on t h e amounts o f goods. A s a n o t h e r example, one may t a k e t h e t e m p e r a t u r e

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and h u m i d i t y , compared from t h e p o i n t of view o f t h e i r pleasantness

.

A s a n example o f p h y s i c s , one can t h i n k o f comparing momentums, depending on masses and v e l o c i t i e s .

To f o r m a l l y d e s c r i b e s i t u a t i o n s t o which t h e problems o f c o n j o i n t measurement a p p l i e s , l e t A1 and A2 d en o t e t h e s e t s o f p o s s i b l e " l e v e l s " o f t h e two f a c t o r s . I n t h e s e q u e l , t h e l e t t e r s a,b,c,... w i l l s t a n d f o r elements o f A1, and l e t t e r s p,q,r,... w i l l s t a n d f o r elements o f A2. Moreover, t h e e l e m e n t s o f t h e s e t A1 X A*, which c o n s t i t u t e " o b j e c t s " o f measurement, w i l l be den o t e d by a p , bq,...; such n o t a t i o n i s more co n v en i en t than ( g , ~ ) , (b,q)

,... .

>

The p r i m i t i v e n o t i o n here i s a b i n a r y r e l a t i o n on A1 % A 2 , i n t e r p r e t e d as a j o i n t e f f e c t o f two f a c t o r s : ap bq means t h a t an o b j e c t w i t h l e v e l s a and p o f t h e two f a c t o r s i s "not l a t e r t h a n " from t h e p o i n t o f view o f t h e j o i n t a t t r i b u t e , t h a n t h e o b j e c t w i t h l e v e l s b and q o f t h e two f a c t o r s . The r e l a t i o n s

>

-

and o f s t P i c t i n e q u a l i t y and i n d i f f e r e n c e are d e f i n e d i n t h e u s u a l way, as shown i n s e c t i o n 4.

I n t h e t h r e e examples g i v e n above, t h e symbol a p & bq * means t h a t : 1. A b a s k e t w i t h amount a of goods 1, and amount p o f goods 2, i s p r e f e r r e d t o a b a s k e t c o n t a i n i n g amount b ' o f goods 1, and amount q o f goods 2. 2. The atm o sp h e ri c s i t u a t i o n w i t h t e m p e r a t u r e a and


49 5

h u m i d i t y p is t a k e n as more p l e a s a n t t h a n t h e atmosph-

e r i c s i t u a t i o n w i t h t e m p e r a t u r e b and h u m i d i t y q;

3 . The o b j e c t w i t h mass a and v e l o c i t y p has l a r g e r momentum t h a n o b j e c t w i t h mass b and v e l o c i t y q.

2

Formally, we assume t h a t t h e r e l a t i o n i s d e f i n e d on t h e whole s e t A1 8 A2. The problem w i l l l i e i n f i n d i n g c o n d i t i o n s implying t h e e x i s t e n c e o f measurement s c a l e s b o t h f o r A1 and A*, a s w e l l a s f o r A 1 y A2. T h i s explai n s t h e term " c o n j o i n t measurement". The assumption t h a t t h e r e l a t i o n

&

i s d e f i n e d on t h e A 2 imposes some r e s t r i c t i o n s on t h e c l a s s

whole s e t A1)< o f e m p i r i c a l s i t u a t i o n s t o which t h e t h e o r y o f c o n j o i n t measurement a p p l i e s . It i s t h e r e f o r e worth w h i l e t o c h a r a c t e r i z e such s i t u a t i o n s .

t h e o b s e r v a t i o n o f r e l a t i o n & may concern Firstly, n o t t h e whole s e t A 1 % A 2 , b u t o n l y i t s s u b s e t , because

of t e c h n i c a l d i f f i c u l t i e s . For i n s t a n c e , u s i n g t h e example i n p h y s i c s , t h e o b s e r v a t i o n s o f momentum may be d i f f i c u l t i n t h e domain o f large masses and large v e l o -

cities. Secondly, t h e o b s e r v a t i o n s o f t h e r e l a t i o n may conc e r n o n l y a p a r t o f t h e set A1% A2 because o f some f u n c t i o n a l r e l a t i o n s between l e v e l s i n A1 and A*, expressed, s a y , by a c e r t a i n p h y s i c a l l a w . As an example, l e t u s imagine t h a t we o b s e r v e p i e c e s o f i c e , where A1 r e p r e s e n t s mass, and A2 r e p r e s e n t s t h e volume. I n t h i s c a s e t h e r e i s no p o s s i b i l i t y o f o b s e r v i n g v a r i o u s masses o f i c e w i t h t h e same volume, or v a r i o u s volumes w i t h t h e same mass, s i m p l y because t h e s e two are r e l a t e d by

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CHAPTER 5

a functional relation. The above two s i t u a t i o n s , e s p e c i a l l y t h e second, are

e l i m i n a t e d from t h e c o n s i d e r a t i o n s , s i n c e f o r m a l l y , a s a l r e a d y s t a t e d , i t i s assumed t h a t t h e r e l a t i o n is d e f i n e d on t h e whole s e t A1 % A2.

&

G e n e r a l l y , t h e problem may be f o r m u l a t e d as f o l l o w s , We a r e t o f i n d c o n d i t i o n s imposed on t h e r e l a t i o n a l s y s t e m , under which t h e r e e x i s t f u n c t i o n s f on A1 and g on A 2 , and a f u n c t i o n o f two v a r i a b l e s F ( x , y ) , one-to-one w i t h r e s p e c t t o each o f i t s arguments, such t h a t

I n o t h e r words, w e want t o have numerical s c a l e s on e a c h o f t h e s e t s A1 and A 2 , and a r u l e F o f c o n n e c t i n g t h e s c a l e v a l u e s , which would p r e s e r v e t h e o r d e r

F.

T h i s g e n e r a l f o r m u l a t i o n may b e f u r t h e r s p e c i f i e d by

imposing some c o n d i t i o n s on t h e form o f F. I n p a r t i c u l a r , t h e most developed and a l s o most important from t h e p r a c t i c a l p o i n t o f view, i s t h e t h e o r y o f a d d i t i v e conj o i n t measurement, when F ( x , y ) = x t y. I n t h i s c a s e , t h e problem l i e s i n f i n d i n g , f o r a given r e l a t i o n a l s y s t e m t h e f u n c t i o n s f and g, d e f i n e d on A1 and A 2 r e s p e c t i v e l y , such t h a t (6.1)

ap

bq

i f , and o n l y i f

SELECTED TOPICS LN MEASUREMENT THEORY

6.1.

497

Necessary c o n d i t i o n s

A s i n t h e p r e c e d i n g c a s e s , t o o b t a i n t h e n e c e s s a r y ax-

ioms, l e t u s assume t h a t t h e f u n c t i o n s f and g which s a t i s f y c o n d i t i o n (6.1) e x i s t , and l e t u s i n v e s t i g a t e t h e consequences o f t h i s assumption. Each o f t h e s e cons eq u e n c es i s , o f c o u r s e , a s i n e qua non c o n d i t i o n f o r t h e e x i s t e n c e o f c o n j o i n t measurement. Thus, c o n d i t i o n (6.1) implies, f i r s t of a l l ASSUMPTION 1.

i s a weak o r d e r on A1 x A2.

T h i s assumption i s , t h e r e f o r e , t h e f i r s t o f t h e neces-

s a r y conditions.

Next, under t h e assumption of e x i s t e n c e o f an a d d i t i v e r e p r e s e n t a t i o n (6.11, w e may write t h e f o l l o w i n g c h a i n of equivalences:

Consequently, we must have ASSUMPTION 2.

( A d d i t i v e independence). bq. f o r e v e r y q w e have a l s o aq

&

If a p &

bp, t h e n

An an a lo g o u s c o n d i t i o n must a l s o h ol d f o r t h e second c o o r d i n a t e , so t h a t we have

49 8

CHAPTER 5

ASSUMPTION 2'. bP)

If

ap @ a q , t h e n f o r e v e r y b w e have

m.

Thus, a n o r d e r i n g o f two o b j e c t s w i t h a common v a l u e o f one a t t r i b u t e d o e s n o t change under t h e change o f this attribute. I n g e o m e t r i c i n t e r p r e t a t i o n ( s e e Fig. 6.1) t h e f i r s t c o n d i t i o n means simply t h a t i f p o i n t s P = ap and Q = bp a r e such t h a t P ) Q , t h e n t h e same r e l a t i o n must be s a t i s f i e d f o r a l l c o r r e s p o n d i n g p a i r s o f p o i n t s P ' and Q' on t h e l i n e s crc, and

13 .

The second assumption h a s analogous i n t e r p r e t a t i o n . o(

'A2

q

1 I I I

I I I

- - - - r - - - 1""' Q '1

P '1

I

P

PI

I

I

I

I I

I I

- - - - r - - -1""' PI

Q '

I I

I

a

b

I

Fig. 6.1. Geometric i n t e r p r e t a t i o n o f t h e axiom of a d d i t i v e independence (see explanation i n t e x t ) .

One o f t h e main consequences o f t h e assumption o f add-

499


i t i v e independence 2 and 2 ' i s t h a t t h e y a l l o w us t o define ordering relations and %2 on A1 and A2 as follows.

k1

DEFINITION. We p u t a ap

&

bp, and p

b2 q

b1 b ,

if f o r e v e r y P E A2 w e have

i f f o r e v e r y a -C A1 we have ap

& aq.

According t o a s s u m p t i o n s 2 and 2', t h e words " f o r e v e r y " i n t h e above d e f i n i n i t i o n may b e r e p l a c e d by " f o r some". It f o l l o w s from t h e above d e f i n i t i o n t h a t : ( i ) +i i s

a weak o r d e r o f t h e s e t Ai ( i = 1,2);

(iii) i f a t l e a s t one o f t h e i n e q u a l i t i e s i n ( i i ) i s

s t r i c t (i.e. (iv)

if

a

k1 b or

a P w bq,

p >2 q ) , t h e n

then a k1 b

ap

> bq;

i f and o n l y i f q

fi

----

9 -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I........................

........................ '...................*.*** ........................ I........(................

........I................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P . . . . . . . . . . . . . . . . . . . . . . . . .

P '

----

I i

-

J.r...L.d.L.L.L.L.L.L.L,L

I I

I I

I

I

a

Fig. 6.2.

b I l l u s t r a t i o n of c o n d i t i o n

-

' (ii)

A1

g2 p.

500

CHAPTER 5

F o r g e o m e t r i c a l i l l u s t r a t i o n , assume t h a t t h e o r d e r s

i n t h e s e t s A1 and A2 are r e p r e s e n t e d by t h e o r d e r i n g s o f p o i n t s on t h e r e a l l i n e s . C o n d i t i o n ( i i ) means t h a t i f t h e i n e q u a l i t i e s on b o t h a x e s go i n t h e same d i r e c t ions, then the j o i n t inequality (according t o t h e relati o n + ) a l s o goes i n t h e same d i r e c t i o n . I n o t h e r words, a l l p o i n t s (see F i g . 6.2) i n t h e shaded area a r e i n t h e same r e l a t i o n w i t h t h e c o r n e r P. C o n d i t i o n ( i v ) , i m p l i e d d i r e c t l y by ( i i ) , s t a t e s t h a t t h e p o i n t s of i n d i f f e r e n c e must l i e on a c u r v e w i t h n e g a t i v e s l o p e ( s e e F i g . 6.3).

I I

I I

a

b

’

A1

Fig. 6 . 3 . I n d i f f e r e n c e curve

a p t u bq, t h e n t h e r e l a t i o n s between a and b, and between p and q , must be g o i n g i n t h e o p p o s i t e d i r e c t ions. If

The n e x t n e c e s s a r y c o n d i t i o n s f o r t h e c o n j o i n t measure-


501

ment c o n cer n c a n c e l l a t i o n . These c o n d i t i o n s may be o b t a i n e d as fo l l o ws . Assuming t h e e x i s t e n c e o f c o n j o i n t measurement, i.e. assuming c o n d i t i o n (6.11, we may w r i t e

Adding t h e s e i n e q u a l i t i e s we g e t

or

Consequently, a n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e o f c o i n j o i n t measurement i s : (Double c a n c e l l a t i o n ) . For a l l a , b , c C A1 and p , q , r E A*, if a r cq and cp 8 b r , t h e n ap @ bq. ASSUMPTION 3 .

2

The meaning o f t h i s c o n d i t i o n are b e s t i l l u s t r a t e d i f one c o n s i d e r s some o f i t s consequences. If a r & cq and

&-

T h i s means (see Fig. 6.4) t h a t i f t h e p o i n t s P = a r and Q = cq l i e on one i n d i f f e r e n ce cu r v e , and p o i n t s P t = cp and Q t = b r a l s o l i e on

cp

b r , t h e n a p & bq.

one i n d i f f e r e n c e c u rv e , t h e n t h e p o i n t s PI1 = ap and Q" = bq must a l s o l i e on one i n d i f f e r e n c e curve. Thus, assumption 3, c a l l e d a l s o t h e axiom of double

502

CHAPTER 5

c a n c e l l a t i o n , imposes some c o n s t r a i n t s on t h e f a m i l y o f i n d i f f e r e n c e c u rv e s. The t e r m "double c a n c e l l a t i o n " i s connected w i t h t h e f a c t t h a t i n t h e premises o f assumpt i o n 3 t h e r e a p p e a r two i n d e q u a l i t i e s i n v o l v i n g t h e p o i n t s c and r , which " c a n c e l " t o one i n e q u a l i t y , f r e e of b o t h c and r.

7

b

C

a

A1

Fig. 6.4. I l l u s t r a t i o n of t h e axiom o f double c a n c e l l a t i o n . F i n a l l y , t h e last n e c e s s a r y axiom is t h e Archimedean axiom. D e s p i t e t h e l a c k o f t h e o p e r a t i o n of c o n c a t e n a t i o n , one can b u i l d s t a n d a r d sequences a l o n g each of t h e a x e s , i.e. s t a n d a r d sequences o f e l emen t s o f A1 and A*. Because of t h e symmetry, we s h a l l p r e s e n t here o n l y t h e c o n s t r u c t i o n of a s t a n d a r d sequence a l , a2,.. o f elements o f A1. I n t u i t i v e l y sp e a k i n g , t h e " d i s t a n c e s " b e t -

.

SEL.EClED TOPICS IN MEASUREMENT THEORY

SO3

ween a l and a2, between a 2 and a3,... e t c . must be ident i c a l . T h i s may be e x p r e s s e d by r e q u i r i n g t h a t t h e r e exA2 such t h a t p k2 q and alp ~ a ~ f o+r e~ v e r yq i s t p,q i = 1,2,... T h i s i s i l l u s t r a t e d on Fig. 6.5.

.

I

I

al

Fig. 6.5.

I

I

a2

a3

I

I

a5

’

A1

C o n s t r u c t i o n o f a s t a n d a r d sequence

The above d e f i n i t i o n concerns i n c r e a s i n g sequences.

If

q k2 p a one o b t a i n s i n a similar way a d e c r e a s i n g s t a n d a r d sequence. The s t a n d a r d sequences o f e l e m e n t s o f t h e s e t A2 are d e f i n e d i n a s i m i l a r way.

Now, t h e Archimedean axiom may be formulated as f o l l o w s . ASSUMPTION 4.

Each bounded s t a n d a r d sequence has a t most a f i n i t e number o f elements.

{anla

where a n € Ai Boundedness o f a s t a n d a r d sequence (1 = 1,2) means t h a t t h e r e e x i s t elements x and y i n Ai such t h a t x an Zi y f o r a l l n; t h e i n d e x i e q u a l s 1 or 2 , depending whether w e c o n s i d e r sequences i n A1 or A2.

504

6.2.

CHAPTER 5

Sufficient conditions

The above f o r m u l a t e d n e c e s s a r y c o n d i t i o n s (assumption

8

that i s a weak o r d e r , assumption o f a d d i t i v e independence, double c a n c e l l a t i o n axiom, and Archimedean axiom) are n o t s u f f i c i e n t t o e n s u r e t h e e x i s t e n c e o f c o n j o i n t measurement. It t u r n s o u t t h a t i t i s n e c e s s a r y t o impose some r e q u i r e m e n t s c o n c e rn i n g s o l v a b i l i t y o f t h e eq u at i o n a p d bq f o r g i v e n any t h r e e among a,b ,p ,q , namely: ASSUMPTION 5.

( s o l v a b i l i t y ) . For any t h r e e among a , b € A1 and p,q Ci A2 t h e r e e x i s t s f o u r t h element such t h a t ap hl bq. For a n i l l u s t r a t i o n , assume t h a t we have a , b and p (see Fig. 6.6).

I F i g . 6.6.

I

I

I

I

b

a

I l l u s t r a t i o n o f s o l v a b i l i t y axiom.

The s o l v a b i l i t y assumption asserts t h a t t h e i n d i f f e r e n c e c u r v e which p a s s e s t h r o u g h t h e p o i n t ap w i l l i n t e r s e c t t h e l i n e d ; t h e coordinate of t h i s p o i n t o f i n t e r -


505

s e c t i o n i s t h e desired p o i n t q. F i n a l l y , t h e l a s t assumption e l i m i n a t e s t r i v i a l S i t u a t i o n s . It may be fo rm u l a t e d as f o l l o w s : ASSUMPTION 6. Each o f t h e c o o r d i n a t e s i s e s s e n t i a l , i . e .

t h e r e e x i s t a,b

e A1 and

p,q G A2 such t h a t a

h1 b and

p >2 q *

Under as s u m p t i o n s 1-6 t h e c o n j o i n t measurement e x i s t s . More p r e c i s e l y , we have t h e f o l l o w i n g theorem ( s e e Krantz e t a l . , 1971):

&}

THEOREM 6.1. If t h e s y s t e m 0 and

e

>

- p1,

p2

Thus, t h e axioms f o r c o n j o i n t measurement imply a l s o t h e e x i s t e n c e o f measurements on t h e s e t s A1 and A 2 sep a r a t e l y , and t h e s e measurements are on a n i n t e r v a l s c a l e . What is c h a r a c t e r i s t i c here i s t h a t on b o t h a x e s t h e u n i t s must be i d e n t i c a l ( w h i l e t h e z e r o s may be arbitrary),

6.3.

G e n e r a l i z a t i o n : m u l t i n o m i a l c o n j o i n t measurement

There e x i s t s i t u a t i o n s i n which t h e p a r t i c u l a r f a c t o r s i n f l u e n c e t h e f i n a l r e s u l t , b u t n o t i n an a d d i t i v e way.

CHAPTER 5

506

Such s i t u a t i o n s o c c u r most o f t e n i n p h y s i c s , where one has laws which are n o t l i n e a r . F o r i n s t a n c e , s u c h f a c t ors as mass and v e l o c i t y determine t h e k i n e t i c energy E e q u a l Jmv2, e t c . P a s s i n g t o examples from psychology, one may mention here s i m i l a r i t y i n d i c e s b u i l t on t h e basis of v a r i o u s a t t r i b u t e s , d e f i n e d as n

are t h e vecwhere 5 = (xl,...,xn) and y = (yl,...,yn) t o r s o f v a l u e s o f t h e a t t r i b u t e s o f 5 and y. A s s t i l l a n o t h e r example from psychology, w e may t a k e t h e m o t i v a t i o n model o f Atkinson (1963), where it i s assumed t h a t t h e a t t r a c t i v e n e s s o f a s i t u a t i o n o f r i s k i s e x p r e s s e d as A = IsPsMs

-

IfPfMf,

where I and If are t h e s o - c a l l e d i n c e n t i v e v a l u e s of S s u c c e s s and f a i l u r e , P s and Pf are s u b j e c t i v e probab i l i t i e s o f s u c c e s s and f a i l u r e , and Ms and Mf a r e t h e g e n e r a l m o t i v e s towards achievement and towards avoidi n g f a i l u r e . Atkinson p o s t u l a t e s t h a t Pf = 1 Ps and Is = 1 Ps, If = -P Consequently, t h e a t t r a c t i v e n e s s S' o f a g i v e n s i t u a t i o n becomes

-

-

A s a n o t h e r example of l a c k o f a d d i t i v i t y of f a c t o r s one may t a k e t h e well-known s t u d i e s of C l i f f (19591,

507


who i n v e s t i g a t e d t h e m u l t i p l i c a t i v e v a l u e s o f a d v e r b s o f q u a n t i t y , such as v e r y , r a t h e r , somewhat, a l i t t l e , e t c . C l i f f analysed the perceived values of various combinations o f t h e above adverbs i n t h e i r c o n n e c t i o n w i t h a d j e c t i v e s such as p l e a s a n t , bad, e t c . ( s o t h a t t h e combinations were "very p l e a s a n t " , " r a t h e r bad", e t c . ) . Because o f t h e obvious e x i s t e n c e o f t h e z e r o of t h e s c a l e , the a c t i o n o f adverbs differs f o r p o s i t i ve and n e g a t i v e domains, f o r i n s t a n c e v e r y good ) . r a t h e r good very pleasant > r a t h e r p l e a s a n t ,

(6.2)

but (6.3)

r a t h e r unpleasant

very unpleasant.

I n t h i s c a s e one may p o s t u l a t e t h a t t h e r e e x i s t s a scal e f l f o r adverbs of q u a n t i t y , w i t h values f l being always p o s i t i v e , and s c a l e f2 f o r e v a l u a t i o n s , w i t h v a l u e s b o t h p o s i t i v e and n e g a t i v e . The combined e f f e c t i s m u l t i p l i c a t i v e , i. e. f ( r a t h e r u n p l e a s a n t ) = fl(rather)f2(unpleasant),

f ( v e r y good) = f l ( v e r y ) f 2 ( g o o d ) , e t c . The change o f d i r e c t i o n o f i n e q u a l i t y i n (6.2) and (6.3) i s connected w i t h t h e f a c t t h a t f l > 0 f o r a l l v a l u e s o f t h e argument, w h i l e f 2 ( u n p l e a s a n t ) 0, and f2(good) 0, etc.

>

0 , and p s u f f i c i e n t l y c l o s e t o 1, t h e f u n c t i o n I s ( p , r ) i s r e p r e s e n t a b l e a s a sum (b)

(8.4)

(8.5)

5 44

CHAPTER 5

H(r)

--+0 as r -+ 0 .

(8.7)

C o n d i t i o n ( a ) of H y p o t h e s i s 1 i s o f t e c h n i c a l c h a r a c t e r and s c a r c e l y r e q u i r e s any comments: i t a s s e r t s t h a t t h e r a t e a t which t h e i n c e n t i v e o f s u c c e s s changes w i t h t h e change of p r o b a b i l i t y o f s u c c e s s remains bounded. Observe t h a t t h i s c o n d i t i o n does n o t a s s e r t a n y t h i n g about t h e s i g n of t h e d e r i v a t i v e w i t h r e s p e c t t o p . C o n d i t i o n ( b ) of H y p o t h e s i s 1 s t a t e s g e n e r a l l y t h a t t h e v a r i a b l e s p and r i n t h e i n c e n t i v e I ( p , r ) o f s u c c e s s S s e p a r a t e a d d i t i v e l y i n t h e neighbourhood o f t h e p o i n t p = 1, r = 0 . It s a y s t h e r e f o r e t h a t t h e p s y c h o l o g i c a l

and economical e f f e c t s a r e a d d i t i v e , a t l e a s t i n t h e domain o f small rewards and h i g h l y p r o b a b l e s u c c e s s e s , i . e . i n t h e domain- where t h e s e e f f e c t s a r e weak. I n s t i l l o t h e r words, t h i s a s s u m p t i o n s t a t e s t h a t p and r do n o t i n t e r a c t i n t h e i n c e n t i v e v a l u e of s u c c e s s i n t h e neighbourhood o f p = 1, r = 0. The j u s t i f i c a t i o n o f t h e t h r e e s p e c i f i c a s s u m p t i o n s (8.5), ( 8 . 6 ) and ( 8 . 7 ) i s as f o l l o w s . The i n c e n t i v e v a l u e o f s u c c e s s i s (or may b e ) c o n n e c t e d w i t h overcomi n g some d i f f i c u l t i e s , use of s k i l l , knowledge, e t c . I n any s u c h c a s e , t h e l e s s l i k e l y i s t h e s u c c e s s , t h e more s a t i s f a c t i o n i t b r i n g s . T h i s means, i n t u r n , t h a t I ( p , r ) s h o u l d d e c r e a s e w i t h t h e i n c r e a s e of p , hence S i t s d e r i v a t i v e w i t h r e s p e c t t o p s h o u l d be n e g a t i v e . I n f a c t , Hypothesis 1 r e q u i r e s t h a t t h e i n c e n t i v e of s u c c e s s s h o u l d d e c r e a s e w i t h p o n l y i n t h e neighbour-


54s

hood o f p = 1 ( c o n d i t i o n ( 8 . 6 ) ) ; no m o n o t o n i c i t y r e q u i rement i s imposed f o r o t h e r v a l u e s o f p . I n t e r p r e t i n g t h e f u n c t i o n s G and H i n c o n d i t i o n ( 8 . 4 ) as t h e e f f e c t s on i n c e n t i v e o f s u c c e s s due r e s p e c t i v e l y t o t h e p r o b a b i l i t y of s u c c e s s ( h e n c e from o c c u r r e n c e of " s u c c e s s as s u c h " , r e g a r d l e s s o f t h e economical reward r ) , and t o t h e economical reward r , one may s a y t h e f o l l o w i n g . C o n d i t i o n ( 8 . 5 ) s t a t e s t h a t t h e satTisf a c t i o n from " s u c c e s s as s u c h " , s e p a r a t e l y from t h e e c o n o m i c a l r e w a r d , becomes n e g l i g i b l e when t h e s u c c e s s becomes c e r t a i n . However, t h e r a t e of change of t h i s e f f e c t as p t e n d s t o 1 i s n o t n e g l i g i b l e ( c o n d i t i o n ( 8 . 6 ) ) : a s p becomes l a r g e r , t h e p s y c h o l o g i c a l s a t i s f a c t i o n from s u c c e s s becomes s m a l l e r a t a r a t e which a p p r o a c h e s some n e g a t i v e v a l u e . F i n a l l y , c o n d i t i o n ( 8 . 7 ) a s s e r t s t h a t t h e e f f e c t on t h e i n c e n t i v e o f s u c c e s s from t h e reward r becomes n e g l i g i b l e when t h e reward a p p r o a c h e s z e r o . The f o l l o w i n g a s s u m p t i o n a b o u t I S ( p , r ) w i l l be needed i n c a s e o f one theorem o n l y ; t h i s e x p l a i n s why i t i s l i s t e d separately. HYPOTHESIS 2 .

If r

> 0 , then

I n t u i t i v e l y , t h i s hypothesis a s s e r t s t h a t t h e incentive v a l u e of s u c c e s s which i s v e r y i m p r o b a b l e , and which b r i n g s a p o s i t i v e r e w a r d , i s p o s i t i v e . A moment o f reflection i s sufficient t o r e a l i z e that condition (8.8) i s v e r y n a t u r a l and p l a u s i b l e .

546

CHAPTER 5

L e t u s now t u r n t o t h e h y p o t h e s e s about t h e i n c e n t i v e

v a l u e I a f ( p , d ) o f a v o i d i n g f a i l u r e . It a p p e a r s t h a t f o r t h e e x i s t e n c e of non-normal r i s k a r e a s , t h e a s s u m p t i o n s about t h e f u n c t i o n Iaf a r e s t i l l weaker t h a n t h o s e a b o u t t h e f u n c t i o n I s . Again, we s e p a r a t e t h e c o n d i t i o n which i s needed i n one o f t h e theorems o n l y .

HYPOTHESIS 3 . The f u n c t i o n I a f ( p , d ) i s d i f f e r e n t i a b l e w i t h r e s p e c t t o p f o r e v e r y d , and t h e d e r i v a t i v e 9 1 /Jp i s bounded i n t h e c l o s e d i n t e r v a l [O,l]. af HYPOTHESIS 4 . For a l l d

>0

we have

(8.9) Both of t h e s e h y p o t h e s e s h a r d l y need any i n t u i t i v e j u s t i f i c a t i o n . H y p o t h e s i s 3 i s a g a i n of a t e c h n i c a l c h a r a c t e r , a s s e r t i n g simply a c e r t a i n d e g r e e o f smoothn e s s of f u n c t i o n I ( p , d ) . H y p o t h e s i s 4 s t a t e s t h a t af f o r p s u f f i c i e n t l y c l o s e t o 1, hence f o r f a i l u r e s which a r e h i g h l y improbable, t h e v a l u e I a f ( p , d ) i s n e g a t i v e (at l e a s t f o r f a i l u r e s which r e s u l t i n a loss d > 0 ) . To p u t i t d i f f e r e n t l y : i f d > 0 , t h e f a i l u r e a l w a y s " h u r t s " , and i t h u r t s more when i t i s i m p r o b a b l e . T h i s , i n c i d e n t a l l y , s u g g e s t s t h a t a n a t u r a l requirement on Iaf would be t h a t i t i s a d e c r e a s i n g f u n c t i o n o f p . We do n o t s t a t e i t as a n a s s u m p t i o n , however, s i n c e i t w i l l n o t be needed i n e s t a b l i s h i n g t h e p r o o f o f t h e t heorem. We s h a l l now prove two theorems about t h e e x i s t e n c e of non-normal r i s k a r e a s ,


541

THEOREM 2 . Assume t h a t t h e i n c e n t i v e f u n c t i o n s I s ( p , r )

and I a f ( p , d ) s a t i s f y r e s p e c t i v e l y Hypotheses 1 and 3. Then for e v e r y F gr& d t h e r e e x i s t s Mo s u c h t h a t whene v e r M > M and r i s s u f f i c i e n t l y s m a l l , t h e r i s k a r e a 0i s non-normal f o r s u i t a b l e reward b f o r r i s k l e s s o p t i o n . I n symbolic n o t a t i o n :

(M > M o ) 3 ( 3 r o ) ( V r ) C ( cr r o ) j

( 3 b ) : r i s k area i s

non-normall. What t h i s theorem a s s e r t s , i n e s s e n c e , i s t h a t i f t h e

motive M t o a c h i e v e s u c c e s s i s s u f f i c i e n t l y h i g h , and t h e reward r o f s u c c e s s i s s u f f i c i e n t l y small, t h e n one c a n f i n d s u c h a reward f o r t h e r i s k l e s s o p t i o n N t h a t t h e r i s k area w i l l be non-normal: t h e d e c i s i o n maker w i l l p r e f e r t h e r i s k y o p t i o n R f o r some p , b u t when P t h e chances p of s u c c e s s w i l l be s u f f i c i e n t l y c l o s e t o 1, he w i l l p r e f e r t h e r i s k l e s s o p t i o n N

.

P r o o f . It s u f f i c e s t o show t h a t t h e v a l u e f u n c t i o n f ( p ) of t h e r i s k y o p t i o n R i s not monotonically i n c r e a s i n g . P F o r t h a t , i n t u r n , i t i s enough t o show t h a t t h e r e e x i s t s a p o i n t a t which t h e d e r i v a t i v e f ' ( p ) i s n e g a t i v e . We s h a l l p r o v e t h a t s u c h a p o i n t e x i s t s i n t h e neighbourhood of p = 1. Using ( 8 . 1 ) and ( 8 . 3 ) w e o b t a i n

L e t u s choose p c l o s e t o 1 and r c l o s e t o 0 , s o t h a t

548

CHAPTER 5

t h e r e l a t i o n ( 8 . 4 ) of Hypothesis 1 holds. S u b s t i t u t i o n

of ( 8 . 4 ) i n t o (8.10) y i e l d s

N e x t , l e t us f i x F and d , and l e t (8.12)

Mo = Fqd/c

where c i s t h e c o n s t a n t a p p e a r i n g i n ( 8 . 6 ) , and q d i s the l i m i t i n c o n d i t i o n ( 8 . 9 ) . A t p r e s e n t , i t i s n o t req u i r e d t h a t q d be n e g a t i v e .

S i n c e c c: 0 by a s s u m p t i o n (8.6), w e h a v e , f o r a l l M exceeding Mo t h e r e l a t i o n MC

- Fqd 4 0 .

(8.13)

D i f f e r e n t i a t i o n of (8.11) y i e l d s

+ (1-P)F ? I a f / 3 P .

>

L e t us now f i x M with p

--+1 and

r

(8.14)

Mo and p a s s i n (8,14)

+0.

t o the l i m i t

We have t h e n M G ( p )

-+

0 by

(8.5), M H ( r ) --+. 0 by ( 8 . 7 ) , and ( 1 - p ) F 2 1 a f / a p + 0 by t h e a s s m D t i o n o f boundedness of t h e d e r i v a t i v e o f Iaf ( H y p o t h e s i s 3 ) . N e x t , p M G 7 ( p ) 4 Me by ( 8 . 6 ) , and f i n a l l y , F I a f ( p , d ) 4 qd by ( 8 . 9 ) . C o n s e q u e n t l y , t h e l i m i t o f t h e d e r i v a t i v e ( 8 . 1 4 ) a s p -3 1 and r --3 0 i s e q u a l Mc - F q d y a q u a n t i t y which i s n e g a t i v e i n view o f ( 8 . 1 3 ) . T h i s c o m p l e t e s t h e p r o o f .


549

We s h a l l now prove THEOREM 3 . I f , i n a d d i t i o n t o t h e assumptions o f Theo-

rem 2 , t h e i n c e n t i v e s Is and Iaf s a t i s f y r e s p e c t i v e l y Hypotheses 2 and 4 , t h e n one can f i n d v a l u e s F and d

>0

z c h t h a t f o r a s u i t a b l e b y t h e r i s k a r e a i s non-

normal and c e n t r a l . P r o o f . By t h e p r e c e d i n g theorem, t h e v a l u e f u n c t i o n f ( p ) i s n o t monotonically i n c r e a s i n g : it d e c r e a s e s i n t h e neighbourhood o f t h e p o i n t p = 1 f o r s u i t a b l e v a l u e s o f t h e p a r a m e t e r s . To prove t h e a s s e r t i o n o f t h e theorem, i t s u f f i c e s t o show t h a t f ( p ) i s n o t monoton i c a l l y d e c r e a s i n g i n t h e whole i n t e r v a l [0,1]. We s h a l l show t h a t f o r s u i t a b l e c h o i c e o f F and d , t h e d e r i v a t i v e f ' ( p ) i s p o s i t i v e i n t h e neighbourhood of p = 0 .

>

Mo d e f i n e d by ( 8 . 1 2 ) f o r ( F o , d o ) , and t h e n choose M 7 Mo and r s u c h

L e t us f i r s t f i x Fo and d o

0 , choose

t h a t f ' ( p ) i s n e g a t i v e i n t h e neighbourhood of p = 1,

a c c o r d i n g t o Theorem 2 . L e t us d i f f e r e n t i a t e ( 8 . 1 0 ) ; we g e t

Let c

O

F,

i . e . when t h e motive t o

a c h i e v e s u c c e s s dominates o v e r t h e f e a r o f f a i l u r e , t h e b r a n c h e s of t h e p a r a b o l s ( 8 . 2 7 ) a r e d i r e c t e d downwards, and f ( p ) w i l l be m o n o t o n i c a l l y i n c r e a s i n g i n t h e i n t e r v a l [O,l] i f and o n l y i f t h e maximum o c c u r s t o t h e r i g h t o f t h e p o i n t p = 1. T h i s means t h a t we must have r + d t M - F 2(M

-

F)

21

(8.29)

554

CHAPT.ER 5

which red-uces t o P

If M

F t l whenever t 5 t ' .

THEOREM.

Let

#

. * >t .

T(a) = t

a h Ft f o r a l l t

Then a C Ft f o r a l l t

- t ] T:

(9.26)

2 tj - t

has t h e p r o p e r t y P[Y

Q

( t )
a J! b !J and b i c i >- ,b j' c j' imply a c 2 % ! c ! ( m o n o t o n i c i t y ) . There i s a n a d d i t i o n a l i i J J axiom o f s o l v a b i l i t y : f o r any a i , b .i E A 1 . there exist c l , d l E A1 s u c h , t h a t a . b . 2 c l d l . T h i s means t h a t e v e r y biai

1 1

o f s t i m u l i on continuum A may be matched b y a i r a t i o o f some s t i m u l i on continuum A 1' The r e m a i n i n g

ratio

CHAPTER 5

618

two axioms a s s e r t t h a t t h e o r d e r i n g 2 r e s t r i c t e d t o a f i x e d continuum Ai s a t i s f i e s t h e s o l v a b i l i t y and Archimedean axiom. The a s s e r t i o n o f t h e theorem i s what was d i s c o v e r e d e m p i r i c a l l y b y S t e v e n s , namely t h a t t h e r e e x i s t s p o s i t i v e f u n c t i o n s u l,..., u m' d e f i n e d on r e s p e c t i v e s e t s A i , such t h a t

These f u n c t i o n s a r e d e t e r m i n e d u n i q u e l y up t o a power u' a r e some o t h e r funct r a n s f o r m a t i o n , i . e . i f ui m t i s n s s a t i s f y i n g t h e above c o n d i t i o n , t h e n t h e r e e x i s t p o s i t i v e c o n s t a n t s d l ,. . . , d m and 8 s u c h t h a t

,...,

u'i =

diui

lr f o r

i = 1,2,

... ,m.

A t t h e end, l e t us mention t h a t d e s p i t e t h e s o l u t i o n

o f m e a s u r e m e n t - t h e o r e t i c a l problems c o n n e c t e d w i t h S t e v e n s ' a p p r o a c h , h i s laws s t i l l c a u s e some c o n t r o v e r s i e s . They a r e r e l a t e d , f i r s t o f a l l , w i t h t h e f a c t t h a t h i s laws were o b t a i n e d by means o f s c a l i n g p r o c e d u r e s , and n o t by c h e c k i n g t h a t t h e axioms o f t h e c o r r e s p o n d i n g s y s t e m s h o l d . The p r o c e d u r e s a r e based on e s t i m a t e s o f r a t i o s f r o b samples o f s u b j e c t s . S i n c e a psychophys i c a l s c a l e i s t o r e f l e c t t h e p e r c e p t i o n of a s i n g l e p e r s o n , it i s n o t c l e a r how t o i n t e r p r e t t h e r e s u l t s o b t a i n e d from a v e r a g i n g t h e answers o f groups o f p e r s o n s . F i n a l l y , more p r e c i s e a n a l y s e s i n d i c a t e t h e e x i s t ence o f some s y s t e m a t i c d e v i a t i o n s i n S t e v e n s ' laws: t h e s c a l e v a l u e s a r e approximated w e l l by t h e laws of t h e form u ( x ) = a ( x - c ) b or u ( x ) = axb t c , b u t n o t b y


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t h e law u ( x ) = a x b . Thus f a r , t h e r e i s no s a t i s f a c t o r y e x p l a n a t i o n f o r t h e s e d e v i a t i o n s ( f o r some h y p o t h e s e s see K r a n t z e t a l . , 1971, p . 5 1 9 ) .

1 0 . 3 . 3 . R a t i n g s c a l e s , and c l a s s i f i c a t i o n i n t o e q u a l l y s p a c e d c a t e g o r i e s . T h i s t y p e o f s c a l i n g t e c h n i q u e s was o r i g i n a t e d b y T i t c h e n e r ( 1 9 0 5 ) , under t h e name of method of s i n g l e s t i m u l i . T h e y a r e a p p l i c a b l e t o s t i m u l i which a r e o r d e r e d on a continuum, and one may e x p e c t t h a t t h e answers w i l l r e f l e c t i n some way t h i s o r d e r i n g . The e s s e n t i a l f e a t u r e i s t h a t t h e s t i m u l i a r e p r e s e n t e d t o t h e s u b j e c t one by one. I n c a s e of r a t i n g s c a l e s , t h e s u b j e c t i s asked t o c l a s s i f y t h e s t i m u l u s i n t o one o f t h e c a t e g o r i e s , d e s c r i b e d i n t e r m s of some a d j e c t i v e s , s u c h as v e r y q u i e t , q u i e t , l o u d , v e r y l o u d , o r p l e a s a n t , n e u t r a l , unpleasant, e t c . In case of the c l a s s i f i c a t i o n i n t o e q u a l l y s p a c e d c a t e g o r i e s , t h e p e r s o n i s asked t o c l a s s i f y t h e s t i m u l u s i n t o c a t e g o r i e s which are d e s c - r i b ed by numbers, s u c h as from -3 t o i-3, from 1 t o 1 0 0 , e t c .

The e m p i r i c a l d a t a have t h e form of f r e q u e n c i e s of c l a s s e s f o r t h e same s t i m u l u s under r e p e t i t i o n . T h e y a l l o w t o estimate t h e p r o b a b i l i t y p ( k \ s ) t h a t s t i m u l u s s w i l l be c l a s s i f i e d t o c a t e g o r y numbered k ( r e g a r d l e s s o f t h e a c t u a l names of c a t e g o r i e s , one may always assume t h a t t h e y a r e numbered from 1 t o m). The s i m p l e s t s c a l i n g t e c h n i q u e , t h e s o - c a l l e d mean c a t e g o r y s c a l e , c o n s i s t s of d e f i n i n g t h e s c a l e v a l u e o f t h e s t i m u l u s s as

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A more c o m p l i c a t e d t e c h n i q u e i s based on a g e n e r a l i z a t -

i o n o f t h e law o f c o m p a r a t i v e judgment o f T h u r s t o n e . Let us c o n s i d e r , f o r e a c h s t i m u l u s s , t h e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n o f i t s c l a s s i f i c a t i o n to v a r i o u s categories, that is P(r,s)

=

f

p(k\s).

k= 1

Here P ( r , s ) i s t h e p r o b a b i l i t y t h a t s t i m u l u s s w i l l be

c l a s s i f i e d t o one of t h e c a t e g o r i e s 1 , 2 , . . . , r . N e x t , one a s s u m e s t h a t w i t h e a c h s t i m u l u s s and w i t h t h e upper l i m i t o f c a t e g o r y r , t h e r e a r e c o n n e c t e d two random v a r i a b l e s , S and T r . The s t i m u l u s s w i l l be

c l a s s i f i e d t o one o f t h e c a t e g o r i e s 1 , 2 , . . . , r Thus , P(P,s)

= P[Tr

- S

>

if S

< Tr.

01.

A s f o r p a i r comparisons,

one assumes t h a t t h e random v a r i a b l e s S and Tr have normal d i s t r i b u t i o n , w i t h means i d e n t i f i e d respectively with the scale value of the s t i m u l u s , and t h e b o r d e r o f t h e c a t e g o r y , and some v a r i a n c e s and c o r r e l a t i o n s . To perform e f f e c t i v e s c a l i n g , i . e . to d e t e r m i n e t h e s c a l e v a l u e s o f s t i m u l i , one needs to make Some a s s u m p t i o n s a b o u t v a r i a n c e s and c o r r e l a t i o n s , a n a l o g o u s as t h e a s s u m p t i o n s i n c a s e V of Thurstone. An i m p o r t a n t s p e c i a l c a s e o f T h u r s t o n e ' s t e c h n i q u e i s t h e s o - c a l l e d method o f s u c c e s s i v e i n t e r v a l s , s u g g e s t e d by Adams and Messick ( 1 9 5 8 ) . One assumes h e r e t h a t t 2, between s u c c e s s i v e c l a s s e s t h e boundaries t

.. .

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are n o t random; t h e s t i m u l u s i s r e p r e s e n t e d , as b e f o r e , by a random v a r i a b l e S ( d i s c r i m i n a n t p r o c e s s ) , w i t h normal d i s t r i b u t i o n . The mean o f t h i s d i s t r i b u t i o n i s t h e s c a l e v a l u e of t h e s t i m u l u s , and t h e v a r i a n c e i s unknown. Thus, t h e p r o b a b i l i t i e s p ( r l s ) a r e e q u a l t o p r o b a b i l i t i e s of o b s e r v i n g a v a l u e S i n t h e i n t e r v a l (tr+J Y that is +

where u(s) and 6S are t h e mean and s t a n d a r d d e v i a t i o n of t h e d i s c r i m i n a n t p r o c e s s S . Adams and Messick (1958) gave n e c e s s a r y and s u f f i c i e n t

c o n d i t i o n s under which t h e r e e x i s t common v a l u e s ( f o r a l l s t i m u l i ) t l , t 2 , . . . o f b o u n d a r i e s of c a t e g o r i e s . These c o n d i t i o n s a r e e x p r e s s e d t h r o u g h p r o b a b i l i t i e s p ( r [ s ) of categories. I n c o n n e c t i o n w i t h s c a l i n g t e c h n i q u e s based on e q u a l l y p l a c e d c a t e g o r i e s and t e c h n i q u e s based on magnitude s c a l i n g one s h o u l d mention a v e r y i n t e r e s i n g phenomenon d i s c o v e r e d when t h e s e t e c h n i q u e s are a p p l i e d t o v a r i o u s continua. S t e v e n s ( 1957) i n t r o d u c e d a p a r t i t i o n of p s y c h o p h y s i c a l c o n t i n u a i n t o p r o t e t i c and m e t a t e t i c . The f i r s t c a t e g o ry comprises c o n t i n u a where t h e c o r r e s p o n d i n g p h y s i c a l continuum a l l o w s measurement on a r a t i o s c a l e ( e , g . l e n g t h , w e i g h t , e t c . ) . The second c a t e g o r y , o f m e t a t e t i c c o n t i n u a , comprises t h o s e where t h e p h y s i c a l c o n t i n u u m ’ a l l o w s measurement o n l y on a n i n t e r v a l s c a l e ( e . g . t e m -

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perature, pitch, etc. ) It a p p e a r s ( s e e S t e v e n s and G a l a n t e r 1957) t h a t when one compares t h e r e s u l t s o f s c a l i n g b y two t e c h n i q u e s

(magnitude s c a l i n g , and c a t e g o r y r a t i n g s ) , t h e y a r e d i f f e r e n t f o r p r o t e t i c and m e t a t e t i c c o n t i n u a . F o r t h e l a t t e r t h e dependence i s l i n e a r , w h i l e f o r p r o t e t i c cont i n u a we have t h e f o l l o w i n g p i c t u r e : i f t h e s c a l e v a l u e s o b t a i n e d by magnitude s c a l i n g a r e r e p r e s e n t e d b y t h e h o r i z o n t a l a x i s , and t h o s e by c a t e g o r y s c a l i n g - b y t h e v e r t i c a l a x i s , t h e r e a p p e a r s a curve w i t h "negative a c c e l e r a t i o n " , as on F i g . 1 0 . 3 .

magnitude r a t i o s c a l e Fig. 10.3. T h i s phenomenon a p p e a r s i n a l l c a s e s of p r o t e t i c con-

t L n u a , and i t s n a t u r e has n o t y e t been e x p l a i n e d adequately. One o f t h e most s e r i o u s problems c o n n e c t e d w i t h c a t e g o r y s c a l i n g i s t h a t o f i n v a r i a n c e o f t h e s c a l e under change of e x p e r i m e n t a l c o n d i t i o n s . It t u r n s o u t t h a t t h e s c a l e depends i n some r e l a t i v e l y low d e g r e e on t h e number of c a t e g o r i e s , b u t i s h i g h l y s e n s i t i v e t o t h e changes i n t h e s e t o f s t i m u l i and o r d e r o f p r e s e n t a t i o n . We


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have h e r e t h e e f f e c t t h a t t h e s c a l e v a l u e s i n c r e a s e more s t e e p l y i n t h e neighbourhood of t h o s e i n t e n s i t i e s which a r e p r e s e n t e d more f r e q u e n t l y . A q u a l i t a t i v e e x p l a n a t i o n , which t a k e s i n t o a c c o u n t t h e

m o t i v a t i o n a l f a c t o r s , i s r e l a t i v e l y s i m p l e : i f t h e maj o r i t y of p r e s e n t e d s t i m u l i come from a fragment of t h e s c a l e c o m p r i s i n g r e l a t i v e l y few c a t e g o r i e s ( s o t h a t n e i g h b o u r i n g c a t e g o r i e s a r e empty, or have o n l y few c l a s s i f i c a t i o n s ) , t h e n t h e n a t u r a l tendency t o p r e c i s e d i s c r i m i n a t i o n w i l l c a u s e t h e usage of t o o many c a t e g o r i e s . T h i s w i l l l e a d t o a s t e e p e r i n c r e a s e of s c a l e val u e s ( h e n c e more e x a c t d i s c r i m i n a t i o n ) i n t h e neighbourhood o f more f r e q u e n t s t i m u l i . T h u s f a r , t h e r e i s no s a t i s f a c t o r y t h e o r y which would t a k e i n t o a c c o u n t b o t h t h e m o t i v a t i o n a l f a c t o r s and t h e e f f e c t s of f r e q u e n c y o f s t i m u l i on s c a l e v a l u e s . The problem l i e s i n a c o n s t r u c t i o n o f a n a d e q u a t e c h o i c e model f o r c l a s s i f i c a t i o n of s t i m u l i , which would l e a d t o s c a l e v a l u e s t h a t a r e i n v a r i a n t under v a r i a t i o n s and m o d i f i c a t i o n s of e x p e r i m e n t a l p r o c e d u r e s . A s d i s t i n c t from measurement t h e o r y , s c a l i n g t h e o r y

l a c k s i n most c a s e s a s a t i s f a c t o r y t h e o r e t i c a l foundat i o n s , which would a c c o u n t a d e q u a t e l y for v a r i o u s phenomena o b s e r v e d i n u s i n g d i f f e r e n t s c a l i n g t e c h n i q u e s . I n a b s t r a c t p r e s e n t a t i o n , t h e main i s s u e s h e r e a r e a s f o l l o w s . It i s assumed t h a t t o e a c h s t i m u l u s i n t e n s i t y t h e r e c o r r e s p o n d s a s c a l e v a l u e , which r e p r e s e n t s t h e p r o p e r t i e s of p e r c e p t i o n of t h i s s t i m u l u s . Each s c a l i n g t e c h n i q u e u s e s t h e d a t a e l i c i t e d from s u b j e c t s ' respons e s , s p e c i f i c for a g i v e n t e c h n i q u e . The problem l i e s ,

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most g e n e r a l l y ,

CHAPTER 5

i n c o n s t r u c t i o n of a model which would

c o n n e c t t h e i n t e n s i t i e s of s t i m u l i , t h e i r v a l u e s on p s y c h o p h y s i c a l s c a l e , and p r o b a b i l i t i e s of d i f f e r e n t responses. I n o t h e r words, one wants a model which would l e a d t o p r o b a b i l i t i e s of r e s p o n s e s for s t i m u l i o f a g i v e n i n t e n s i t y ( i . e . p r o b a b i l i t i e s p ( r j s ) i n cases of category r a t i n g , p r o b a b i l i t i e s p ( a , b ) i n c a s e of p a i r comp a r i s o n s , e t c . ) . These p r o b a b i l i t i e s s h o u l d depend on t h e v a l u e s on p h y c h o p h y s i c a l s c a l e , i n a way p e r m i t t i n g t h e e s t i m a t i o n of t h e s e v a l u e s on t h e b a s i s of e s t i m a t es of p r o b a b i l i t i e s . Moreover, a s a t i s f a c t o r y model should be u n i v e r s a l , i n t h e sense t h a t i t should lead t o t h e same p s y c h o p h y s i c a l s c a l e , r e g a r d l e s s o f t h e s c a l i n g t e c h n i q u e used. Some a t t e m p t s i n t h i s d i r e c t i o n a r e due t o Luce (1959), who i n t r o d u c e d v a r i o u s c h o i c e models. I n p a r t i c u l a r , f o r p s y c h o p h y s i c a l phenomena ( n o t o n l y s c a l i n g , b u t a l s o d e t e c t i o n and d i s c r i m i n a t i o n ) he i n t r o d u c e d t h e concept o f g e n e r a l i z a t i o n f u n c t i o n , which e x p r e s s e s t h e p r o b a b i l i t y t h a t s t i m u l u s s w i l l appear " l i k e s t i mulus t " . The second n o t i o n o f Luce i s t h a t of b i a s f u n c t i o n , d e f i n e d on t h e s e t of r e s p o n s e s , d e s c r i b i n g t h e s y s t e m a t i c d i s t o r t i o n s o f r e s p o n s e s . Thus, Luce assumes two p s y c h o l o g i c a l mechanisms which d e t e r m i n e t h e response: w i t h a c e r t a i n p r o b a b i l i t y , stimulus s i s p e r c e i v e d as s t i m u l u s t ( " g e n e r a l i z e d t o s t i m u l u s t " ) , w i t h v a l u e u ( t ) on t h e a p p r o p r i a t e p s y c h o p h y s i c a l s c a l e . T h i s p e r c e p t i o n , i n t u r n , l e a d s t o r e s p o n s e r, d i s t o r t e d by bias f u n c t i o n b . Under c e r t a i n a s s u m p t i o n s one can d e r i v e t h e o r e t i c a l


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f o r m u l a e for p r o b a b i l i t y of r e s p o n s e r , g i v e n t h e s t i mulus s . However, w i t h few e x c e p t i o n s , t h e s e f o r m u l a e do n o t a l l o w e s t i m a t i o n o f p s y c h o p h y s i c a l f u n c t i o n . Moreover, L u c e ' s model d o e s n o t t a k e i n t o a c c o u n t s u c h f a c t o r s as i n f l u e n c e of i n s t r u c t i o n s , o r d e r and f r e q u ency of p r e s e n t a t i o n s , e t c . , s o t h a t t h e q u e s t i o n o f a n a d e q u a t e model i s s t i l l open. It i s n o t even c l e a r what i s t h e n e c e s s a r y number o f p s y c h o l o g i c a l mechanisms i n v o l v e d ( i . e . if t h e two mechanisms of Luce, o f gener a l i z a t i o n and b i a s , are s u f f i c i e n t ) .

1 0 . 4 . P e r c e p t i o n o f s c a l e o f s o c i a l power

Most t y p i c a l examples o f s c a l i n g concern t h e c o n t i n u a where t h e s t i m u l u s i n t e n s i t y i s d e t e r m i n e d b y i t s phys i c a l p r o p e r t i e s , and t h e s c a l e v a l u e s r e f l e c t p e r c e p t i o n o f t h e s e p r o p e r t i e s . The s i t u a t i o n becomes more c o m p l i c a t e d , b o t h t h e o r e t i c a l l y and e m p i r i c a l l y , when t h e u n d e r l y i n g continuum has a more " e l u s i v e " c h a r a c t e r I n t h i s s e c t i o n , we s h a l l p r e s e n t a n example o f s u c h s i t u a t i o n , when t h e p e r c e p t i o n c o n c e r n s t h e s c a l e o f s o c i a l power, and i n p a r t i c u l a r , t h e c o n d i t i o n s which tend t o d i s t o r t t h i s perception. O b v i o u s l y , t h e s t u d y o f d i s t o r t i o n s r e q u i r e s t h e knowl e d g e o f " t r u e " powers, t o be compared w i t h t h e a c t u a l c h o i c e s by t h e p e r c e i v i n g s u b j e c t s . T h i s r e q u i r e m e n t f o r c e d t h e r e s t r i c t i o n of c o n s i d e r a t i o n s t o some i d e a l i z e d s i t u a t i o n s , namely t o v o t i n g committees d e s c r i b e d i n terms of t h e i r c o n s t i t u t i o n s , where t h e power i n d e x of e v e r y member c o u l d be computed a p r i o r i . These i n d i -

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ces p r o v i d e t h e r e f e r e n c e base i n t h e experiments. The r e s u l t s i n d i c a t e t h a t p e r c e p t i o n of c e r t a i n power yielding privileges, e.g. the r i g h t t o veto, tend t o be s e r i o u s l y d i s t o r t e d .

1 0 . 4 . 1 . The c o n c e p t o f power. The concept o f power i s one of t h e c e n t r a l n o t i o n s i n t h e s o c i a l s c i e n c e s . A t t h e same t i m e , it a p p e a r s t o b e one o f t h e more e l u s i v e c o n c e p t s , and i n most c a s e s , d e f i e s t h e a t t e m p t s o f g i v ing it a precise d e f i n i t i o n . T h i s s e c t i o n w i l l p r o v i d e a b r i e f r e v i e w of some o f t h e

d e f i n i t i o n s of s o c i a l power, and i n d i c a t e t h e r e a s o n s why t h e r e seems t o be o n l y one s u g g e s t e d t h u s f a r , which i s a p p l i c a b l e f o r t h e p u r p o s e of t h e s t u d y o f d i s t o r t i o n s o f p e r c e p t i o n of s o c i a l power, namely t h e v o t i n g power i n d e x o f Shapley and Shubik (1954). Before p r e s e n t i n g t h i s i n d e x , it i s worth while t o o u t l i n e some o t h e r a t t e m p t s t o d e f i n e power. The non-mathematical a p p r o a c h e s t o power y i e l d n o t h i n g which c o u l d be u s e f u l for t h e p r e s e n t p u r p o s e , i n t h e s e n s e t h a t t h e n u m e r i c a l i n d e x may be e l i c i t e d from them, n o r can one p r e c i s e l y d e t e r m i n e who i s more p o w e r f u l i n a given s i t u a t i o n . To i l l u s t r a t e t h e d i f f i c u l t i e s e n c o u n t e r e d i n mathematic a l d e f i n i t i o n s , l e t us c o n s i d e r two p e r s o n s , A and B , who may be " f o r " o r " a g a i n s t " a c e r t a i n i s s u e . The f i n a l d e c i s i o n depends a l s o on some o t h e r f a c t e o r s , i . e . o t h e r members of t h e group, o r some random e f f e c t s may i n f l u e n c e t h e outcome, e t c .


627

The s i t u a t i o n may be r e p r e s e n t e d i n t h e form o f t h e following matrix. P r o b a b i l i t y of a p o s i t i v e outcome:

for

against

p2 1

p22

I n t h i s c a s e , a l l d e f i n i t i o n s ( s e e Wittman 1 9 7 7 f o r a r e v i e w of t h e l i t e r a t u r e on t h e s u b j e c t ) a g r e e : A i s more p o w e r f u l t h a n B y i f P12

>

p21.

T h i s , o f c o u r s e , i s a n a t u r a l and e a s i l y d e f e n s i b l e de-

f i n i t i o n : A i s more p o w e r f u l t h a n B y i f t h e p r o b a b i l i t y of p o s i t i v e outcome i s g r e a t e r if A i s for and B i s a g a i n s t i t , t h a n if A i s a g a i n s t and B i s f o r . I n more g e n e r a l s i t u a t i o n s , t h e m a t r i x a l l o w s for more a c t i o n s of b o t h A and B y and t h e d e f i n i t i o n s a r e based on s u c h p r o p e r t i e s as t h e a b i l i t y of one p e r s o n t o res t r i c t t h e r a n g e of p o s s i b i l i t i e s o f t h e o t h e r , r e s t r i c t h i s outcomes, o r on a b i l i t y t o harm. F o r t h e r e v i e w o f l i t e r a t u r e s e e Wittman 1977; f o r o t h e r s o u r c e m a t e r i a l , see f o r i n s t a n c e Dahl 1957, H a r s a n y i 1956, March 1957, or Nagel 1975. A l l t h e s e d e f i n i t i o n s a r e d e f e n s i b l e , b u t for p r a c t i c a l

a p p l i c a t i o n s t h e c r u c i a l p o i n t i s t o de’;ermine t h e res-

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p e c t i v e p r o b a b i l i t i e s P i j . I n d e e d , even i n t h e s i m p l e s t c a s e above, i f A and B a r e members o f a l a r g e r d e c i s i o n making body, i t i s n o t c l e a r how t o d e t e r m i n e P ij

'

Thus, for t h e p u r p o s e o f t h e p r e s e n t s t u d y , where t h e c r u c i a l p o i n t was t o a s s e s s t h e a p r i o r i powers, which c o u l d be compared w i t h t h e c h o i c e s o f t h e s u b j e c t , t h e c o n s i d e r a t i o n s had t o be r e s t r i c t e d t o v o t i n g committees.

V o t i n g power. Assume t h a t we have a v o t i n g bod y , e . g . a committee, board o f d i r e c t o r s , e t c . We a s s u 10.4.2.

me t h a t e a c h member may v o t e " y e s " o r "no" , t h a t i s , he may be " f o r ' ' o r " a g a i n s t " some i s s u e , and t h a t t h e r e e x i s t s a w e l l d e f i n e d r u l e which s p e c i f i e s how t h e v o t es o f d i f f e r e n t members a r e a g g r e g a t e d i n t o t h e f i n a l decision. The s i m p l e s t example o f s u c h a r u l e i s t h e m a j o r i t y

r u l e , or 2 / 3 m a j o r i t y r u l e , when e a c h member has j u s t one v o t e , and t h e i s s u e p a s s e s when t h e r e i s a t l e a s t

a c e r t a i n number of v o t e s " y e s " . I n s u c h a c a s e , o f c o u r s e , a l l members have t h e same a p r i o r i power because o f symmetry. N a t u r a l l y , i n a c t u a l s i t u a t i o n s , v a r i o u s members o f s u c h committees may have d i f f e r e n t r e a l powers, depending on who t h e y a r e , whom t h e y know, how p e r s u a s i v e t h e y may b e , e t c .

These a s p e c t s , however, are n o t e a s i l y r e p r e s e n t a b l e n u m e r i c a l l y , and t h e r e f o r e t h e c o n s i d e r a t i o n s had t o b e r e s t r i c t e d t o t h e a p r i o r i power connected w i t h t h e p o s i t i o n w i t h i n a committee, r a t h e r t h a n w i t h t h e a c t -

SELECZ'ED TOPICS IN MEASUREMENT THEORY

629

u a l p e r s o n who o c c u p i e s t h i s p o s i t i o n . The c a s e when t h e r e i s a s y m m e t r y i s of l i t t l e i n t e r e s t . However, t h e r e a r e c a s e s of i n t e r e s t when t h e r e i s no symmetry, and t h e a p r i o r i v o t i n g powers o f v a r i o u s members a r e d i f f e r e n t . I n p a r t i c u l a r , t h e f o l l o w i n g

f e a t u r e s a r e common i n p r a c t i c e , and t h e r e f o r e w o r t h studying:

-- v a r i o u s members may have d i f f e r e n t numbers o f v o t e s ;

---

some members may have v e t o power; v o t i n g may b e i n d i r e c t : for i n s t a n c e , t h e r e may b e s e v e r a l subcommittees, and t h e f i n a l d e c i s i o n depends on t h e r e s u l t s of v o t i n g s i n t h e subcommittees; -- a p e r s o n may b e l o n g t o s e v e r a l subcommittees, and t h e r e f o r e may i n e f f e c t have s e v e r a l i n d i r e c t v o t e s . The q u e s t i o n i s t o d e v i s e a measure which would adequat e l y r e p r e s e n t t h e v o t i n g power of d i f f e r e n t members of t h e committee, as g u a r a n t e e d t o them by t h e c o n s t i t u t i o n of it. T h i s q u e s t i o n was s o l v e d by Shapley and Shubik

(1954).

The i n d e x of s o c i a l power d e v i s e d by them i s based on t h e concept of minimal winning c o a l i t i o n , i . e . minimal group of p e r s o n s who may e n s u r e t h e f i n a l d e c i s i o n " y e s " r e g a r d l e s s of t h e v o t e s of t h e r e m a i n i n g members. We p r e s e n t t h e d e f i n i t i o n o f Shapley and Shubik i n t h e

form which a t t h e same t i m e p r o v i d e s a method o f numeri c a l d e t e r m i n a t i o n o f v o t i n g power ( s e e , a l s o Kemeny, S n e l l and Thompson 1957, o r Luce and R a i f f a 1957). C o n s i d e r a f i x e d p e r m u t a t i o n o f v o t e r s , and imagine t h a t t h e members v o t e ' ' y e s " one a f t e r a n o t h e r , accord-

CHAPTER 5

630

i n g t o t h i s p e r m u t a t i o n . Then t h e r e w i l l b e a moment when t h e i s s u e w i l l p a s s , even b e f o r e a l l t h e v o t e s a r e c a s t . The l a s t member, whose v o t e was d e c i s i v e i s c a l l ed t h e p i v o t i n t h i s p e r m u t a t i o n . O b v i o u s l y , t h e p i v o t p l u s a l l members who p r e c e d e him i n t h e g i v e n permutat i o n form a minimal winning c o a l i t i o n . Thus, e a c h p e r m u t a t i o n has e x a c t l y one p i v o t , and t h e power of a v o t i n g member i s d e f i n e d a s t h e number o f p e r m u t a t i o n s i n which he a p p e a r s as p i v o t . For normal-i z a t i o n , one d i v i d e s t h e t o t a l number o f p e r m u t a t i o n s i n which x i s a p i v o t by n! ( t h e t o t a l number o f permut a t i o n s of n members o f a c o m m i t t e e ) . Thus, i f d ( x ) i s t h e v o t i n g power i n d e x o f x , t h e n

# of p e r m u t a t i o n s i n which x i s a p i v o t

d(x) =

-

n! To u s e a s i m p l e example, c o n s i d e r t h e c a s e o f a committ e e w i t h f o u r members, A , B , C and D . Assume t h a t A has 2 v o t e s , and t h a t o t h e r members have one v o t e e a c h , s o t h a t t h e r e a r e a l t o g e t h e r 5 v o t e s . The i s s u e i s dec i d e d by a s i m p l e m a j o r i t y o f v o t e s - a t l e a s t t h r e e v o t e s "yes" , b u t w i t h t h e a d d i t i o n a l p r o v i s i o n t h a t B has v e t o power: h i s v o t e must be " y e s " , or o t h e r w i s e t h e i s s u e does n o t p a s s . The l i s t of all 2 4 p e r m u t a t i o n s , w i t h p i v o t a l members u n d e r l i n e d , i s g i v e n i n t h e t a b l e below.

For example, i f t h e p e r m u t a t i o n i s A B C D , t h e p i v o t i s B : a f t e r A g i v e s two v o t e s , and B g i v e s h i s one v o t e , t h e i s s u e pasSes, s i n c e t h e r e a r e a l r e a d y t h r e e v o t e s


63 1

" y e s " and v o t e o f B among them. S i m i l a r l y , i n t h e p e r m u t a t i o n B C D A , t h e p i v o t i s D -- he c a s t s t h e t h i r d v o t e , and B h a s a l r e a d y v o t e d l t y e s l l . T a b l e of p e r m u t a t i o n s w i t h p i v o t a l members A B C D

B A C D

C A B D

D A B C

A B D C

B A D C

C A D B

D A C E

A C E D

B C A D

C B A D

D B A C

A C D B

B C D A

C B D A

D B C A

A D B C

B D A C

C D A B

D C A B

A D C B

B D C A

C D B A

D C B A

A l t o g e t h e r , A a p p e a r s 6 t i m e s as a p i v o t , B - 1 4 t i m e s , and C and D e a c h 2 t i m e s . The powers a r e t h e r e f o r e d ( A ) = 6 / 2 4 = 1/4,

d ( B ) = 1 4 / 2 4 = 7/12

d ( C ) = d ( D ) = 2 / 2 4 = 1/12.

A s may be s e e n , d e s p i t e h a v i n g o n l y t w i c e as many v o t e s

as C o r D , member A has t h r e e times as much power. A l s o , t h e r i g h t o f v e t o g i v e s v e r y h i g h power: B h a s o n l y one v o t e , s i m i l a r l y a s C and D , b u t s e v e n t i m e s as much power.

1 0 . 4 . 3 . The h y p o t h e s e s . The e x p e r i m e n t s i n v o l v e d subj e c t i v e e v a l u a t i o n s o f s o c i a l power o f d i f f e r e n t memb e r s o f h y p o t h e t i c a l committees p r e s e n t e d t o t h e subj e c t . Each committee was d e s c r i b e d by s p e c i f y i n g i t s c o n s t i t u t i o n , i . e . by c h a r a c t e r i z i n g how t h e v o t e s o f

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d i f f e r e n t members i n t e r v e n e i n t h e f i n a l d e c i s i o n , To f a c i l i t a t e g r a s p i n g t h e power s t r u c t u r e o f a committee, i t was a l s o p r e s e n t e d i n form o f a diagramme ( s e e t h e following s e c t i o n ) . The s u b j e c t was asked t o a s s e s s which o f t h e two spec i f i e d members of a committee has more power, and a l s o g i v e h i s numerical e v a l u a t i o n of t h e r a t i o of t h e i r powers. I n t h e second s t a g e o f t h e e x p e r i m e n t , t h e committees were p r e s e n t e d i n p a i r s , and t h e s u b j e c t was a s k e d t o choose t h a t committee, which he f e l t more l i k e l y t o v o t e i n h i s f a v o u r . The c o n t r o l l e d f a c t o r i n t h i s s t a g e o f experiment was t h e i n f o r m a t i o n about t h e v o t e s o f some ( b u t n o t a l l ) members o f t h e commitee. Thus, i n e f f e c t , t h e c h o i c e was made n o t s o much between commit t e e s , as between p a i r s c o n s i s t i n g o f a commitee and s p e c i f i c i n f o r m a t i o n ( p o s s i b l y none) a b o u t t h e v o t e s o f some o f i t s members. To f o r m u l a t e t h e b a s i c h y p o t h e s e s , suppose t h a t B are two members o f t h e committee which are t o a s s e s s e d , and l e t d(A) and d ( B ) be t h e i r s o c i a l F u r t h e r , l e t p(A,B) be t h e p r o b a b i l i t y t h a t t h e

A and be

powers. subject

w i l l p o i n t o u t A as more p o w e r f u l i n t h e p a i r ( A , B ) , and l e t r ( A , B ) be t h e s u b j e c t i v e l y e v a l u a t e d r a t i o d ( A ) / d ( B ) o f powers o f A and B .

HYPOTHESIS 1. P e r c e p t i o n o f s o c i a l power i s o r d e r - c o r r e c t , i n t h e s e n s e t h a t p(A,B) 7 3 i f and o n l y i f we have d(A) '7 d ( B ) .

-

HYPOTHESIS 2 . T h e r e e x i s t s a t h r e s h o l d i n p e r c e p t i o n


633

of s o c i a l power, i n t h e s e n s e t h a t p(A,B) i s s i g n i f i c a n t l y greater t h a n $ only i f t h e d i f f e r e n c e d(A) - d ( B ) exceeds t h e t h r e s h o l d . HYPOTHESIS 3. I n g e n e r a l , p ( A , B ) t e n d s to be l a r g e r for l a r g e r v a l u e s o f t h e d i f f e r e n c e d(A) - d ( B ) , but the r e l a t i o n i s n o t f u n c t i o n a l : p(A,B) depends n o t o n l y on d(A) - d ( B ) , b u t a l s o on t h e complexity o f t h e committee. HYPOTHESIS 4. P e r c e p t i o n of s o c i a l power i s b i a s e d , i n t h e s e n s e t h a t t h e e x p e c t a t i o n of r ( A , B ) t o the r a t i o d(A)/d(B).

i s not equal

HYPOTHESIS 5 . The b i a s t e n d s t o be l a r g e r f o r some t y p e s o f s o c i a l r o l e s ; i n p a r t i c u l a r , power due t o t h e r i g h t o f v e t o , and t h e power due t o t h e r i g h t o f c a s t i n g more t h a n one v o t e , t e n d t o be u n d e r e s t i m a t e d . T o f o r m u l a t e f u r t h e r h y p o t h e s e s , l e t Q ( C , x , J ) be t h e

p r o b a b i l i t y t h a t t h e committee C w i l l d e c i d e i n f a v o u r o f some i s s u e , g i v e n t h a t t h e members s p e c i f i e d i n J w i l l v o t e as s p e c i f i e d , and t h a t t h e r e m a i n i n g members w i l l v o t e randomly and i n d e p e n d e n t l y , w i t h p r o b a b i l i t y x o f v o t e Ityes". To u s e a s i m p l e example, l e t C s t a n d for t h e committee d e s c r i b e d i n t h e preceding s e c t i o n ( c o n s i s t i n g of f o u r members A , B , C and D , o f which A has two v o t e s , B has v e t o power, and t h e i s s u e i s d e c i d e d by s i m p l e m a j o r i t y ) . I f J s t a n d s f o r t h e i n f o r m a t i o n "A w i l l v o t e a g a i n s t t h e i s s u e " , t h e n Q ( C , x , J ) = x3, s i n c e t h e i s s u e may p a s s o n l y i f B , C and D w i l l a l l v o t e " y e s " ( a s t h e r e a r e a l r e a d y two v o t e s a g a i n s t ) . S i m i l a r l y , i f J

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stands f o r t h e information t h a t B w i l l vote "yes", then Q(C,x,J) = x + ( 1 - x ) x 2 I n d e e d , i f B v o t e s " y e s " , t h e i s s u e w i l l pass i f e i t h e r A ( w i t h 2 v o t e s ) w i l l v o t e l r f o r l r ( p r o b a b i l i t y x ) , i n which c a s e C and D may v o t e i n an a r b i t r a r y way, or e l s e , i f A v o t e s no ( p r o b a b i l i t y 1-x), and b o t h C and D v o t e " y e s " ( p r o b a b i l i t y x 2 ) .

.

L e t now ( C , J ) and ( C 1 , J 1 ) be two committees w i t h g i v e n i n f o r m a t i o n about t h e v o t e s . Denote by p ( C , J ; C ' , J ! ) t h e p r o b a b i l i t y t h a t t h e s u b j e c t w i l l p o i n t o u t ( C , J ) as more f a v o u r a b l e committee ( t h e one he f e e l s would be more l i k e l y t o v o t e i n h i s f a v o u r ) . HYPOTHESIS 6 . I n a s s e s s i n g committees a c c o r d i n g t o t h e i r " f a v o u r a b l e n e s s " , t h e s u b j e c t s t e n d t o e v a l u a t e them a s i f assuming t a c i t l y t h a t t h e members o f t h e committee a r e u n p r e j u d i c e d and do n o t form c l i q u e s : p ( C , J ; C 1 , J 1 ) > 4 i f and o n l y i f Q ( C , ; , J ) Q(Ct,$,J1), and i n general, p ( C , J ; C 1 , J 1 ) i s an i n c r e a s i n g f u n c t i o n of t h e d i fference Q(C,$,J) - Q(C1,$,J1).

>

The c o n s i s t e n c y o f t h e c h o i c e s w i t h t h e o r d e r i n d u c e d by Q ( C , x , J ) c o r r e s p o n d s t o t h e e x p e c t a t i o n o f t h e l a c k of c l i q u e s , s i n c e Q ( C , x , J ) i s c a l c u l a t e d under t h e a s s u m p t i o n o f independence o f v o t e s o f members o f t h e committee. The c o n s i s t e n c y w i t h Q ( C , x , J ) f o r t h e p a r t i c u l a r value x = $ corresponds t o t h e expectation t h a t members o f t h e committee w i l l be u n p r e j u d i c e d ( w i l l b e e q u a l l y l i k e l y v o t i n g " f o r " as " a g a i n s t " ) . HYPOTHESIS 7 . The c h o i c e p r o b a b i l i t i e s p ( C , J ; C ' , J 1 ) a r e c o n s i s t e n t i n t h e s e n s e o f weak s t o c h a s t i c t r a n s i t i v i t y ( s e e for i n s t a n c e Coombs, Dawes and Tversky

1970).


635

HYPOTHESIS 8 . The c h o i c e p r o b a b i l i t i e s s a t i s f y t h e cond i t i o n of s t r o n g s t o c h a s t i c t r a n s i t i v i t y , i f r e s t r i c t e d t o c h o i c e s from t r i p l e t s ( C , J ) , ( C l , J ' ) , ( C ' ' , J ' ' ) with c = C ' = C " , i . e . i f e v a l u a t i o n s concern t h e same commit e e under d i f f e r e n t i n f o r m a t i o n . HYPOTHESIS 9. The p e r c e p t i o n o f s o c i a l power i s b i a s e d i n the sense t h a t there exist t r i p l e t s ( C , J ) , ( C 1 , J 1 ) , -( C ' ' , J ' ' ) of s u f f i c i e n t l y complex s t r u c t u r e , s u c h t h a t t h e c h o i c e p r o b a b i l i t i e s do n o t s a t i s f y s t r o n g s t o c h a s t i c transitivity. The e x p e r i m e n t s r e p o r t e d below c o n c e r n o n l y t h e hypot h e s e s 1-5; t h e r e m a i n i n g h y p o t h e s s e s 6-9 , as w e l l as t h e H y p o t h e s i s 1 0 below, r e q u i r e d more e l a b o r a t e d experiments. The n a t u r a l c o n t i n u a t i o n o f t h e r e s e a r c h on h y p o t h e s e s

1-9 i s t o s t u d y t h e p e r c e p t i o n o f "composite" s o c i a l power, due t o membership i n more t h a n one decision-maki n g body. The e x p e r i m e n t s here c o n c e r n c h o i c e s made b y g r o u p s o f s u b j e c t s f a c i n g t h e t a s k o f c h o o s i n g one o f s e v e r a l members of h y p o t h e t i c a l v o t i n g committees. Each commit t e e makes t h e " v o t i n g " , where t h e member s e l e c t e d b y t h e group ( " b r i b e d " ) v o t e s ' ' y e s " , and t h e r e m a i n i n g v o t e s a r e d e t e r m i n e d b y some chance mechanism. I f t h e v o t e comes o u t f a v o u r a b l y f o r t h e s u b j e c t s , t h e y r e c e i v e some p r i z e . The chance mechanism h e r e i s a c o n t r o l v a r i a b l e : i n som e e x p e r i m e n t s i t i s simply t o s s i n g a c o i n f o r e a c h member s e p a r a t e l y , c o r r e s p o n d i n g t o e x p e c t a t i o n of l a c k o f p r e j u d i c e and l a c k of c l i q u e s . I n o t h e r v e r s i o n s o f

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e x p e r i m e n t , t h e i n f o r m a t i o n a b o u t t h e c l i q u e s and t h e p r o b a b i l i t i e s a r e fuzzy. The c r u c i a l p o i n t i s t h a t t h e committees a r e n o t d i s -

j o i n t , w h i l e t h e c h o i c e o f t h e member who v o t e s ''YeS" for t h e group i s t o be made for a l l committees a t o n c e . The s u b j e c t s f a c e t h e r e f o r e a dilemma, s i n c e a c a n d i d a t e may p l a y more i m p o r t a n t r o l e i n some committees, and a l e s s i m p o r t a n t r o l e i n o t h e r committees. The h y p o t h e s i s t e s t e d a s s e r t s t h a t s o c i a l power, a s

measured by t h e i n d e x of Shapley and Shubik, i s e a s i e r t o p e r c e i v e t h a n p r o b a b i l i t i e s o f complex e v e n t s , and t h a t t h e s o c i a l power r e s u l t i n g from membership i n s e v e r a l v o t i n g committees i s p e r c e i v e d a d d i t i v e l y ( i f t h e i s s u e s t o be d e c i d e d by a l l committees a r e of t h e same i m p o r t a n c e ) . To f o r m u l a t e t h e h y p o t h e s i s p r e c i s e l y , l e t C 1' C2>". s t a n d for t h e committees. Suppose t h a t t h e c h o i c e i s made between p e r s o n s A and B , e a c h of them belongand i n g t o a l l t h e committees.. L e t d l ( A ) , d 2 ( A ) , dl(B), d2(B), be t h e s o c i a l powers of A and B i n t h e committees C1, C2, , and l e t p l ( A ) , p 2 ( A ) , and be t h e p r o b a b i l i t i e s of v o t e " y e s " b y pl(B), p2(B), committees C 1, C2, under t h e scheme o f d e t e r m i n i n g t h e v o t e s used i n p a r t i c u l a r v e r s i o n o f t h e e x p e r i m e n t . Thus, i f J = "member A w i l l v o t e y e s " , and t h e chance mechanism b e i n g i n d e p e n d e n t c o i n t o s s i n g , we have pi(A) = Q ( C i , 2, J).

... ...

...

...

... ...

...

Finally, l e t D(A) = dl(A) t d2(A) t and D(B) = d l ( B ) t d2(B) t be t h e sums o f s o c i a l powers o f A and B i n a l l cominittees. L e t P ( A ) = pl(A) t p 2 ( A ) +

...

...


and P ( B ) = p l ( B ) t p 2 ( B ) t

...

631

b e t h e sums o f p r o b a b i l i -

t i e s o f p o s i t i v e d e c i s i o n s . Thus, P ( A ) and P ( B ) a r e prop o r t i o n a l t o t h e expected t o t a l p r i z e s f o r a l l v o t i n g s , i f e a c h committee v o t e s f o r t h e same p r i z e , and i f A ( r e s p . B ) i s chosen b y t h e group. HYPOTHESIS 1 0 . I n c h o o s i n g between members A and B y t h e d e c i s i v e q u a n t i t i e s are t h e i r s o c i a l powers D ( A ) and D ( B ) , and n o t t h e e x p e c t e d p r i z e s P ( A ) and P ( B ) . As s i b l e e x c e p t i o n o c c u r s i n t h e c a s e when (1) t h e probab i l i t i e s o f v o t e s " y e s ' ' are g i v e n e x p l i c i t l y , ( 2 ) memb e r s do n o t form complex c l i q u e s , and ( 3 ) a l l committees are o f a v e r y s i m p l e s t r u c t u r e . T h i s h y p o t h e s i s may be t e s t e d by o b s e r v i n g c h o i c e s b e t -

ween A and B i n s i t u a t i o n s when D ( A ) 7 D ( B ) and P ( A ) < P ( B ) or v i c e v e r s a . The s o c i a l power i s s t r o n g l y c o r r e l a t e d w i t h t h e probab i l i t y o f p o s i t i v e f i n a l d e c i s i o n ( g i v e n t h a t t h e member i n question votes "yes"). Therefore t h e sets of committees for which t h e sums of powers D ( A ) and D ( B ) and e x p e c t e d p r i z e s P ( A ) and P ( B ) s a t i s f y o p p o s i t e i n e q u a l i t i e s , a r e n o t e a s y t o c o n s t r u c t . However, some s u c h p a i r s have been f o u n d , and may s e r v e as a b a s i s f o r t e s t i n g Hypothesis 1 0 .

The e x p e r i m e n t a l m a t e r i a l . I n t h i s s e c t i o n w e s h a l l p r e s e n t t h e s i x committees which were used i n t h e experiments 10.4.4.

.

Committee 1. The d e s c r i p t i o n of t h e v o t i n g r u l e i s here as f o l l o w s :

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1. The committee c o n s i s t s of 6 members, among them A and B; 2 . A i s t h e chairman;

3. B has t h e r i g h t of c a s t i n g 3 v o t e s , w h i l e t h e r e m a i n i n g members, i n c l u d i n g A , c a s t s one v o t e e a c h ;

4. The d e c i s i o n

i s t a k e n by s i m p l e m a j o r i t y ( t h e r e

are 8 votes altogether), with the provision that i n the c a s e of s p l i t y 4-4 t h e c h a i r m a n ' s v o t e i s d e c i s i v e . The scheme of t h e committee i s as f o l l o w s .

t h e v o t e of A i s decisive

'Ti ;i

,I ,! :

A

B

j

x

x

x

x

Scheme o f committee 1

Thus, i n committee 1, t h e power o f A l i e s i n h i s r i g h t t o b r e a k t i e s , w h i l e t h e power of B l i e s i n h i s r i g h t t o cast three votes. The s u b j e c t s were asked whether A o r B has more power, and g i v e t h e i r a s s e s s m e n t o f t h e powers o f A and B , i f one assumes t h a t t h e power o f e a c h o f t h e r e m a i n i n g members (marked x ) i s t a k e n as 1.


639

I t i s r e l a t i v e l y easy t o e v a l u a t e t h e powers o f members

of t h i s committee. Member A a p p e a r s as a p i v o t i n 1 4 4 p e r m u t a t i o n s , B - i n 288 p e r m u t a t i o n s , and e a c h o f t h e r e m a i n i n g members - i n 7 2 p e r m u t a t i o n s . S i n c e t h e r e a r e

6!

=

720 p e r m u t a t i o n s , t h e powers a r e

T h u s , B has t w i c e as much power as A , w h i l e t h e powers o f A and B e x p r e s s e d i n u n i t s e q u a l d ( x ) a r e r e s p e c t i -

v e l y 2 and 4 . L e t now P A ( z ) be t h e p r o b a b i l i t y of p o s i t i v e d e c i s i o n

o f Committee 1, i f member A v o t e s " y e s " , and t h e remaini n g members v o t e ''yes" i n d e p e n d e n t l y w i t h p r o b a b i l i t y z . We have t h e n P,(r)

=

z

t

(l-2)[z4

4-

423 ( 1 - z ) l .

I n d e e d , i f A v o t e s " y e s " , t h e r e are two p o s s i b i l i t i e s : e i t h e r B v o t e s a l s o " y e s " ( p r o b a b i l i t y z ) , i n which c a s e t h e i s s u e p a s s e s , as t h e r e are a l r e a d y f o u r v o t e s " y e s " , and t h e chairman A h a s t h e r i g h t t o b r e a k a t i e , or B v o t e s "no" ( p r o b a b i l i t y 1 - z ) . I n t h e l a t t e r c a s e , one needs e i t h e r t h r e e o r f o u r v o t e s o f t h e r e m a i n i n g members. P r o b a b i l i t y o f a l l f o u r v o t e s b e i n g " y e s " i s z 4 , w h i l e p r o b a b i l i t y o f t h r e e v o t e s "yes" and one v o t e "no" i s 42 3 ( 1 - z ) . T h i s p r o v e s t h e formula f o r

S i m i l a r l y , we have

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The argument i s s i m i l a r as b e f o r e . I f B v o t e s r l y e s r l , and s o d o e s A ( p r o b a b i l i t y z) t h e i s s u e p a s s e s , which g i v e s t h e f i r s t t e r m . I f A v o t e s "no" ( p r o b a b i l i t y

1-z), t h e n a t l e a s t two o f t h e r e m a i n i n g members must v o t e ''yes'' f o r t h e i s s u e t o p a s s . The t h r e e t e r m s i n t h e square bracket give respectively t h e probability o f 4 , 3 and 2 v o t e s " y e s " . We s e e t h a t i n t h i s c a s e we have a l w a y s P,(z)

< P,(z),

which a g r e e s w i t h t h e f a c t t h a t B h a s more power t h a n A : t h e p r o b a b i l i t y o f a p o s i t i v e d e c i s i o n i s always h i g h e r when B v o t e s ' r f o r t r t h a n when A v o t e s " f o r " . Committee 2 The d e s c r i p t i o n o f t h e v o t i n g r u l e i s as f o l l o w s . 1. The committee c o n s i s t s of 6 members, among them C and D ; 2 . C i s t h e chairman;

3 . The r e m a i n i n g f i v e members form two s u b c o m i t t e e s o f t h r e e p e r s o n s e a c h ; D b e l o n g s t o b o t h subcommittees;

4 . I n e a c h subcommittee, t h e d e c i s i o n r e q u i r e s simpl e majority ( a t least 2 out of 3 v o t e s ) ;

5 . The f i n a l d e c i s i o n r e q u i r e s t h e m a j o r i t y o f t h r e e v o t e s : t h a t of t h e chairman, and t h o s e o f t h e subcommittees. The scheme o f t h i s committee i s p r e s e n t e d below. Here t h e power o f C i s due t o t h e f a c t t h a t h i s v o t e e n t e r s d i r e c t l y i n t o t h e f i n a l d e c i s i o n , w h i l e t h e power of D l i e s i n t h e f a c t t h a t he b e l o n g s t o two sub-


641

c o m m i t t e e s , hence he h a s i n e f f e c t two i n d i r e c t v o t e s .

I

I

I

decides

X

1

majority

C

I

D

X

dec:

X

X

Scheme o f Committee 2

I n t h i s c a s e , t h e d e c i s i o n whether C o r D has more power i s much more d i f f i c u l t t h a t i n t h e c a s e o f Committee 1, and most o f t h e s u b j e c t made a wrong c h o i c e ( c l a i m i n g t h a t t h e chairman i s more p o w e r f u l t h a t D ) . I n f a c t , t h e s i t u a t i o n i s j u s t t h e o p p o s i t e : enumeration o f p e r m u t a t i o n s i n which C , D o r x i s t h e p i v o t y i e l d s t h e powers : d(C) = 0.2,

d(D)

=

4/15 = 0.2666, d ( x ) = 0.1333.

Thus, i f power o f x i s t a k e n as 1, t h e n t h e power o f chairman C i s 1 . 5 , w h i l e power of member D i s 2 .

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CHAPTER 5

I n t h i s c a s e , when C v o t e s ''yes", we have

P,(z)

=

Z[l-(l-Z) 41

t

(1-z)(2z 2 - z 4) .

Here t h e r e a s o n i n g i s as f o l l o w s . If D v o t e s ''yestr ( p r o b a b i l i t y z i n t h e f i r s t f a c t o r ) , one needs j u s t one more v o t e of t h e r e m a i n i n g members, s i n c e t h e n a t l e a s t one subcommittee w i l l have a m a j o r i t y . The probab i l i t y of a l l f o u r r e m a i n i n g members v o t i n g "no" i s

( l - ~ ) ~s,o t h a t w i t h p r o b a b i l i t y 1 -

at l e a s t one v o t e w i l l be p o s i t i v e . T h i s e x p l a i n s t h e f i r s t t e r m 0-2)~

i n t h e formula. I f D v o t e s "no" ( p r o b a b i l i t y 1-z as a f a c t o r i n t h e second t e r m ) , one needs two v o t e s "yes" i n a t l e a s t one subcommittee. P r o b a b i l i t y o f two v o t e s " y e s r r i n a subcommittee i s z2, and d e n o t i n g by U and V t h e e v e n t s t h a t t h e s e subcommittees v o t e we have P(U u V) 2 = P(U) t P(V) - P(U n V). Here P(U) = P(V) = z , as a l r e a d y s t a t e d , w h i l e P(U n V) i s z4, s i n c e for t h e v o t e t o be "yes" i n b o t h subcommittees, a l l f o u r memb e r s must v o t e " y e s " . T h i s g i v e s t h e whole f o r m u l a . Next, we have

The f i r s t t e r m i s e x p l a i n e d a s b e f o r e : i f b o t h C and D v o t e ''yes" ( p r o b a b i l i t y z ) , t h e n one needs a t l e a s t one m r e v o t e " y e s " . On t h e o t h e r hand, i f C v o t e s "no"

we need a t l e a s t one v o t e "yes" i n e a c h o f t h e two subcommittees. The p r o b a b i l i t y o f a n e g a t i v e v o t e i n a subcommittee i s ( l - ~ ) ~s , o that l-(l-~)~ i s t h e probabi


643

l i t y of a p o s i t i v e v o t e , and s q u a r i n g w e g e t t h e prob a b i l i t y of p o s i t i v e v o t e s i n b o t h subcommittees.

Committee 3 The v o t i n g r u l e h e r e i s as f o l l o w s . 1. The committee c o n s i s t s of 5 members, among

them E and F

.

2 . E has t h e r i g h t o f c a s t i n g t h r e e v o t e s , w h i l e

e a c h of t h e r e m a i n i n g members c a s t s one v o t e e a c h ;

3. F h a s t h e v e t o power; 4 . The d e c i s i o n r e q u i r e s s i m p l e m a j o r i t y : t h e i s s u e p a s s e s i f a t l e a s t f o u r v o t e s o u t of s e v e n , among them t h e vote of F, support

the issue.

The scheme of t h e committee i s as f o l l o w s .

majority decides;

1

positive vote of F i s necessary

E

F

X

X

I

X

Scheme of committee 3

The power of E i s due t o t h e r i g h t o f c a s t i n g t h r e e v o t e s , w h i l e t h e power of F i s due t o h i s r i g h t of v e t o .

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Here t h e powers a r e e q u a l d(E) = 0.3, d ( F ) = 0.55, d ( x ) = 0.05. T h i s means t h a t if t h e power of x i s t a k e n as a u n i t , t h e n t h e power of E e q u a l s 6 , w h i l e t h e power of F i s e q u a l 11. I n o t h e r words, a member w i t h t h r e e t i m e s as many v o t e s t h a n " o r d i n a r y " member h a s 6 t i m e s as much power, w h i l e a member w i t h v e t o power i s 11 t i m e s as p o w e r f u l as o r d i n a r y member. I n c a s e o f t h i s committee we have

s i n c e i f E v o t e s " y e s " , a l l one needs i s a p o s i t i v e v o t e o f F (who h a s t h e v e t o p o w e r ) . I f F v o t e s "no", t h e i s s u e cannot p a s s . On t h e o t h e r hand, P,(z)

=

z

t (1-z)z 3

.

I n d e e d , i f F v o t e s " y e s " , we need e i t h e r t h e t h r e e vot e s o f E ( p r o b a b i l i t y z ) , o r i f E v o t e s "no" ( p r o b a b i l i t y 1-z), we need a l l t h r e e v o t e s of t h e r e m a i n i n g

members. Thus, we have here P,(z) v a l u e of z .

ma-) , and -mA = f x : x < mA], s o t h a t

1.:

-ma & -mA

=

ix: m a

ations)

>

e

AT

T i f h(t)

(2.3)

g

A for

t ' (here T i s t h e end o f t h e p e r i o d o f observ-

.

S i m i l a r l y , i f h ( t ' ) E A , w e d e f i n e t h e time of f i r s t e x i t from A a f t e r t ' a s

th(A,t')

= inf

f

t

t ' , h ( t ) d A[.

(2.4)

O b v i o u s l y , i f h ( t ' ) E A , t h e f i r s t e x i t from A a f t e r t ' i s t h e same as t h e f i r s t e n t r a n c e a f t e r t ' t o t h e complement of A . Thus, t h e c o n c e p t of t h e f i r s t e x i t i s r e d u c i b l e t o t h e c o n c e p t of f i r s t e n t r a n c e ; accordi n g l y , we s h a l l c o n c e n t r a t e on t h e l a t t e r . Generally, a t r i p l e (A,t',h) with h ( t ' ) A w i l l be c a l l e d a n e v e n t A a f t e r t l i n h i s t o r y h . We s h a l l s a y t h a t t h i s event occurs, i f

i . e . i f the f i r s t entrance t o A a f t e r t ' occurs i n h i s t o r y h.

If t k ( A , t t )

>

t " , t h e n t h e e v e n t s ( A , t ' , h ) and ( A , t " , h )

w i l l be i d e n t i f i e d .

I n t h e s e q u e l , we s h a l l s i m p l i f y t h e c o n s i d e r a t i o n s t f = 0; t h e n a n e v e n t w i l l be i d e n t i f i e d w i t h a s e t A s u c h t h a t h ( 0 ) Q A , and w i l l be i n t e r p r e t e d simply as t h e f i r s t e n t r a n c e t o A i n h i s t o r y h; s u c h a n e v e n t w i l l b e denoted simply by S'. We s h a l l a l s o d e n o t e by A- t h e e v e n t d e s c r i b e d as " f i r s t e x i t by t a k i n g

FORMAL THEORY OF ACTIONS

73 1

from A a f t e r t = 0 " . The c h a r a c t e r i s t i c f e a t u r e of t h e t e v e n t s A and A- i s t h a t i f t h e y o c c u r , t h e n t h e i r t i mes o f o c c u r r e n c e a r e w e l l d e f i n e d . The g e n e r a l i z a t i o n t o t h e c a s e t ' ward and w i l l be o m i t t e d .

>

0 i s straightfor-

A l s o , f o r s i m p l i f i c a t i o n of t h e s u b s e q u e n t d e f i n i t i o n s , t i t w i l l be c o n v e n i e n t t o d e n o t e t h e time t h ( A , O ) of t h e t f i r s t e n t r a n c e t o A i n h by t ( A , h ) , and l e t t ( A - , h ) t be t h e t i m e t h [ A , t h ( A , O ) ] o f t h e f i r s t e x i t from A a f t e r t h e first entrance t o A. DEFINITION. We s h a l l s a y t h a t t h e e v e n t A i s t h i n , or instantenuous, i n h, i f t(At,h) = t(A-,h)

(2.6)

t

whenever t ( A , h ) < T . Otherwise, t h e e v e n t A w i l l be called thick, or lasting. DEFINITION. The e v e n t A w i l l be c a l l e d t h i n , or i n s t a n t e n u o u s , i f i t i s t h i n i n e v e r y h i s t o r y from H i n which i t o c c u r s , i . e . s u c h t h a t h ( 0 ) d A and t ( A t , h ) < T . Thus, a s e t A ( e v e n t A ) i s t h i n i n h , i f t h e e v e n t associated w i t h i t (provided i t occurs) i s instantenuous: t h e h i s t o r y h s a t i s f i e s t h e c o n d i t i o n h ( t ) € A o n l y a t scme i s o l a t e d p o i n t o r p o i n t s , and n o t i n a n i n t e r v a l o f moments of t i m e . On t h e o t h e r hand, i f A i s thick, then h ( t ) c A i n the i n t e r v a l [t(At,h),t(A-,h)l o f some p o s i t i v e d u r a t i o n . Assume now t h a t h(0) d A for a l l h G H . The e v e n t A w i l l be c a l l e d n e c e s s a r y , o r u n a v o i d a b l e , i f

732

CHAPTER 6

s u p F ( At , h ) : h E H I 0

and f ( q )

f(a),

which y i e l d s f ( c )

>

0,

and

which g i v e s a c o n t r a d i c t i o n . We may now i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n . DEFINITION. A time-event

q with f ( q )

d e s i r a b l e , and an e v e n t q w i t h f ( q ) undesirable.

>

0 . Let f

*

be some o t h e r a d d i t i v e r e p r e -

s e n t a t i o n o f D . By a s s u m p t i o n , f o r any s e t A which d o e s n o t c o n t a i n q , we have ( A U

0 , and c o m p l e t e s t h e p r o o f .

DEFINITION. A t i m e - e v e n t

q i s called neutral with res-

p e c t t o g o a l G which a l l o w s a d d i t i v e r e p r e s e n t a t i o n , i f f ( q ) = 0. O b v i o u s l y , as b e f o r e w e have THEOREM. N e u t r a l i t y i s i n v a r i a n t w i t h r e s p e c t t o t h e

choice of r e p r e s e n t a t i o n f .

4.1.2.

Goals and t h e i r i d e a l s . L e t u s c o n s i d e r a g a i n

t h e c a s e o f one c r i t e r i o n D . T h i s t i m e , however, we do n o t assume t h a t t h e s e t Q

*

*

is finite.

*

*

w e have A DA f o r a l l * A c Q , we s h a l l s a y t h a t g o a l G i s c l o s e d , and any * A w i t h t h e above p r o p e r t y w i l l be c a l l e d t h e i d e a l o f t h e g o a l G . O t h e r w i s e , t h e g o a l G w i l l be c a l l e d o p e n . DEFINITION. I f f o r some A

C Q

We have THEOREM.

If

A

*

B

*

* *

I n d e e d , w e have A DB

* A

and

* B

* *

are two i d e a l s o f G , t h e n A I B

* *

and B I A

.

by t h e a s s u m p t i o n t h a t

are i d e a l s , which i m p l i e s t h e a s s e r t i o n .


765

Next, w e have a l s o THEOREM.

If

Q

*

i s f i n i t e , then the goal G i s closed.

I n d e e d , l e t A1 be a r b i t r a r y , and l e t

G

S1 = $ A

*

Q : AIDA].

* -

S1 = 0, t h e n A1 i s t h e i d e a l , and t h e g o a l i s * c l o s e d . O t h e r w i s e , t a k e A 2 e Q - S l y and d e f i n e If Q

S2

=iA

*

G Q : A~DA]

.

By t r a n s i t i v i t y , w e have S1 C S2 and S1 Z S2. Again,

*

-

S2 = 0, t h e g o a l i s c l o s e d , s i n c e A 2 i s an i d e * a l . O t h e r w i s e , we can choose A 3 G Q - S 2 and p r o c e e d a s b e f o r e . I n t h i s way, we c o n s t r u c t a n i n c r e a s i n g se* quence S c S C . E Q w i t h a l l elements d i s t i n c t . 2, * I n c a s e when Q i s f i n i t e , we must have Q = Sk f o r some k , and t h e a s s e r t i o n f o l l o w s . if Q

..

4 . 1 . 3 . The e f f e c t o f m u l t i d i m e n s i o n a l i t y . L e t us now c o n s i d e r t h e c a s e r > 1, i . e . t h e c a s e o f more t h a n one c r i t e r i o n . N a t u r a l l y , t h e s i t u a t i o n i s of i n t e r e s t only i f t h e c r i t e r i a a r e , a t l e a s t ; p a r t i a l l y , inconsistent. W e s t a r t from t h e followi.ng d e f i n i t i o n :

The p o i n t A i f the conditions

DEFINITION.

BD A , k = 1, k

6

. . ,r

Q

*

i s c a l l e d Pareto optimal,

CHAPTER 6

766

with

BDiA

f o r a t l e a s t one k , imply t h a t B d Q

*.

Thus, a c o n g i g u r a t i o n A of t i m e - e v e n t s i s P a r e t o o p t i m a l , i f one cannot f i n d any c o n f i g u r a t i o n which would be a t l e a s t as good w i t h r e s p e c t t o a l l c r i t e r i a , and be s t r i c t l y b e t t e r on a t l e a s t one c r i t e r i o n . I n o t h e r words, any improvement a l o n g a l l c r i t e r i a must lead o u t s i d e t h e r a n g e of a d m i s s i b i l i t y . Let

*

Q,

A i s Pareto optimal

= {A:

j

.

We have t h e n t h e f o l l o w i n g t h e o r e m .

If

THEOREM.

* and

A , B E Qp

for all j .

AD.B for a l l j , then A 1 . B J J

DEFINITION. The g o a l G = d Q , D A ,

*

closed, i f there e x i s t s A G Q

*

A D A k

f o r e v e r y k = 1,.

...,

Dr)

i s s a i d t o be

such t h a t

. . ,r

and e v e r y

A6 Q

*

.

C l e a r l y , i f a g o a l i s c l o s e d , t h e n any s u b g o a l < Q , D k ) c o r r e s p o n d i n g t o one c r i t e r i o n i s also c l o s e d . The converse i s , i n general, f a l s e , t h a t i s , closure of a l l subgoals

-

>sot'

i n d i v i d u a l and s o c i a l s t r i c t p r e f e r e n c e xJ and a r e defined i n the usual way). indifference 4 XJ

SOC

a ) Lack of d i c t a t o r . No member of t h e s o c i e t y i s a d i c t a t o r , i n t h e s e n s e t h a t h i s p r e f e r e n c e s dominate o v e r t h e p r e f e r e n c e s o f o t h e r s . Formally, x i s a d i c t a t o r ,

>sot

i f t h e c o n d i t i o n A >x B i m p l i e s A B regardless of p r e f e r e n c e s of o t h e r members of t h e s o c i e t y . b ) S o v e r e i g n t y . The group may d e c i d e t h a t A

>sot

By and i t may a l s o d e c i d e t h a t B A by a p p r o p r i a t e votes ( t h a t i s : it i s not t r u e t h a t A B y even i f B x) A f o r a l l members x; a n a l t e r n a t i v e f o r m u l a t i o n of t h i s p o s t u l a t e i s t h a t t h e s o c i a l c h o i c e f u n c t i o n

>sot

>sot

i s n o t imposed).

c ) C o n s i s t e n c y between i n d i v i d u a l and s o c i a l c h o i c e . Suppose t h a t f o r some s e t of p r e f e r e n c e s w e have A )sot B y and t h a t f o r some r e a s o n t h e v o t i n g i s r e p e a t e d . Assume t h a t i n t h e new v o t i n g , a l l who v o t e d A > x B s t i c k to t h e i r o p i n i o n , w h i l e some of t h o s e

785


who p r e v i o u s l y v o t e d B > x A a r e now v o t i n g A

>x B.

Then i n new v o t e , t h e s o c i a l c h o i c e s h o u l d a g a i n be

A

>

SOC

B.

A r u l e which does n o t meet t h i s c o n d i t i o n a p p e a r s

h i g h l y u n n a t u r a l : i t s h i f t s from A

>sot

B to B

>sot

A

when t h e number o f v o t e s f o r A i n c r e a s e s . d)

Independence of i r r e l e v a n t a l t e r n a t i v e s . Suppose

t h a t some v o t i n g gave t h e r e s u l t A

>sot

B , and t h a t

f o r some r e a s o n t h e v o t i n g i s r e p e a t e d , w i t h new o p t i o n C added. Suppose t h a t i n t h e new v o t i n g , e v e r y p e r s o n who p r e v i o u s l y v o t e d A

> x B v o t e s t h e same way ( w i t h

o p t i o n C b e i n g e i t h e r b e t t e r t h a n A , worse t h a n B , o r between A and B ) . S i m i l a r l y , assume t h a t e v e r y p e r s o n who p r e v i o u s l y v o t e d B > x A s t i l l v o t e s t h e same w ay, w i t h C p l a c e d i n a n a r b i t r a r y w a y . Then i n t h e s o c i a l

c h o i c e f u n c t i o n w e s h o u l d a g a i n have A

>

SOC

B.

I n o t h e r words, t h e mutual o r d e r i n g o f A and B i n t h e s o c i a l c h o i c e f u n c t i o n s h o u l d depend o n l y on t h e o r d e r i n g s o f A and B i n i n d i v i d u a l o r d e r i n g s , and n o t on t h e o r d e r i n g s o f any o t h e r a l t e r n a t i v e . e ) Freedom. There a r e a t l e a s t t h r e e p e r s o n s i n t h e soc i e t y , and a t l e a s t t h r e e a l t e r n a t i v e s , which may be o r d e r e d i n any p o s s i b l e way. The theorem o f I m p o s s i b i l i t y o f Arrow a s s e r t s t h a t t h e p o s t u l a t e s a ) - e ) are mutually i n c o m p a t i b l e , t h a t i s , e v e r y s o c i a l c h o i c e f u n c t i o n must v i o l a t e a t l e a s t one o f t h e s e condi.tions. The consequences of Arrow theorem for t h e problem of

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CHAPTER 6

i n d i v i d u a l c h o i c e based on e v a l u a t i o n s on s c a l e s o f m o t i v a t i o n a l space a r e a s f o l l o w s . A s a l r e a d y mentioned, we may i n f e r t h a t a t l e a s t one o f t h e p o s t u l a t e s o f Arrow theorem must be v i o l a t e d . It a p p e a r s t h a t t h e l i k e l y c a n d i d a t e s for t h a t a r e p o s t u l a t e s a ) , b) and d ) , i . e . l a c k o f d i c t a t o r s h i p , s o v e r e i g n t y and i n d e pendence o f i r r e l e v a n t a l t e r n a t i v e s . The r e m a i n i n g p o s t u l a t e s are p r o b a b l y s a t i s f i e d , a t l e a s t i n d e c i s i o n s of r a t i o n a l p e r s o n s . One can now t r y t o d e s c r i b e t h e d e c i s i o n p r o c e s s e s o f p e r s o n s f o r whom t h e d e c i s i o n f u n c t i o n s do n o t s a t i s f y t h e p o s t u l a t e s a ) , b ) or d ) . I n c a s e when a ) i s v i o l a t e d , t h e d e c i s i o n f u n c t i o n i s d i c t a t o r i a l , i n t h e s e n s e t h a t one s c a l e dominates o v e r t h e r e m a i n i n g o n e s . I n o t h e r words, t h e f i n a l p r e f e r e n t i a l o r d e r i n g c o i n c i d e s w i t h t h e o r d e r i n g on t h e dominating scale. There a r e two n a t u r a l c a n d i d a t e s f o r such d o m i n a t i n g

continuum, namely t h e s c a l e s r e p r e s e n t i n g Superego, and s c a l e s r e p r e s e n t i n g I d . One can t h e r e f o r e e x p e c t t h a t among p e r s o n s whose c h o i c e f u n c t i o n s do not s a t i s f y a ) one may d i s t i n g u i s h two c a t e g o r i e s : p e r s o n s f o r whom t h e d o m i n a t i n g continuum b e l o n g s t o Superego f a m i l y , r e p r e s e n t e d b y f u n c t o r s s u c h a s "I ought t o " , "I must", e t c . , and p e r s o n s for whom t h e d o m i n a t i n g continuum b e l o n g s t o t h e I d f a m i l y , r e p r e s e n t e d b y f u n c t o r s such as "I want", "I p r e f e r " , e t c . The f i r s t group w i l l comprise p e r s o n s g e n e r a l l y d e s c r i b e d a s r e l i a b l e , r i g h t e o u s , h o n e s t , e t c . , w h i l e t h e second group compr i s e s h e d o n i s t s . From t h e c l i n i c a l p o i n t o f view, t h e f i r s t group w i l l c o n t a i n compulsive p e r s o n s , and t h e second - p s y c h o p a t s .


787

The second p o s s i b i l i t y i s t h a t t h e c h o i c e f u n c t i o n d o e s n o t s a t i s f y p o s t u l a t e b ) o f s o v e r e i g n t y . Thus , t h e d e c i s i o n ( a t l e a s t w i t h r e s p e c t t o some a l t e r n a t i v e s ) i s imposed. T h i s means t h a t t h e c h o i c e a l w a y s c o i n c i d e s w i t h some a c c e p t e d s t a n d a r d , even i f a l l o r d e r i n g s on

p s y c h o l o g i c a l continua are Contrary t o t h i s s t a n d a r d . It seems t h a t i n t h i s c a t e g o r y one may f i n d o n l y e x t r e -

me c a s e s , of p e r s o n s w i t h weak p e r s o n a l i t y , whose s t r e n g t h i s d e r i v e d from b l i n d o b e d i e n c e ( e . g . SS-men). F i n a l l y , t h e r e may e x i s t p e r s o n s whose c h o i c e f u n c t i o n does n o t s a t i s f y t h e p o s t u l a t e d ) o r independence from i r r e l e v a n t a l t e r n a t i v e s . From p s y c h o l o g i c a l p o i n t of view, t h i s grpup w i l l comprise n e u r o t i c s , who a r e prone t o change t h e d e c i s i o n when a new a l t e r n a t i v e becomes a v a i l a b l e , even i f t h i s a l t e r n a t i v e i s n o t p r e f e r r e d on any continuum: j u s t t h e mere p r e s e n c e o f new o p t i o n , however u n d e s i r a b l e , makes a p e r s o n t o change h i s mind. I n extreme c a s e s , w e have here a t y p i c a l " a l l o r n o t h i n g " a t t i t u d e o f some n e u r o t i c s . The above groups r e p r e s e n t p e r s o n s f o r whom t h e c h o i c e f u n c t i o n g i v e s t r a n s i t i v e and c o n n e c t e d f i n a l p r e f e r e n c e s . I n o t h e r words, t h e theorem o f Arrow i n t h e present case a s s e r t s t h a t a l i n e a r ordering of a l t e r n a t i v e s may be a c h i e v e d o n l y a t t h e c o s t o f v i o l a t i n g one o f t h e p o s t u l a t e s (most l i k e l y a ) , b ) o r d ) ) . There remains a p o s s i b i l i t y o f p e r s o n s f o r whom t h e c h o i c e f u n c t i o n does n o t y i e l d a l i n e a r o r d e r i n g , o n l y a p a r t i a l o r d e r , or i n t r a n s i t i v i t y . P s y c h o l o g i c a l l y , not

much may be s a i d about t h i s group, e x c e p t t h a t , as i n d i c a t e d by e x p e r i m e n t s of Tversky ( 1 9 6 9 ) -- t h e group o f p e r s o n s w i t h i n t r a n s i k i v e p r e f e r e n c e s may be i n f a c t more numerous t h a n one might e x p e c t .

788

CHAPTER 6

One s h o u l d mention t h a t t h e e a s e w i t h which one o b t a i n s t h e above consequences o f Arrow theorem i s c o n n e c t e d w i t h i t s n e g a t i v e c h a r a c t e r . I n d e e d , t h e only assumption which i s used h e r e i s t h a t t h e j o i n t e v a l u a t i o n i s based on some e v a l u a t i o n s on p a r t i c u l a r s c a l e s , e a c h c o n c e r n i n g some a s p e c t of t h e a l t e r n a t i v e s . The e x a c t n a t u r e of t h e s e a s p e c t s i s i r r e l e v a n t f o r t h e a s s e r t i o n , and one needs no e m p i r i c a l t e s t s t o v e r i f y t h e consequences. I n o t h e r words, i t t u r n s o u t t h a t t h e above i n t e r p r e t a t i o n y i e l d s c e r t a i n i m p o r t a n t p s y c h o l o g i c a l consequenc e s about t h e p r o c e s s o f d e c i s i o n making, s i m p l y b y o b s e r v i n g t h a t p e o p l e must d i v i d e i n t o v a r i o u s c a t e g o r i e s , depending on which o f t h e f i v e p o s t u l a t e s of Arrow t h e y v i o l a t e . Each o f t h e s e c a t e g o r i e s , i n t u r n , a c q u i r e s a n i n t e r p r e t a t i o n i n p s y c h o l o g i c a l terms. The t h e o r e t i c a l and e m p i r i c a l i m p o r t a n c e o f i d e n t i f i c a t i o n of i n d i v i d u a l and group d e c i s i o n , and t h e r e s u l t i n g a p p l i c a t i o n o f Arrow theorem i s as f o l l o w s . F o r t h e o r e t i c a l r e s e a r c h i t s t r e s s e s t h e importance o f o r d i n a l t y p e s o f c o n t i n u a , which may i n t e r v e n e i n t h e p r o c e s s of d e c i s i o n , and which a r e t r a d i t i o n a l l y l e f t o u t i n d e c i s i o n models. S e c o n d l y , i t l e a d s (among o t h e r s ) t o t h e d e f i n i t i o n and taxonomy o f i n t e r n a l c o n f l i c t s mentioned above. For e m p i r i c a l r e s e a r c h , i t suggests ( a ) t o i d e n t i f y t h e c l a s s e s of s i t u a t i o n s when t h e c r i t e r i a a r e n o t a g g r e g a t e d i n t o a t r a n s i t i v e r e l a t i o n , and ( b ) i f t h e y a r e so aggregated, t o search f o r those p a r t i c u l a r axioms o f Arrow which have been v i o l a t e d i n o r d e r to a t t a i n t r a n s i t i v i t y . The u l t i m a t e p u r p o s e o f t h e l a t t e r


789

s t u d y i s t o i d e n t i f y t h e t y p e s o f d e c i s i o n problems which a r e most l i k e l y t o l e a d t o v i o l a t i o n o f a g i v e n axiom; such knowledge would, i n t u r n , a l l o w t o r e s t r i c t t h e c l a s s o f d e c i s i o n c r i t e r i a which c o u l d b e employed by t h e d e c i s i o n maker.

5.2.

Motivational calculus

If one l o o k s a t l i n g u i s t i c r e p r e s e n t a t i o n o f t h e s c a l e s of m o t i v a t i o n a l s p a c e , one o b t a i n s a fragment o f n a t u r a l l a n g u a g e . The a n a l y s i s o f t h i s fragment i s o f cons i d e r a b l e i n t e r e s t f o r i t s own s a k e ; e s p e c i a l l y importa n t here a r e t h e r u l e s o f i n f e r e n c e , which c o n s t i t u t e the motivational calculus. A s r e g a r d s m o t i v a t i o n a l c a l c u l u s , i t p r o v i d e s schemes of' i n f e r e n c e from s e n t e n c e s i n n a t u r a l language c o n t a i n -

i n g m o t i v a t i o n a l f u n c t o r s . The b a s i c p r i m i t i v e n o t i o n o f t h e t h e o r y i s t h e c o n c e p t o f i n a d m i s s i b i l i t y of a

s e n t e n c e . I n t u i t i v e l y , an e x p r e s s i o n i s i n a d m i s s i b l e , i f t h e r e i s no c o n t e x t i n which i t could be meaningfull y a p p l i e d , or' i f it c o u l d be used o n l y i n " u n n a t u r a l "

s i t u a t i o n s . T h i s c o n c e p t a l l o w s us t o d e f i n e t h e n o t i o n o f semantic i m p l i c a t i o n : A 2 B ( A implies semantically B), i f t h e s e n t e n c e "A and n o t B" i s i n a d m i s s i b l e . M o t i v a t i o n a l c a l c u l u s c o m p r i s e s a number o f s e m a n t i c i m p l i c a t i o n s tletween s e n t e n c e s c o n t a i n i n g m o t i v a t i o n a l f u n c t o r s . These i m p l i c a t i o n s a l l o w i n f e r e n c e from t h e s e n t e n c e s i n n a t u r a l l a n g u a g e . To g i v e some examples, "I a m g l a d t h a . t p" i m p l i e s t h a t t h e p e r s o n e i t h e r knows t h a t p , or he t h i n k s t h a t p ( i s t r u e ) . S y m b o l i c a l l y , t h i s may be w r i t t e n as Gp Kp J T p . A s a n o t h e r ex-

s/

790

CHAPTER 6

ample o n e may t a k e i m p l i c a t i o n s s u c h a s W(p&q) >S Wp & Wq, where W s t a n d s f o r "I w a n t " , o r D ( p & q ) S> Dp V Dq, where D s t a n d s f o r "I d o u b t t h a t " . S i m i l a r l y , t h e l a t t e r f u n c t o r D may be r e l a t e d t o o t h e r e p i s t e m i c f u n c t o r t h r o u g h t h e i m p l i c a t i o n Dp --+ T ( - p ) & - C r ( - p ) , S w i t h C r s t a n d i n g f o r "I a m c e r t a i n t h a t " . T h e s e i m p l i c a t i o n s a s s e r t t h a t i f someone w a n t s a conj u n c t i o n o f two e v e n t s , t h e n he w a n t s e a c h o f them separately , too; t h e implications involving t h e functor "I d o u b t " s t a t e t h a t i f someone d o u b t s a c o n j u n c t i o n , t h e n he d o u b t s a t l e a s t o n e o f i t s c o m p o n e n t s , a n d a l s o when someone d o u b t s ( t h e t r u t h o f ) p , t h e n he t h i n k s t h a t not-p, but i s n o t c e r t a i n t h a t not-p. By c o n s t r u c t i n g a s e t - t h e o r e t i c a l m o d e l , i t i s shown i n Nowakowska (1973) t h a t t h e o b t a i n e d c a l c u l u s (comp r i s i n g some 40 i m p l i c a t i o n s ) i s l o g i c a l l y c o n s i s t e n t . I n e f f e c t , t h e s e semantic i m p l i c a t i o n s ( i n form o f t r a n s f o r m a t i o n r u l e s from s e n t e n c e s i n v o l v i n g m o t i v a t i o n a l f u n c t o r s ) c o n s t i t u t e r u l e s of approximate reasoni n g , t h e a p p r o x i m a t i o n r e s u l t i n g from t h e f u z z i n e s s o f t h e underlying concept of semantic a d m i s s i b i l i t y . I n the construction of motivational calculus, i n order t o a c h i e v e g r e a t e r g e n e r a l i t y , t h e t r a n d i t i o n a l way o f l i s t i n g t h e p r i m i t i v e s a n d l a y i n g down t h e r u l e s o f c o n s t r u c t i o n o f c o m p o s i t e s e n t e n c e s was d e l i b e r a t e l y o m i t t e d . It s h o u l d b e c l e a r , h o w e v e r , t h a t g i v i n g s u c h r u l e s would r e s t r i c t t h e c o n s i d e r a t i o n s t o a s p e c i f i c l a n g u a g e , and would a l s o r e q u i r e a s p e c i f i c a t i o n o f the sense of motivational functors (the l a t t e r t a s k i s

s t i l l n o t c o m p l e t e l y s o l v e d f o r any s i n g l e f u n c t o r , l e t


791

a l o n e f o r t h e whole s e t o f f u n c t o r s a t o n c e ) . It d i d a p p e a r more p r o m i s i n g t o p r o v i d e d i r e c t l y a s e t o f i n f e r e n c e r u l e s , an d p r o v e t h e c o n s i s t e n c y o f t h i s s e t . One c a n a r g u e t h a t s u c h an a p p r o a c h i s c l o s e r t o t h e n a t u r a l l a n g u a g e , s i n c e l a y i n g down t h e c o n s t r u c t i o n r u l e s f o r c o m p o s i t e s e n t e n c e s wo u la a l l o w l o n g s e n t e n c e s , acceptable l o g i c a l l y but not l i n g u i s t i c a l l y . The laws o f t h e m o t i v a t i o n a l c a l c u l u s a l l o w f o r m u l a t i o n

o f t h e laws o f m o t i v a t i o n a l c o n s i s t e n c y , by c o mb in in g d e c i s i o n s w i t h u t t e r a n c e s ( t h i s t o p i c w i l l be analysed i n n e x t s u b s e c t i o n ) . Such laws l e a d t o c l a s s e s o f conc l u s i o n s ( u t t e r a n c e s o r a c t i o n s ) w h ic h a r e c o n s i s t e n t w i t h g i v e n u t t e r a n c e , h en ce a l s o a l l o w some p r e d i c t i o n s o f b e h a v i o u r on t h e b a s i s o f u t t e r a n c e s . A n o t h e r p o s s i b i l i t y i s t o u s e t h e laws o f m o t i v a t i o n a l c o n s i s t e n c y and m o t i v a t i o n a l c a l c u l u s f o r d i r e c t i n f e r e n c e a b o u t m o t i v a t i o n . Knowing t h e p e r s o n ' s u t t e r a n c e s and h i s d e c i s i o n s ( a c t i o n s ) , an d a s s u m i n g t h a t he i s m o t i v a t i o n a l l y c o n s i s t e n t , t h e problem l i e s i n adding t a c i t premise:; t o h i s u t t e r a n c e s i n s u c h a way t h a t h i s d e c i s i o n s become m o t i v a t i o n a l l y c o n s i s t e n t w i t h h i s u t t e r a n c e s ( t a c i t and n o t t a c i t ) . I n s u c h c a s e , t h e added p r e m i s e s r e f l e c t t h e ( u n - u t t e r e d ) m o t i v e o f a p e r s o n . C l e a r l y , t h e l e s s freed o m i n s u c h a n a d d i t i o n of p r e m i s e s , t h e more c e r t a i n i s t h e i n f e r e n c e a b o u t p e r s o n ' s motives. The r u l e s of i n f e r e n c e o u t l i n e d abo v e may p r o v i d e forma l methods o f i n f e r e n c e i n p s y c h o l o g i c a l d i a g n o s i s .

It i s also w o r t h w h i l e t o m e n t i o n t h a t t h e s t u d y of l a n g u a g e o f m o t i v a t i o n an d m o t i v a t i o n a l c a l c u l u s , b a s e d

792

CHAPTER 6

on common and everyday s e n s e and i n t u i t i o n , i s a neces s a r y p r e r e q u i s i t e f o r t h e c o n s t r u c t i o n of a model o f

language u s e r . To sum up, t h e c e n t r a l i d e a o f m o t i v a t i o n a l c a l c u l u s ,

which seems t o have some i m p o r t a n c e f o r s e m a n t i c and psycholinguistic s t u d i e s , i s the introduction of l i n g u i s t i c r e p r e s e n t a t i o n o f s i t u a t i o n a b o u t which one wants t o make t h e i n f e r e n c e , and s u c c e s s i v e t r a n s f o r m a t i o n s which a r e a p p l i e d t o t h i s s i t u a t i o n , based on l i n g u i s t i c competence and i n t u i t i o n . It i s assumed h e r e t h a t t h e l e x i c a l u n i t s have c e r t a i n a d m i s s i b l e l o g i c a l p r o p e r t i e s , which a l l o w o b s e r v a t i o n and s t u d y o f d i f f e r e n c e s between l o g i c a l and l i n g u i s t i c s c o p e o f inference. Finally, t h e studies of l i n g u i s t i c inference s u p p l y arguments a g a i n s t s y n t a c t i c c o n c e p t i o n s o f t h e l a n g u a g e , b y s t r e s s i n g t h e f a c t t h a t i n t h e dynamics o f l i n g u i s t i c transformations t h e important r o l e i s

p l a y e d by b o t h s y n t a c t i c and s e m a n t i c a s p e c t s , o r -- i n o t h e r words

--

i n t e r a c t i o n s between l o g i c a l and seman-

t i c properties. I n Nowakowska ( 1 9 8 3 ) m o t i v a t i o n a l c a l c u l u s was e n r i c h e d by i n c l u s i o n o f t e m p o r a l a s p e c t s , i . e . l i n g u i s t i c rep r e s e n t a t i o n o f time. This i n t r o d u c e s an a d d i t i o n a l d i m e n t i o n t o a l r e a d y m u l t i d i m e n s i o n a l concept o f l i n g u i s t i c r e p r e s e n t a t i o n o f m o t i v a t i o n , and a l l o w s f o r a more dynamic t r e a t m e n t of d e c i s i o n s p a c e , i d e n t i f i e d ( t h r o u g h Arrow Theorem) w i t h c o g n i t i v e o r m o t i v a t i o n a l s p a c e , i n which one makes t h e judgments about c o u r s e s of a c t i o n s i n d e c i s i o n s i t u a t i o n s . I n t h i s w a y , t h e s t r i n g s o f a c t i o n s are combined w i t h c o g n i t i v e p r o c e s s e s and l i n g u i s t i c d e s c r i p t i o n s i n t o


193

one u n i f i e d system o f t h o u g h t and a c t i o n , i n which one c o n s i d e r s r a t h e r r e a s o n s f o r a c t i o n s t h a n more e l u s i v e c a u s e s of a c t i o n s . T h i s e x t e n d e d a c t i o n system i s based on t h e c o n c e p t i o n

-- d e r i v e d from n a t u r a l language -- of l i n g u i s t i c measurement ( s e e Chapter 5 l , which a l l o w s as e x t e n s i o n of m e a s u r e m e n t - t h e o r e t i c a l c o n c e p t s t o s u c h n o t i o n s as measurement by a n a l o g y , e t c .

5.3. Linguistic representation of time i n motivational calculus The t o p i c o f t i m e i s d i s c u s s e d e x t e n s i v e l y i n C h a p t e r

5 , h e r e w e c o n s i d e r o n l y some s p e c i a l a s p e c t s , namely t h e l i n g u i s t i c r e p r e s e n t a t i o n of t i m e . We s t a r t h e r e from t h e g e n e r a l c o n c e p t u a l framework o f

f u z z y a t t r i b u t e s which change i n t i m e , similar t o t h a t used i n C h a p t e r 5. I n t h e s e q u e l , t h e symbol t w i l l s t a n d f o r a moment of t i m e , w i t h t h e c o n v e n t i o n t h a t t = 0 s i g n i f i e s t h e p r e s e n t , being t h e time of t h e u t t e r a n c e of t h e sentenc e . P o s i t i v e v a l u e s o f t s i g n i f y f u t u r e moments, w h i l e n e g a t i v e v a l u e s of t denote p a s t . N e x t , w e i n t r o d u c e t h e s e t X o f o b j e c t s under c o n s i d e r a t i o n , i t s elements b e i n g x , y , z , . .

. .

We s h a l l be concerned w i t h a t t r i b u t e s of o b j e c t s i n X ;

t h e s e a t t r i b u t e s may be f u z z y , and a l s o may change i n time.

794

CHAPTER 6

To i n t r o d u c e t h e a p p r o p r i a t e c o n v e n i e n t f o r m a l i s m , l e t U , V,

...

be t h e a t t r i b u t e s under c o n s i d e r a t i o n . W i t h e a c h a t t r i b u t e we may a s s o c i a t e t h e s e t of i t s p o s s i b l e v a l u e s , s a y AU, AV, ...

.

The b a s i c c o n c e p t which w i l l be used f o r e x p l i c a t i n g various concepts concerning l i n g u s t i c representation o f t i m e w i l l be t h a t o f p o s s i b i l i t y , as p e r c e i v e d by AU o f t h e o b s e r v e r s , t h a t o b j e c t x has v a l u e xu a t t r i b u t e U a t t i m e t . We s h a l l assume t h a t t h i s p o s s i b i l i t y i s g r a d e d , and a c c o r d i n g l y , d e n o t e i t by t h e symbol f S ( x , t , x u ) . I n t h e s e q u e l , we s h a l l c o n s i d e r t h e c a s e o f a f i x e d o b s e r v e r s , and t o s i m p l i f y t h e n o t a t i o n s , we s h a l l o f t e n omit t h e s u b s c r i p t s , w r i t i n g s i m p l y f ( x , t , x U ) . Thus,

-

0 c f(x,t,xu)

f

1,

with f ( x , t , x ) = 1 representing the subjective certainty

U

t h a t t h e v a l u e of a t t r i b u t e U o f o b j e c t x a t t i m e t

e q u a l s xu. C o n s i d e r now an e x p r e s s i o n " x i s u a t t i m e t " . Here u i s some d e s c r i p t i o n , p e r t a i n i n g t o a t t r i b u t e U . As a n example, we may t a k e as U t h e a t t r i b u t e s u c h a s h e i g h t , and as u t h e e x p r e s s i o n s such a s " t a l l " , e t c . Now, w i t h e v e r y d e s c r i p t o r u w e may a s s o c i a t e a f u z z y s u b s e t o f t h e s e t AU o f a t t r i b u t e v a l u e s o f U. Let t h i s fuzzy s u b s e t be d e s c r i b e d by membership f u n c t i o n g u . We may t h e n d e f i n e t h e p o s s i b i l i t y f ( x , t , u ) t h a t x i s u a t t i m e t , as

795


f(x,t,u) =

sup

min C f ( x , t , x U ) , g u ( x , ) l .

(5.1)

xu (5 AU Given t h e f u n c t i o n f ( x , t , u ) , we may d e f i n e , for e v e r y 0 5 o( G 1

fixed& with

w i t h t h e conventions t h a t

t (x,u) = -

w

if f ( x , t , u ) l o
S

(a,€')

b e P , b u t d i d n o t know t h a t a can n e v e r

be P" i m p l i e s t h a t t h e s p e a k e r i s now c e r t a i n t h a t a can n e v e r be P , and t h a t he s h o u l d n o t want now a t o be P ) . T h i s i m p l i c a t i o n shows how t h e f o u n d a t i o n s of r a t i o n a l i t y of m o t i v a t i o n a r e s t r u c t u r e d , and how t h e d e f e n s e mechanisms i n t h e s e n s e o f Freud a r e formed. They a l l o w

f u n c t i o n i n g i n t h e c o n d i t i o n s of s t r e s s and d e p r i v a t i o n . Finally, Some t i m e ago I wanted a t o be P

5


803

( t h e s p e a k e r d o e s n o t know now whether he wants a t o be P o r n o t ) .

I n t h i s example one may s e e how we u s e ( i n t h e l a n g u a g e ) t h e weak ( o r v a g u e ) n e g a t i o n , where t h e t r u t h condit i o n s a r e r e s t r i c t e d t o e a r l i e r t i m e s and t h e s e n t e n c e does n o t ( b u t may) be t r u e a t l a t e r t i m e s . T h i s i m p l i e s t h e e x p e c t a t i o n o f change connected w i t h s u c h e v e n t s which o c c u r r e d i n t h e meantime, and c o u l d c a u s e changes i n t h e s t r u c t u r e o f knowledge ("I t h o u g h t t h a t a i s PI', which d o e s n o t imply t h a t I now t h i n k s o ) . This implication presupposes t h a t t h e l i s t e n e r i s not

c e r t a i n w h e t h e r now t h e s p e a k e r i s g o i n g t o s u p p o r t h i s p r e v i o u s s t a t e o f mind, p r e f e r e n c e , e t c . or deny i t , o r announce t h e s t a t e o f c o n f u a i o n , r e q u i r i n g c l a r i f i c a t i o n and j u s t i f i c a t i o n . Such a s s e r t i o n s u n d e r l i e t h e o b s e r v e d and e x p e c t e d c h a n g e a b i l i t y . A s s e r t i n g a g i v e n v a l u e s o f a n a t t r i b u t e o f a n o b j e c t i n t h e past p l a y s i n t h e l a n g u a g e t h e r o l e o f something l i k e vague n e g a t i o n , which l a t e r s h o u l d b e and may be s h a r p e n e d i n t h e discourse. L e t u s now remark on t h e p o s s i b l e f u z z i n e s s o f a p p l i -

c a t i o n o f f u n c t o r s t o t h e i r arguments. Thus, l e t F Z , ,... b e f u n c t o r s F , G ,... w i t h z i n d i c a t i n g t h e d e g r e e , s o t h a t F Z p s t a n d s for p i s F i n d e g r e e z . GZ

Such f u n c t o r s may b e i t e r a t e d , i f p i t s e l f i s a f u z z y e x p r e s s i o n , s o t h a t we may have F Z p , F Z ( G Z , p ) , F Z I G Z l ( H Z l , p ) ]and s o on. The n e g a t i o n o f s u c h f u n c t o r s i s o b t a i n e d from t h e r u l e -FZp = F1-z ( - p ) , s o t h a t

CHAPTER 6

804

I f w e have a time-indexed p r o p o s i t i o n p , and T i s t h e

t i m e a x i s , t h e n f o r e a c h p o s s i b l e world W we may cons t r u c t a f a m i l y o f f u n c t i o n s which map pand W i n t o 2 T , i n d e x e d b y v a r y i n g t i m e , namely showing a t which mome n t s t h e p r o p o s i t i o n p was t r u e i n W . Thus, i f P i s a * s e t o f p r o p o s i t i o n s , and W . i s t h e c l a s s o f p o s s i b l e w o r l d s , t h e n w e have f: P x

w*

->2,

T

,

where f(p,W) i s t h e s e t o f p o i n t s i n t i m e when p i s t r u e i n W . T h i s may be c a l l e d a h i s t o r y o f p i n W . I n c a s e o f t h e s e c o n s i d e r a t i o n s , as w e l l as t h o s e o f C h a p t e r 5 , i n s t e a d o f t h e t e r m " t r u t h " , t h e terms " f a i t h f u l n e s s " and " e x a c t n e s s " were i n t r o d u c e d . These n o t i o n s a r e based on t h e a s s u m p t i o n o f e x i s t e n c e o f t r u e s t a t e o f a n o b j e c t , e x p r e s s e d by i t s a t t r i b u t e s , and r e l a t i o n s between them and t h e p r o p e r t i e s o f descr i p t ion. The d e s c r i p t i o n o f an o b j e c t has t o c o v e r t h e a t t r i b u t e s

which are e s s e n t i a l ( f o r t h e p u r p o s e i n q u e s t i o n ) on a n a p p r o p r i a t e l e v e l o f e x a c t n e s s . The g o a l o f t h e d e s c r i p t i o n i s t h e r e f o r e t h e second c o n s t r a i n i n g r e f e r e n t i a l f a c t o r . C o n s e q u e n t l y , one can have d e s c r i p t i o n s w i t h v a r y i n g d e g r e e s o f f a i t h f u l n e s s and e x a c t n e s s , which r e f l e c t s v a r i o u s d e g r e e s o f t r u t h about t h e o b j e c t . One c o u l d c o n s i d e r t h e c o n d i t i o n s imposed on d e s c r i p t i o n s under which one c o u l d s p e a k o f t h e f u l l t r u t h o f d e s c r i p t i o n . The c o n c e p t s o f f a i t h f u l n e s s and e x a c t n e s s are t h e r e f o r e b a s e d on a c e r t a i n o n t o l o g y of the objects.

FORMAL THEORY OF ACTTONS

805

5 . 4 . M o t i v a t i o n a l c o n s i s t e n c y and s t r u c t u r a l p r o p e r t i e s o f s e t s o f s t r i n g s o f a c t i o n s which f u l f i l l o r b r e a k a promise One o f t h e b a s i c e l e m e n t s o f t h e d e s c r i p t i o n o f p e r s o n ' s b e h a v i o u r i s t h e m o t i v a t i o n a l c o n s i s t e n c y (or i t s l a c k ) , u n d e r s t o o d i n most common s e n s e as t h e c o n s i s t e n c y b e t ween p e r s o n ' s u t t e r a n c e s and a c t i o n s . T h i s element o f d e s c r i p t i o n pr ovides an important c l u e t o t h e personali t y of t h e person evaluated. The m o t i v a t i o n a l c o n s i s t e n c y , a f o u n d a t i o n o f p r o s o c i a l

b e h a v i o u r , c a n m a n i f e s t i t s e l f i n v a r i o u s ways, s u c h as t r u t h f u l n e s s , k e e p i n g p r o m i s e s , e t c . , or as t h e a b i l i t y o f r e l i a b l e and c o n v i n c i n g e x p l a n a t i o n o f o n e ' s behaviour. There i s no need t o s t r e s s t h e i m p o r t a n c e o f t h i s con-

c e p t f o r t h e i n t e r p e r s o n a l r e l a t i o n s , i n s c i e n c e as w e l l as o u t s i d e o f i t . C l e a r l y , t o d e t e r m i n e w h e t h e r a g i v e n b e h a v i o u r i s mot i v a t i o n a l l y consistent w i t h a given utterance r e q u i r e s a s e p a r a t e a n a . l y s i s i n e a c h c a s e . There a r i s e s a spec i a l problem, t o which e x t e n t one may c h a r a c t e r i z e t h e s t r u c t u r a l a s p e c t s o f t h e b e h a v i o u r which i s c o n s i s t e n t ( o r i n c o n s i s t e n t ) w i t h a given utterance. In o t h e r words, t h e problem i s t o answer t h e q u e s t i o n whether t h e c l a s s o f a l l b e h a v i o u r c o n s i s t e n t (or i n c o n s i s t e n t ) w i t h a g i v e n u t t e r a n c e can be c h a r a c t e r i z e d t h r o u g h i t s structural properties. Some r e s u l t s i n t h i s d i r e c t i o n a r e g i v e n i n t h i s s e c t i o n . The p r e s e n t a t i o n r e q u i r e s t h e usage o f c e r t a i n

806

CHAPTER 6

n o t i o n s o f m a t h e m a t i c a l l i n g u i s t i c s ( t h e s e n o t i o n s may be f o u n d , f o r i n s t a n c e , i n Chomsky, 1957). The b a s i c i d e a s may be p r e s e n t e d a s f o l l o w s . Suppose t h a t we c o n s i d e r t h e a c t i o n s o f a c e r t a i n p e r s o n , and t h a t t h i s person, p r i o r t o t h e beginning of t h e s t r i n g o f a c t i o n s , u t t e r e d a s e n t e n c e which c o n t a i n s a promis e . A l l subsequent s t r i n g s of a c t i o n s ( p o t e n t i a l l y p o s s i b l e ) may t h e n b e c l a s s i f i e d i n t o t h o s e which f u l f i l l t h e p r o m i s e , and t h o s e which b r e a k t h e promis e . The o b j e c t w i l l be t o c h a r a c t e r i z e t h e s e s e t s o f s t r i n g s of a c t i o n s . Before t h a t , however, one may p r o c e e d somewhat more g e n e r a l l y , and a n a l y s e t h e n o t i o n o f t h e c o n s i s t e n c y between v e r b a l and n o n v e r b a l b e h a v i o u r . We s h a l l t r y t o d i s t i n g u i s h i n t h e s e n t e n c e and i n t h e b e h a v i o u r , t h o s e e l e m e n t s which are r e s p o n s i b l e f o r

c o n s i s t e n c y or f o r i t s l a c k ; t h e c l a s s e s o f f u n c t o r s of m o t i v a t i o n a l c a l c u l u s w i l l p l a y h e r e an e s s e n t i a l role. G e n e r a l i z i n g t h e f o r m a l i s m i n t r o d u c e d i n Nowakowska (1973), one may proceed a s f o l l o w s . L e t M s t a n d f o r t h e * c l a s s of a l l m o t i v a t i o n a l u t t e r a n c e s , and l e t M b e t h e c l a s s of a l l f i n i t e s t r i n g s o f e l e m e n t s o f M . F u r t h e r , l e t L d e n o t e t h e l a n g u a g e of a c t i o n s under c o n s i d e r a t i o n , i . e . L = LT [s(O)], f o r some t i m e h o r i zon T and some i n i t i a l s t a t e s ( 0 ) . p r i m i t i v e notion here i s that o f consistency between an element of M ( m o t i v a t i o n a l s e n t e n c e ) and a s t r i n g o f a c t i o n s . We s h a l l d e n o t e t h i s r e l a t i o n by 6, s o t h a t t h e s y m b o l uhm w i l l mean t h a t t h e s t r i n g of actions u i s consistent w i t h t h e sentence m. The


807

We p u t

c-l(m)

= {u

c

L : u$mJ

,

s o t h a t C-'(m) i s t h e c l a s s o f a l l s t r i n g s of a c t i o n s which are c o n s i s t e n t w i t h t h e u t t e r a n c e m . The main point; i s t h a t some s e n t e n c e s i n M r e f e r t o t h e p a s t , some r e f e r t o t h e f u t u r e , and some r e f e r t o b o t h

past and f u t u r e . I f t h e dynamic a s p e c t s o f m o t i v a t i o n a l

c o n s i s t e n c y a r e t o be c o n s i d e r e d , one must r e q u i r e t h a t a s e n t e n c e r e l ' e r r i n g t o t h e f u t u r e , s a y , be c o n s i s t e n t w i t h t h e s t r i n g o f a c t i o n which f o l l o w s t h e u t t e r a n c e of t h i s s e n t e n c e . S i m i l a r r e s t r i c t i o n s concern t h e p a s t , and i t i s c l e a r t h a t one must a n a l y s e c o n s i s t e n c y b e t ween a s e n t e n c e and a s u b s t r i n g o f t h e s t r i n g of a c t i o n s i n which t h i s s e n t e n c e i s embedded. To e x p r e s s i t f o r m a l l y , and a t t h e same t i m e l a y t h e f o u n d a t i o n s f o r t h e theorems which d e s c r i b e t h e r e g u l a r i t i e s o f s e t s o f s t r i n g s o f a c t i o n s which a r e c o n s i s t e n t w i t h some u t t e r a n c e s , one may p r o c e e d a s f o l l o w s . Let M = M

+

t J

M- u M- be t h e p a r t i t i o n o f t h e c l a s s o f

a l l m o t i v a t i o n a l s e n t e n c e s i n t o t h o s e which r e f e r t o t

t h e f u t u r e o n l y (M ) , t o t h e p a s t o n l y (M-), and t o t b o t h p a s t and f u t u r e (M-).

*

L e t us c o n s i d e r t h e monoid (M u A ) , where A i s t h e c l a s s o f a c t i o n s under c o n s i d e r a t i o n . Thus, e l e m e n t s o f t h i s monoid a r e s t r i n g s o f a c t i o n s and u t t e r a n c e s of s e n t e n c e s from M . With e v e r y s t r i n g from ( M

*

one may a s s o c i a t e t h e s t r i n g of a c t i o n s o b t a i n e d by removing a l l u t t e r a n c e s , I/

A)

808

CHAPTER 6

as w e l l as s t r i n g of u t t e r a n c e s , o b t a i n e d b y o m i t t i n g

a l l a c t i o n s . These two s t r i n g s , for a g i v e n s t r i n g z * i n ( M u A ) , w i l l b e d e n o t e d by a ( z ) and u ( z ) , s o t h a t n * a ( z ) 6 A and u ( z ) G M We s h a l l assume t h a t t h e f i r s t of t h e s e s t r i n g s , a ( z ) , i s a d m i s s i b l e , t h a t i s , b e l o n g s to t h e language o f a c t i o n s L .

.

C l e a r l y , t h e mappings a and u a r e homomorphisms w i t h respect t o concatenation, i . e . i f z = z'z", then a ( z ) = a ( z ( ) a ( z " ) and u ( z ) = u ( z l ) u ( z " ) . -1

We s h a l l c o n s i d e r t h e s e t a ( L ) , i . e . t h e c l a s s o f a l l mixed s t r i n g s , s u c h t h a t t h e s t r i n g of a c t i o n s i s i n L.

w

and l e t MZ be t h e set o f a l l e l e m e n t s o f M which a p p e a r i n z . F o r m E M Z , l e t z = ummvm ' so t h a t um i s t h e s t r i n g p r e c e d i n g m y and vm i s t h e s t r i n g f o l l o w i n g m. Let z C (M u A)

DEFINITION. We s h a l l s a y t h a t t h e s t r i n g z . i s m o t i v a t i o n a l l y c o n s i s t e n t , i f f o r e v e r y m € MZ w e have a ( u m ) & m + + whenever m E M , a ( vm ) d m whenever m E M , and for + m G M- w e have [ a ( u m ) a ( v m ) l G m . T h i s g e n e r a l scheme

must be somewhat s i m p l i f i e d , i n o r d e r to a l l o w some m e a n i n g f u l and i m p o r t a n t c o n c l u s i o n s . We s h a l l namely p r e s e n t theorems ( s e e Nowakowska 1973) c o n c e r n i n g s t r u c t u r a l p r o p e r t i e s of t h e c l a s s o f a l l m o t i v a t i o n a l l y c o n s i s t e n t s t r i n g s i n c a s e of some t y p e s of utterances. We s h a l l namely c o n s i d e r s t r i n g s of t h e form mu, where t

m i s a c e r t a i n promise ( s o t h a t m t M ) , and u i s a s t r i n g o f a c t i o n s f o l l o w i n g m y so t h a t a ( u ) = u.

809


Let

Lm be t h e c l a s s o f a l l s t r i n g s which s a t i s f y t h e

promise m , i . e . s u c h s t r i n g s u t h a t u6m. We s h a l l c o n s i d e r two t y p e s o f p r o m i s e s , t o be c a l l e d p o s i t i v e and n e g a t i v e . A p o s i t i v e promise i s c h a r a c t e r i z e d by t h e e x i s t e n c e o f a t i m e - h o r i z o n ( a n example h e r e would be "I w i l l c a l l him tomorrow"), w h i l e a neg a t i v e promise has t h e p r o p e r t y t h a t i t may o n l y b e b r o k e n , b u t n e v e r f u l f i l l e d ( e . g . "I s h a l l n e v e r do t h a t again'' )

.

I n c a s e o f p o s i t i v e promises w e assume the, f o l l o w i n g property :

If u c

Lm

uu'

c

L,

then

uu'

g

Lm.

What t h i s c o n d i t i o n a s s e r t s i s t h a t once a promise i s

f u l f i l l e d , no s u b s e q u e n t a c t i o n s may change t h i s f a c t . L e t us s a y t h a t t h e s t r i n g u G Lm d i s c h a r g e s t h e o b l i g -

a t i o n contained i n m , i f the c o n d i t i o n s u = ufu" w i t h 1u"i > 0 imply u ' k Lm. Thus, u f u l f i l l s t h e promise m , b u t no s u b s t r i n g o f i t has t h i s p r o p e r t y . F u r t h e r m o r e , l e t L' be t h e c l a s s of a l l s t r i n g s which m d i s c h a r g e t h e o b l i g a t i o n i n m , and l e t h(m) = max

1lul :

u E LA

1 .

The v a l u e h ( m ) w i l l be c a l l e d t h e m o t i v a t i o n a l h o r i z o n o r t h e promise m . One can p r o v e t h e f o l l o w i n g theorem ( s e e Nowakowska 1973): THEOREM. If t h e c l a s s of a l l n o n - p a r a s i t i c s t r i n g s i n

810

CHAPTER 6

is a Lm is L

c o n t e x t - f r e e l a n g u a g e , and h ( m ) i s f i n i t e , t h e n a c o n t e x t - f r e e language.

A s t r i n g u i s c a l l e d p a r a s i t i c i n L , i f uluu" & L f o r any s t r i n g s u l , u " , s o t h a t a p a r a s i t i c s t r i n g cannot be a p a r t of any s t r i n g i n L . F o r a p r e c i s e d e f i n i t i o n o f c o n t e x t - f r e e l a n g u a g e , s e e f o r i n s t a n c e Ginsburg

1968.

,

t h e s i t u a t i o n i s somewhat d i f f e r * e n t . Denote namely by Lm t h e c l a s s o f a l l s t r i n g s o f a c t i o n s i n L which b r e a k t h e promise m , and assume t h e following property:

For n e g a t i v e promises

If u have

*

G Lm,

then f o r a l l u',u"

u'uu" G

* Lm.

such t h a t u l u u "

G

L,

we

I n o t h e r words, i f t h e s t r i n g u b r e a k s t h e p r o m i s e , t h e n any s t r i n g which c o n t a i n s u as a s u b s t r i n g has a l s o t h e same p r o p e r t y ( o b s e r v e t h a t t h i s c o n d i t i o n i s d i f f e r e n t from t h e one f o r p o s i t i v e p r o m i s e s : i n t h e l a t t e r c a s e , o n l y c o n c a t e n a t i o n s from t h e r i g h t were used). We omit t h e d e t a i l s h e r e ; r o u g h l y , one i n t r o d u c e s f i r s t t h e c l a s s of a l l s h o r t e s t s t r i n g s which b r e a k t h e p r o m i s e . The theorem a s s e r t s t h a t i f t h e maximum l e n g t h * o f s u c h s t r i n g s i s f i n i t e , t h e n Lm i s a c o n t e x t - f r e e language. I n a somewhat l o o s e i n t e r p r e t a t i o n , p e r h a p s c l o s e r t o a metaphor t h a n t o bounds imposed by t e c h n i c a l i t i e s o f d e f i n i t i o n s , one may s a y t h a t t h e p r o p e y t i e s d e s c r i b e d by t h e s e theorems c o n s t i t u t e a f o r m a l r e p r e s e n t a t i o n


81 1

of t h e o b s e r v e d f a c t t h a t s o c i a l norms ( e s p e c i a l l y e t h i c a l o n e s ) a r e , as a r u l e , f o r m u l a t e d w i t h o u t r e f e r e n c e t o s p e c i f i c s i t u a . t i o n s which might s e r v e as c o n t e x t s of t h e i r a p p l i c a b i l i t y . Thus , t a k i n g r e l i g i o n as an example, a s i n i s d e f i n e d as a s p e c i f i c a c t , and i t s p r e s e n c e i n a s t r i n g of a c t i o n s d e t e r m i n e s t h e q u a l i f i c a t i o n of t h i s s t r i n g ( w i t h f e w e x c e p t i o n s ) r e g a r d l e s s of t h e p r e c e d i n g and s u b s e q u e n t a c t i o n s , i . e . r e g a r d l e s s o f t h e c o n t e x t i n which t h e s i n o c c u r s . Law may p e r h a p s b e t h o u g h t o f as somewhat more f l e x i b l e ,

i n t h e s e n s e t h a t i t o c c a s i o n a l l y p r o v i d e s c l a s s e s of c o n t e x t s i n which a g i v e n a c t i o n c o n s t i t u t e s a n o f f e n s e , and c l a s s e s of c o n t e x t s i n which i t i s n o t s o . T h i s amounts t o s p e c i f y i n g s t r i n g s o f a c t i o n s t o g e t h e r w i t h t h e s u r r o u n d i n g c i r c u m s t a n c e s as o f f e n s e s , r a t h e r t h a n i s o l a t e d a c t i o n s . Again, however, t h e p r e s e n c e of a s t r i n g c o n s t i t u t i n g a n o f f e n s e q u a l i f i e s t h e s t r i n g of a c t i o n s i n which t h e o f f e n s e i s embedded. The p r e c e d i n g o r s u b s e q u e n t a c t i o n s may s e r v e as a f a c t o r d e t e r m i n i n g t h e punishment, b u t as a r u l e , n o t f o r t h e q u a l i f i c a t i o n of t h e o f f e n s e as s u c h . The most s e r i o u s o f f e n s e s , s u c h as murder, r a p e , a r s o n , e t c . a r e q u a l i f i e d r e g a r d l e s s of t h e c o n t e x t i n which t h e y a p p e a r . I n t h i s s e n s e , one can s a y t h a t t h e above theorems r e f l e c t c e r t a i n gen e r a l r e g u l a r i t i e s i n t h e t r e a t m e n t o f t h e norms of s o c i a l o r d e r , namely t h e r e l a t i v e u n i v e r s a l i t y of cont e x t s i n which t h e y a p p l y . These r e s u l t s may t h e r e f o r e be of some i n t e r e s t f o r d e o n t i c l o g i c .

6 . STRUCTURE OF SETS OF TIME-EVENTS I n t h i s s e c t i o n we s h a l l a n a l y s e t h e s t r u c t u r e o f c l a s s es o f t i m e - e v e n t s , i n p a r t i c u l a r c l a s s e s of t h e form

CHAPTER 6

812

R ( u ) , i . e . . s e t s of t i m e - e v e n t s of a c t i o n s u.

caused b y a g i v e n s t r i n g

More s p e c i f i c a l l y , l e t u s c o n s i d e r some f i x e d f a m i l y F of s u b s e t s of t h e s t a t e s p a c e , s a y F = {El,. ' * Y E m j and a s s o c i a t e d w i t h them t i m e - e v e n t s . We s h a l l namely s a y t h a t t h e time-event ( E i , t ) o c c u r r e d , i f t h e s t a t e o f t h e system a t t i m e t i s i n t h e s e t Ei. We s h a l l be a n a l y s i n g t h e s e t R F ( u ) , d e f i n e d as t h e c l a s s o f a l l pairs (Ei,t),

E.

1

F , which o c c u r i f u i s p e r f o r m e d .

Let Horn u = ( E :

(E,t)

e

R F ( u ) f o r some t

1

s o t h a t Hom u i s t h e c l a s s o f a l l e v e n t s i n F which o c c u r a t some t i m e i f u i s performed. Using l i n g u i s t i c a n a l o g i e s , a c c o r d i n g t o which a s t r i n g o f a c t i o n s p l a y s t h e r o l e o f an e x p r e s s i o n and t h e outcomes p l a y t h e r o l e o f t h e meaning o f a n e x p r e s s i o n , t h e s i z e o f t h e s e t Hom u r e f l e c t s t h e llhomonymity" o f e x p r e s s i o n u: if t h i s s e t i s l a r g e , t h e n u i s more "ambiguous". I n a s i m i l a r way, f o r a f i x e d E h F we may d e f i n e TrEu = { t :

(E,t)

E

RF(u)l

and

Tr u

=

(t:

We have t h e n

(E,t)

E R F ( u ) f o r some E

i Fj

.


813

The s e t TrEu i s t h e “ t r a c e ” o f e v e n t E when u i s p e r formed, i . e . t h e s e t of a l l moments a t which E o c c u r s ( E may o c c u r more t h a n once as a r e s u l t o f u; t o g i v e a n example, u may be “ p a y i n g a s u b s c r i p t i o n “ , and E may b e t h e e v e n t “ r e c e i v i n g t h e s u b s c r i b e d j o u r n a l ” ) . C l e a r l y , t h e most i n t e r e s t i n g s e t s h e r e a r e Horn u and T r u. The f i r s t o f t h e s e d i s r e g a r d s t h e t e m p o r a l asE

p e c t s , and a1l.o.w~ comparison o f s t r i n g s o f a c t i o n s w i t h r e s p e c t t o t h e “ s c o p e ” o f t h e i r outcomes. The second s e t g i v e s t h e t e m p o r a l c h a r a c t e r i s t i c s o f t h e occurrences of t h e event E . The t i m e - t r a c e s o f e v e n t s a r e o f s p e c i a l i m p o r t a n c e f o r t h o s e everits which o c c u r more t h a n once as a r e s u l t

o f p e r f o r m i n g t h e s t r i n g of a c t i o n s u. Here one may d i s t i n g u i s h some c l a s s e s o f e v e n t s , u s i n g t h e p r o p e r t i e s o f t h e t r a c e and i t s l o c a t i o n w i t h r e s p e c t t o t h e moment o f t e r m i n a t i o n o f t h e s t r i n g u . Thus, i f

t h e n t h e f i r s t o c c u r r e n c e of t h e e v e n t E l i e s a f t e r t h e t e r m i n a t i o n o f t h e s t r i n g u. We may c a l l s u c h a n e v e n t E d e l a y e d . On t h e o t h e r hand, i f t h e l a s t o c c u r ence of t h e event E l i e s bef or e t h e t e r m i n a t i o n of t h e s t r i n g u. t h a t i s max

it: t

t TrEu’f C l u \ ,

t h e e v e n t E ma.y be c a l l e d t r a n s i e n t .

814

CHAPTER 6

I n t h e r e m a i n i n g c a s e , when f o r some t l , t 2 : i n T r Eu w e have t l < IuJ< t 2 , we s p e a k o f p e r s i s t e n t e v e n t s (cons e q u e n c e s of u)

.

These n o t i o n s d e s c r i b e o n l y t h e l o c a t i o n o f t h e s e t T r u w i t h r e s p e c t t o t h e moment of t e r m i n a t i o n o f u.

E

They do n o t , however , p r o v i d e any i n f o r m a t i o n about t h e s t r u c t u r e o f t h e t r a c e o f u. A s r e g a r d s t h e l a t t e r , t h e most i m p o r t a n t and i n t e r e s t i n g c o n c e p t i s t h a t o f

periodicity. S i n c e we d e a l h e r e m o s t l y w i t h f i n i t e s e t s , t h e n o t i o n of p e r i o d i c i t y has t o be d e f i n e d i n a s p e c i a l w a y . I n o t h e r words, t h e u s u a l d e f i n i t i o n o f p e r i o d i c i t y , as i n v a r i a n c e under some t r a n s l a t i o n s , has t o be appropr i a t e l y modified. We s h a l l p r e s e n t h e r e t h e r e l e v a n t concept and r e s u l t s from Nowakowska (1973). Given any s e t o f numbers B , l e t S t ( B ) d e n o t e i t s t r a n s l a t i o n by t, i . e . St(B) = { s :

s = t

-k

x, x

B].

We have t h e n St [S, ( B ) 1 = St +t (B). 2 1 1 2 A s e t B w i l l be c a l l e d p e r i o d i c , i f it i s r e p r e s e n t a b l e

as a u n i o n o f a c e r t a i n number o f t r a n s l a t i o n s o f a g i v e n s e t , c a l l e d c a r r i e r of p e r i o d i c i t y . F o r m a l l y , a s e t B of t i m e moments i s s a i d to be p e r i o d i c , i f t h e r e e x i s t s a s e t N , and two numbers, k > 1 and T 7 0 s u c h t h a t


81 5

provided t h a t max i t : t

c

N\

< min It:

t E s ~ ( N ), ~

I n s u c h a c a s e , t h e s e t N i s c a l l e d t h e c a r r i e r of per i o d i c i t y , and T i s c a l l e d t h e p e r i o d . We a l s o s h a l l s a y t h a t t h e s e t i s of t h e t y p e ( T , k ) . N a t u r a l l y , t h e s e t N i t s e l f may b e p e r i o d i c a c c o r d i n g t o t h e above d e f i n i t i o n . I n p a r t i c u l a r , any s e t c o n s i s t i n g o f j u s t two p o i n t s i s p e r i o d i c . It i s o f i n t e r e s t t o a n a l y s e c o n d i t i o n s under which

a u n i o n o f two p e r i o d i c s e t s i s a l s o p e r i o d i c . The t h r e e theorems below p r o v i d e s u f f i c i e n t c o n d i t i o n s under which a union o f p e r i o d i c s e t s i s p e r i o d i c . F o r t h e p r o o f s , s e e Nowakowska (1973). THEOREM. Assume t h a t t h e s e t s B1

and

B2 a r e p e r i o d i c ,

and -

(1) B1

and

B 2 have t h e same p e r i o d T ;

( 2 ) The c a r r i e r N N

of 1-

B1

Q c

>

Then t h e s e t B1

of B

i s a t r a n s l a t i o n of t h e c a r r i e r

2 2 0 , t h a t i s , N 2 = Sc(N1).

3 B2

i s p e r i o d i c , i f i n a d d i t i o n one

of t h e fol l owing c o n d i t i o n s holds:

( a ) max B1 4 min E and t h e s e t s B1,B2 a r e o f t h e 2’ same t y p e ( T , k ) ; ( b ) max B 7 min B and t h e number c i s a m u l t i p l e of 1/ 2

816

CHAPTER 6

t h e common p e r i o d T .

B2 i s B1, t h e p e r i o d e q u a l s c , and t h e number o f r e p e t i t i o n s i s 2 . I n c a s e ( a ) , t h e c a r r i e r o f t h e sum B1

I n c a s e ( b ) , t h e c a r r i e r i s N1,

t~

t h e p e r i o d r e m a i n s equ-

a l T , and t h e number o f r e p e t i t i o n s depends on t h e numbers kl,

k2 and c .

N e x t , we have a l s o t h e f o l l o w i n g THEOREM. N1

and

B2 w i t h c a r r i e r s

If t h e p e r i o d i c s e t s B1

N 2 a r e o f t h e same t y p e ( T , k )

t h e n t h e s m B1

ir

and

B2 i s p e r i o d i c w i t h c a r r i e r N1 u N 2

and o f t y p e ( T , k ) ( i . e .

b o t h t h e p e r i o d and t h e number

of r e p e t i t i o n s a r e t h e same as f o r e a c h component). F i n a l l y , w e have a l s o t h e f o l l o w i n g t h e o r e m . THEOREM.

:i.f B1 and B2

and ( T 2 , k 2 ) ,

a r e periodic of types (Tl,kl)

w i t h c a r r i e r s N1

and

N2 such t h a t

i s an i n t e g e r m u l t i p l e of t h e p e r i o d T2 i . e . T1 = mT2 for some m;

the period

TI,

( b ) The number k l e q u a l s mk2;

( c ) the set N

*

= N 2 d N1

0

ST1(N1)

satisfies t h e condition

J

S2T (N1)LJ...;I S (m- 1)T (N1) 1


max N

*
-a ( k ) ) o f j o i n t a c t i o n s i s s a i d t o be i n e q u i l i b r i u m , i f f o r a l l i K and a l l s t r i n g s si, whenever t h e j o i n t s t r i n g v' = (i-1) a( k ) ) i s admissible, ( 2( 1 ) y . - * , s a(itl) t h e n 2 '2i s ' , where 2 and 2' a r e s t a t e s a t t a i n e d by s t r i n g s 1 and 1'.

,...,

J

I n t u i t i v e l y , a j o i n t s t r i n g of a c t i o n s i s i n e q u i l i b r i u m i f t h e f o l l o w i n g i s t r u e f o r every person: i f a pers o n knows t h a t a l l o t h e r s w i l l p e r f o r m a c t i o n s which a r e components o f 2 , t h e n he may o n l y l o s e ( i . e . a t t a i n somelihing l e s s p r e f e r r e d ) i f he c h o s e s any o t h e r s t r i n g o f a c t i o n s t h a n 2( i )

.

E q u i l i b r i u m i n a g i v e n s i t u a t i o n need n o t be u n i q u e . T h i s i s t r u e i n c a s e o f N-person games i n g e n e r a l ; i n o u r c a s e we have a l s o t h e f o l l o w i n g theorem.

CHAPTER 6

848

THEOREM. Suppose t h a t a l l members o f K form as S1-team,

and l e t G be t h e most p r e f e r r e d g o a l i n S ' . Then any s t r i n g of a c t i o n s which b r i n g s about t h i s g o a l i s i n eq u i 1ib r i um

.

It i s w o r t h w h i l e t o d i s c u s s h e r e t h e r e l a t i o n between t h e t h e o r y o f a c t i o n s o f t h i s c h a p t e r , b o t h of a s i n g l e

p e r s o n and o f groups of p e r s o n s , and t h e d e c i s i o n and game t h e o r y . In g r e a t e s t s i m p l i f i c a t i o n , t h e difference l i e s primar i l y i n t h e emphasis, which t h e t h e o r i e s of t h e p r e s e n t c h a p t e r p u t on t h e s t r u c t u r a l a s p e c t s of t h e s i t u a t i o n , as opposed t o t h e more " g l o b a l " a p p r o a c h e s of t h e c l a s s i c a l d e c i s i o n t h e o r y and t h e o r y o f games. L e t us c o n s i d e r f i r s t i n some d e t a i l t h e r e l a t i o n s b e t ween t h e t h e o r i e s i n q u e s t i o n i n c a s e o f a c t i o n s o f one p e r s o n . A l l known d e c i s i o n t h e o r i e s r e d u c e t h e s i t u a t i o n t o

e s s e n t i a l l y the following elements: t h e s e t of availa b l e o p t i o n s , o f which one and o n l y one m u s t be c h o s e n , t h e s e t o f s t a t e s of "Nature", r e p r e s e n t i n g t h e d e c i s i o n maker's i g n o r a n c e as t o t h e e x a c t s i t u a t i o n which he f a c e s ( i n extreme c a s e s , he knows t h e s i t u a t i o n e x a c t l y ) , s e t o f outcomes, and t h e a s s u m p t i o n s which allow us t o r e p r e s e n t t h e p r e f e r e n c e s of t h e d e c i s i o n maker among t h e outcomes. I n t h e s i m p l e s t c a s e , when t h e d e c i s i o n maker knows e x a c t l y t h e s i t u a t i o n , t h e o n l y u n c e r t a i n t y for h i m i s t h e p o s s i b l y s t o c h a s t i c laws t h a t connect t h e o p t i o n s and outcomes. I n t h i s c a s e , t h e problem l i e s i n


849

d e f i n i n g t h e u t i l i t y i n d e x s o t h a t t h e b e s t outcome w i l l be t h a t which y i e l d s t h e h i g h e s t e x p e c t e d u t i l i t y .

V a r i o u s t h e o r i e s s u g g e s t e d t h u s f a r ( o f which S E U , or S u b j e c t i v e Expected U t i l i t y , i s b e s t known) f a i l e d t o p r o v i d e d e s c r i p t i o n which would be a d e q u a t e i n a l l s i t u a t i o n s ; e a c h of them g i v e s good p r e d i c t i o n s i n some c l a s s e s o f s i t u a t i o n s , and f a i l s i n some o t h e r c l a s s e s . It would a p p e a r t h e r e f o r e , t h a t d i f f e r e n t c l a s s e s o f

s i t u a t i o n s m a y r e q u i r e d i f f e r e n t d e c i s i o n models. T h i s makes i t i m p o r t a n t to p r o v i d e a t h e o r y a l l o w i n g a t least a c l a s s i f i c a t i o n of various decision s i t u a t i o n s . The t h e o r y o f t h e p r e s e n t c h a p t e r , b y s t r e s s i n g t h e a n a l y s i s o f s t r u c t u r a l a s p e c t s of t h e s i t u a t i o n , may provide a solution here. In f a c t , t h e action theory p r e s e n t e d here may a l s o b e termed a t h e o r y o f pre-decisional situations. F i r s t l y , i n s t e a d o f j u s t assuming t h e a d e c i s i o n m a k e r f a c e s a set of o p t i o n s , t h e p r e s e n t t h e o r y a t t e m p t s t o e x p l i c a t e t h e s t r u c t u r e of t h i s s e t , by d i s t i n g u i s h i n g d i s c r e t e u n i t s o f b e h a v i o u r . O p t i o n s a r e namely r e p r e s e n t e d as s t r i n g s of elementary a c t i o n s , t h u s s t r e s s i n g t h e dynamic a s p e c t s o f d e c i s i o n making, and t h i s a l l o w s us to i n t r o d u c e and a n a l y s e s u c h n o t i o n s a s d e c i s i v e moments. Here t h e s t r u c t u r e i s i n d u c e d simply b y t h e f a c t t h a t two o p t i o n s , e a c h o f them bei n g a s t r i n g o f a c t i o n s , may have a common b e g i n n i n g , which a l l o w s t h e d e c i s i o n maker t o p o s t p o n e t h e c h o i c e between them. N e x t , t h e p r e s e n t t h e o r y e x p l a i n s i n some d e t a i l t h e

r e l a t i o n s between o p t i o n s ( s t r i n g s o f a c t i o n s ) and outcomes. The l a t t e r have now t h e form o f s t r i n g s of

CHAPTER 6

850

t i m e - e v e n t s , and a g o a l may be d e f i n e d as a d i s j u n c t i o n o f some c o n f i g u r a t i o n s of t h e l a t t e r . T h i s leads t o an a l g e b r a i c a n a l y s i s o f g o a l s , and t h e c e n t r a l f a c t o f t h e t h e o r y i s t h e isomorphism between t h e s t r u c t u r e o f t h e g o a l and t h e s t r u c t u r e o f t h e s e t o f s t r i n g s o f a c t i o n s which b r i n g about t h i s g o a l . Again, t h i s approach s t r e s s e s d i f f e r e n t a s p e c t s t h a n t h o s e i n " c l a s s i c a l " d e c i s i o n t h e o r y , by i n t r o d u c i n g s t r u c t u r a l a s p e c t s o f b o t h t h e o p t i o n s ( s t r i n g s of a c t i o n s ) and outcomes ( s t r i n g s o f e v e n t s ) . I n c a s e o f a c t i o n s o f many p e r s o n s , t h e d i f f e r e n c e s between t h e o r y o f games and t h e p r e s e n t a p p r o a c h a r e a g a i n s i m i l a r . What i s o f s p e c i a l i m p o r t a n c e i s t h e d i s t i n c t i o n between t h e n o t i o n s o f c o o p e r a t i o n , blocki n g , e t c . i n game t h e o r y and h e r e (where i n t h e l a t t e r c a s e t h e y a r e d e f i n e d t h r o u g h t h e s i t u a t i o n a l constr-v a i n t s , and n o t t h r o u g h u t i l i t i e s ) . To i l l u s t r a t e t h i s p o i n t , c o n s i d e r t h e game s u c h a s c h e s s . The a c t i o n s h e r e a r e c o n s e c u t i v e moves by White and B l a c k . Whether a p a r t i c u l a r move i n a g i v e n s i t u a t i o n i s a l l o w e d or n o t depends on t h e c o n f i g u r a t i o n of p i e c e s on t h e c h e s s b o a r d , and i n o r d e r f o r B l a c k ( s a y ) t o make a c e r t a i n move, i t i s n e c e s s a r y t h a t White moved b e f o r e i n some s p e c i a l way. The a c t i o n s o f White and B l a c k have t h e r e f o r e forms o f s t r i n g s

w1#w2#w3#.

..

and # b l # b 2 # b 3 . .

.

and are c o o p e r a t i v e ( s o t h a t t h e j o i n t s t r i n g wlblw 2 . . . i s a n a d m i s s i b l e game). T h i s s h o u l d n o t be c o n f u s e d w i t h t h e f a c t t h a t c h e s s i s a s t r i c t l y c o m p e t i t i v e game i n which t h e i n t e r e s t s o f p l a y e r s are opposed.


851

I n o t h e r words, t h e p r e f e r e n t i a l s t r u c t u r e o f t h e dec i s i o n s i t u a t i o n i s t r e a t e d s e p a r a t e l y from t h e a c t i o n a l s t r u c t u r e a s l o g i c a l l y i n d e p e n d e n t from i t , and i n t h i s c h a p t e r , t h e main S t r e s s i s p u t on t h e l a t t e r , b o t h i n t h e c a s e o f a c t i o n s y s t e m s f o r one and f o r many persons.

1 0 . P L A N N I N G ACTIONS AND GENERATIVE GRAMMARS The f o r m a l s y s t e m o f t h i s c h a p t e r may s e r v e as a b a s i s

f o r c o n s t r u c t i o n of f o r m a l t h e o r y o f p l a n n i n g and d e s i g n . The main i d e a c o n s i s t s h e r e of a p p l y i n g t h e n o t i o n s o f g e n e r a t i v e grammar, and g e n e r a t i n g s e n t e n c e s by t h e s e grammars. Suppose t h a t we a r e i n t e r e s t e d i n a t t a i n i n g a c e r t a i n g o a l G , descri.bed as a c o n f i g u r a t i o n o f t i m e - e v e n t s . I n such a c a s e , i f w e d e c i d e t o u s e s t r i n g s o f a c t i o n s w i t h a d m i s s i b i l i t y a t l e a s t b , we must choose a s t r i n g o f a c t i o n s from t h e s e t R i l ( G ) . These s e t s , f o r v a r y i n g G and b y form some l a n g u a g e s , and one may a t t e m p t t o

f i n d t h e g e n e r a t i n g grammars o f t h e s e l a n g u a g e s . To c o n s t r u c t a grammar, one n e e d s t o s p e c i f y i t s metaa l p h a b e t and p r o d u c t i o n r u l e s , s o t h a t t h e c l a s s o f a l l s t r i n g i n t e r m i n a l a l p h a b e t (which i n t h i s c a s e c o i n c i d e s w i t h t h e c l a s s o f a l l a c t i o n s ) which may be g e n e r a t ed u s i n g t h e s e r u l e s o f p r o d u c t i o n e q u a l s t h e c l a s s o f a l l a d m i s s i b l e s t r i n g s , i . e . t h e c o n s i d e r e d language of actions.

I n t h e c a s e o f t h e l a n g u a g e s o f t h e form R b l ( G ) , t h e t e r m i n a l a l p h a b e t c o i n c i d e s simply w i t h t h e c l a s s of

852

CHAPTER 6

a c t i o n s . The p r o c e s s of g e n e r a t i n g a c t i o n s t r i n g s from

RL1(G) i s , i n t h i s c a s e , t h e p r o c e s s o f s u c c e s s i v e s p e c i f i c a t i o n of t h e p l a n o f a t t a i n i n g G . The r o l e o f m e t a a l p h a b e t i s p l a y e d by t h e l a b e l s o f d e s c r i p t i o n s o f a c t i o n s o r c l a s s e s o f a c t i o n s . The p r o d u c t i o n r u l e s

depend, o f c o u r s e , on s p e c i f i c f e a t u r e s o f t h e s i t u a t i o n which i s t o be a n a l y s e d . On most c a s e s , g e n e r a t h g s t r i n g s of a c t i o n s may b e i n t e r p r e t e d simply as t h e s u c c e s s i v e s t a d i a of e l a b o r a t i o n of t h e p l a n . F o r m a l l y , t o e a c h element o f t h e a c t i o n s p a c e A one may a s s i g n some l a b e l , which c h a r a c t e r i z e s some f e a t u r e s of t h i s a c t i o n ( d i f f e r e n t a c t i o n s may be a s s i g n e d t h e same l a b e l ) . L a b e l s c o r r e s p o n d t o v a r i o u s l e v e l s o f g e n e r a l i t y . F o r i n s t a n c e many o f t h e a c t i o n s which are performed i n t h e k i t c h e n i n c o n n e c t i o n w i t h p r e p a r a t i o n o f meals may b e termed g e n e r a l l y , e . g . " f r y i n g " , "cooki n g " , e t c . The p l a n may be i d e n t i f i e d w i t h a s u b s e t o f t h e s e t o f a l l l a b e l s , c o n s i s t i n g o f t h o s e which a r e a s s i g n e d t o a c t i o n s i n R-'(G). The s m a l l e r i s t h i s b subset, the l e s s general i s t h e plan. In the particular c a s e , when a one-element

s e t i s chosen ( i . e . a s i n g l e

s t r i n g o f a c t i o n s ) , t h e p l a n i s most s p e c i f i c , i . e .

it

p r e s c r i b e s which a c t i o n s a r e t o be t a k e n , and i n which order. The p r o d u c t i o n r u l e s , i.e. t h e r u l e s which a l l o w r e p l a -

cement o f some l a b e l s by l a b e l s o f more s p e c i f i c s t r k n g s o f a c t i o n s , may be o b t a i n e d by u s i n g t h e r u l e s o f mo-

t i v a t i o n a l consistency:

t h e l a b e l c may be r e p l a c e d b y t h e composite l a b e l b l b 2 , . ,b k ' i f i n e a c h c o n t e x t , two s t r i n g s which d i f f e r o n l y by t h e f a c t t h a t one con-

..

t a i n s l a b e l a , and t h e o t h e r - l a b e l b l b 2 . . . b k , a r e both consistent o r both inconsistent motivationally.


853

One may e x p e c t t o o b t a i n some s p e c i f i c r e s u l t s here i n a s p e c i a l c a s e of s t r i n g s which a r e t o produce some o b j e c t s o u t of t h e i r c o n s t i t u e n t p a r t s . The g o a l i s t o c r e a t e a n o b j e c t w i t h some d e s i r e d p r o p e r t i e s , and i n t h i s case w i l l one may e x p e c t t h a t t h e s e t R-'(G) b b e r e l a t e d i n some way t o t h e s t r u c t u r e of t h e o b j e c t . The c r u c i a l p o i n t h e r e i s c o n n e c t e d w i t h l a n g u a g e s of a c t i o n s of many p e r s o n s , and e s p e c i a l l y t h e p o s s i b i l i t y o f p e r f o r m i n g some a c t i o n s i n p a r a l l e l , and n o t s e r i a l l y . The most i n t e r e s t i n g problem h e r e i s t h a t of opera t i o n s on p l a n s , s o as t o o b t a i n a p l a n which would l e a d t o c r e a t i o n of a p a i r o f o b j e c t s c r e a t e d b y separate plans. One may a l s o e x p e c t t h a t s u c h problems a r e c l o s e l y r e l a t e d t o problems of a r t i f i c i a l i n t e l l i g e n c e , and a l s o a r e u n d e r l y i n g t h e usage o f language and t h o u g h t .

11. AN APPLICATION TO O R G A N I Z A T I O N THEORY

The system of a c t i o n s from t h e p r e c e d i n g s e c t i o n s may be e a s i l y a p p l i e d t o a d e s c r i p t i o n of o p e r a t i o n of org a n i z a t i o n s , u n d e r s t o o d as i n s t i t u t i o n a l i z e d s e t s of p e r s o n s . Formally t h e r e f o r e i t i s t h e same as a n a c t i o n system of many p e r s o n s . One m a y , however, make t h e s i t u a t i o n more s p e c i f i c b y e x h i b i t i n g some s p e c i a l f e a t u r e s of a n o r g a n i z a t i o n , as opposed t o an a c t i o n system o f some o t h e r groups o f p e r s o n s . We s h a l l namel y c o n s i d e r a s p e c i a l c a s e , of two o r g a n i z a t i o n s of which one i s d i r e c t i n g t h e o p e r a t i o n s of a n o t h e r . We s h a l l t h e r e f o r e c o n s i d e r a system c o n s i s t i n g o f a pair

854

CHAPTER 6

p2)

(11.1)

where P1 w i l l be i n t e r p r e t e d as t h e d i r e c t i n g o r g a n i z a t i o n , and P2 -- as t h e o r g a n i z a t i o n which i s b e i n g d i rected. Now, under t h i s i n t e r p r e t a t i o n , i t may be t h a t P1 and P a r e d e p a r t m e n t s o f t h e same i n s t i t u t i o n , e t c . 2

Although e a c h o f t h e components P1 and P2 forms t y p i c a l l y a complex m u l t i p e r s o n s y s t e m , t o d e s c r i b e t h e e s s e n t i a l f e a t u r e s o f d i r e c t i n g , some of t h e i n t r i c a c i e s of i n t e r n a l s t r u c t u r e s o f P1 and P 2 may be d i s r e garded. F i r s t l y , we assume t h a t P1 i s a d e c i s i o n making body, which s e t s t h e p o l i c i e s , t o be c a r r i e d o u t b y P Thus, 2' P1 w i l l have a s p e c i a l a c t i o n a l p h a b e t , namely i t s a c t -

ions will be d e c i s i o n s , o r more s p e c i f i c a l l y , d i r e c t i o n s which a r e t o be c a r r i e d o u t b y P 2 ( o t h e r p o s s i b l e a c t i o n s o f P1 may be d i s r e g a r d e d f o r t h e p r e s e n t a n a l y s i s ) . A s r e g a r d s P 2 , it i s t o be i n t e r p r e t e d as a n a c t i o n s y s t e m ( a s a r u l e , o f many p e r s o n s ) , which i s to f u l -

f i l l t h e d i r e c t i o n s of P

1'

T h i s a c t i o n s y s t e m i s cha-

r a c t e r i z e d by t h e u s u a l components, i . e . s e t o f p e r s o n s , s e t o f s t a t e s , s e t o f a c t i o n s , and a d m i s s i b i l i t y and t r a n s i t i o n f u n c t i o n s . Thus, we may summarize i t b y r e p r e s e n t i n g P as 2

P2 =(K,

LA, O i d S l , E, R)

(11.2)

where K i s t h e s e t o f p e r s o n s i n P each L d i s t h e 2' c l a s s o f a l l k - s t r i n g s o f a c t i o n s which a r e a d m i s s i b l e


855

i n d e g r e e a t l e a s t g c , E i s t h e s e t o f r e s u l t s , and R i s t h e r e l a t i o n c o n n e c t i n g k - s t r i n g s from L w i t h outpc comes from E . Now, t h e i n s t r u c t i o n s o f P1 t o P2 may be p a r t i t i o n e d i n t o t h r e e broad c a t e g o r i e s :

--

t h o s e which s p e c i f y t h e g o a l t o be a t t a i n e d by P2, i . e . t h e y s p e c i f y some c o n f i g u r a t i o n s o f t i m e - e v e n t s ;

--

t h o s e which s p e c i f y s t r i n g s o f a c t i o n s t o be perform-

ed by P 2 ;

--

t h o s e which modify t h e a d m i s s i b i l i t i e s o f s t r i n g s o f a c t i o n s from P 2 . N a t u r a l l y , a n i n s t r u c t i o n may be “mixed” and i n v o l v e two or more o f t h e above components. F o r t h e p u r p o s e o f a n a l y s i s , however, i t i s most c o n v e n i e n t t o i n t r o duce t h e c o n c e p t o f ” e l e m e n t a r y ” i n s t r u c t i o n , and a s s u me t h a t i t may be o n l y of; one o f t h e above t y p e s , w i t h mixed i n s t r u c t i o n s r e p r e s e n t a b l e as s t r i n g s o f elementary instructions. A c c o r d i n g l y , we i n t e r p r e t P1 as a system

where D l Y D2 and D a r e some d i s j o i n t s e t s , i n t e r p r e t e d 3 as s e t s o f e l e m e n t a r y i n s t r u c t i o n s o f t h e above t h r e e types. We s h a l l d e n o t e D = D

1

D 2 V D

3

856

CHAPTER 6

*

and d e n o t e by D t h e monoid o v e r t h e a l p h a b e t Du?#), i . e . the c l a s s of a l l f i n i t e s t r i n g s consisting of i n s t r u c t i o n s from D and p a u s e s # . Now, e a c h element o f D s p e c i f i e s a g o a l G t o be a t t i n 1 ed by P2. C o n s e q u e n t l y , we may a s s i g n t o e a c h element d f D1 t h e s e t C ( d l ) d e f i n e d as R - ' ( G ) , where G i s 1 t h e g o a l s p e c i f i e d by d l . I n o t h e r words, t h e i n s t r u c t i o n d l may b e i n t e r p r e t e d as "perform a s t r i n g o f a c t i o n s from t h e s e t R-'(G)". N e x t , t o e a c h element d2 6 D2 we may a s s i g n a s e t C ( d ) 2 2 of s t r i n g s of a c t i o n s , o f which one i s t o be performed. I n extreme c a s e s , C 2 ( d 2 ) c o n t a i n s j u s t one s t r i n g , i f t h e i n s t r u c t i o n d2 i s s p e c i f i c . F i n a l l y , e l e m e n t s o f D are i n s t r u c t i o n s which modify 3 the admissibilities of s t r i n g s of actions of P This 2. means f o r m a l l y t h a t e a c h s u c h i n s t r u c t i o n s changes t h e f a m i l y of s e t s L, i n t o some o t h e r f a m i l y L b , s o t h a t f o r any l e v e l of a d m i s s i b i l i t y o( , t h e c l a s s o f s t r i n g s a d m i s s i b l e w i t h d e g r e e a t l e a s t d changes from L A t o a new s e t L k . To g i v e a n example, a n i n s t r u c t i o n from D may i n v o l v e 3 a new a l l o c a t i o n o f f u n d s , changes o f o r g a n i z a t i o n a l s t r u c t u r e o f P2, e t c . s o t h a t s t r i n g s o f a c t i o n s t h a t were p r e v i o u s l y i n a d m i s s i b l e ( b e c a u s e , s a y , o f b u d g e t a r y c o n s t r a i n t s ) become now a d m i s s i b l e . C o n s i d e r i n g f o r s i m p l i c i t y o n l y t h e i n s t r u c t i o n s from D1 and D 2 ( p e r t a i n i n g t o some s p e c i f i c p e r i o d o f a c t i v i t y o f P2), one may a s s i g n t o a composite i n s t (n) of r u n c t i o n v which i n v o l v e s e l e m e n t s d l(1),..., d 1 D and e l e m e n t s d 2(1),.. , d:m) o f D2, t h e s e t of s t r i n g s 1

.

FORMAL THEORY OF ACTlONS

857

of actions

m

n

(11.4)

If Q ( v ) n Ld # 0

(11.5)

w e s a y t h a t v i s s a t i s f i a b l e on t h e l e v e l o f a d m i s s i b i l i t y o ( . The l a r g e s t o< f o r which ( 1 1 . 5 ) h o l d s , i . e . dV = s u p i d :

Q W n L&

z

m

(11.6)

i s c a l l e d t h e l e v e l a t which v i s s a t i s f i a b l e .

N a t u r a l l y , s i n c e t h e l a n g u a g e s L, a r e m o n o t o n i c a l l y d e c r e a s i n g w i t h i n c r e a s e o f P(, we have THEOREM. I f v i s s a t i s f i a b l e on a l e v e l d , i t i s a l s o s a t i s f i a b l e on any lower l e v e l o f a d m i s s i b i l i t y .

Two composite i n s t r u c t i o n s v and v ' f o r which Q ( v ) = Q ( v l ) a r e e q u i v a l e n t . I f Q ( v ) C Q ( v ' ) , t h e n v ' i s less s p e c i f i c , o r l e s s r e s t r i c t i v e than v. Alternatively, w e may s a y t h a t v imposes a d d i t i o n a l c o n s t r a i n t s , r e l a t i v e t o v ' , or t h a t v ' r e l a x e s t h e c o n s t r a i n t s , r e l a t i v e t o v . F i n a l l y , i f n e i t h e r o f t h e s e t s Q ( v ) and Q ( v f ) i s conta.ined i n t h e o t h e r , t h e n v and v ' a r e c a l l e d independent. I n t h i s c o n c e p t u a l framework one c o u l d f o r m u l a t e v a r i o u s q u e s t i o n s i n o r g a n i z a t i o n t h e o r y , a n a t t e m p t t o answer them, For i n s t a n c e , imposing some c r i t e r i a o f o p t i m a l i t y , which t y p e s o f i n s t r u c t i o n s a r e most e f f e c t i v e ?

858

CHAPTER 6

What s h o u l d be t h e p r o p o r t i o n o f t a c t i c a l i n s t r u c t i o n s

t o a t t a i n a g i v e n g o a l , where a t a c t i c a l i n s t r u c t i o n i s such t h a t modifies t h e c l a s s o f g o a l s , but otherwise d o e s n o t r e s t r i c t t h e c l a s s o f means t o a t t a i n i t ? What s h o u l d b e t h e p r o p o r t i o n of t a c t i c a l i n s t r u c t i o n s i n an i n s t i t u t i o n w i t h given l e v e l o f complexity o f s t r u c t u r e? If we assume t h a t P c o n s i s t s , i n f a c t , o f s e v e r a l 2 o r g a n i z a t i o n s , a l l s u b j e c t t o P1, a n o t h e r c l a s s o f problems c o n c e r n s c o n c e p t s s u c h as t h e i r h i e r a c h i z a t i o n , e q u i v a l e n c e from t h e p o i n t o f view o f some g o a l s , e t c . One c o u l d c o n s i d e r here some o p e r a t i o n s on o r g a n i z a t i o n s , s u c h as j o i n i n g them, e t c .

F i n a l l y , s t i l l a n o t h e r p o s s i b l e i n t e r p r e t a t i o n o f cons i d e r a t i o n s o f t h i s s e c t i o n i s o b t a i n e d as f o l l o w s . The s e t of a d m i s s i b l e s t r i n g s o f i n s t r u c t i o n s forms a * c e r t a n i n language ( a s u b s e t o f t h e monoid D ) ; t h e " a l p h a b e t " of t h i s language a r e s i m p l y i n s t r u c t i o n s . "Words" a r e h e r e s t r i n g s of i n s t r u c t i o n s , and t o e a c h word t h e r e i s a s s i g n e d s e t o f t h e form Q ( v ) d e f i n e d b y (11.4), which may t h e r e f o r e be regarded as "meaning" o f v . These "meanings" have, i n t u r n , a l s o a l i n g u i s t i c s t r u c t u r e , b e i n g s e t s of s t r i n g s ( o f j o i n t a c t i o n s ) . I n t h i s sense, instructions define a c e r t a i n abstract l a n g u a g e , hence t h e y t h e m s e l v e s are e l e m e n t s o f metal a n g u a g e w i t h r e s p e c t t o l a n g u a g e s Q ( v ) . I n some s i m p l i f i c a t i o n one may s a y t h a t t h e language o f i n s t r u c t i o n s i s a meta-language w i t h r e s p e c t t o a c t i o n l a n guages which a r e c o n s i s t e n t w i t h t h e s e i n s t r u c t i o n s . A l t e r n a t i v e l y , l a n g u a g e s Q ( v ) c o n s t t i t u t e t h e semant i c s for t h e language o f i n s t r u c t i o n s .


859

1 2 . ETHICAL VALUATIONS

I n t h i s s e c t i o n , we s h a l l c o n s i d e r t h e problem o f e t h i c a l v a l u a t i o n s o f outcomes o f a c t i o n s , and a l s o t h e a c t i o n s t h e m s e l v e s . T h i s , n a t u r a l l y , r e q u i r e s an e x t e n s i o n o f t h e f o r m a l s y s t e m , s o as t o i n c l u d e t h e n o t i o n o f e t h i c a l v a l u e s o f a c t i o n s a n d / o r outcomes. Incidentally, despite the f a c t that the considerations w i l l c o n c e r n e t h i c a l v a l u e s , t h e same a n a l y s i s may b e

a p p l i e d t o many o t h e r v a l u a t i o n s , e . g . a e s t h e t i c , e t c . E t h i c a l v a l u a t i o n s a r e one o f t h e major c o n s t r a i n t s on a c t i o n s l e a d i n g t o a g i v e n g o a l . These c o n s t r a i n t s a r e o f g r e a t i n t e r e s t , e s p e c i a l l y i n group d e c i s i o n s i n l a r g e s y s t e m s , under c o l l e c t i v e r e s p o n s i b i l i t y . Moreover, they are closely r e l a t e d t o t h e notions dealing w i t h a l l o c a t i o n s o f goods i n t h e s o c i e t y , t o be a n a l y s ed i n n e x t s e c t i o n . One o f t h e main p r o b l e m s , which w i l l be a n a l y s e d i n t h e present section, i s that o f a j o i n t o v e r a l l evaluation of a s t r i n g of actions, i f t h i s s t r i n g brings outcomes o f which some have p o s i t i v e , and some have negative e t h i c a l value. I n a b s t r a c t f o r m u l a t i o n , t h e problem i s as f o l l o w s . The a c t i v i t y ( o f a p e r s o n X , t o be e v a l u a t e d , s a y ) , b r i n g s a c e r t a i n number o f r e s u l t s ; n e g l e c t i n g f o r s i m p l i c i t y t h e t e m p o r a l a s p e c t s , suppose t h a t t h e e t h i c a l v a l u e s o f t h e s e outcomes a r e e l , e 2 , . , en

..

.

Such a f o r m u l a t i o n p r e s u p p o s e s t h a t e t h i c a l v a l u e s a r e r e p r e s e n t a b l e n u m e r i c a l l y ; we s h a l l assume, i n f a c t t h a t t h e s e v a l u e s r e p r e s e n t measurement on a r a t i o

CHAPTER 6

860

s c a l e ( f o r a d i s c u s s i o n o f measurement problems i n c o n n e c t i o n w i t h e t h i c a l v a l u a t i o n s , s e e Nowakowska 1 9 7 3 ) .

.

We assume t h a t some of t h e v a l u e s e l,.. ,en are p o s i t i -

ve and some a r e n e g a t i v e , Moreover, we assume t h a t t h e i n d i c e s correspond t o t h e o r d e r o f occurrences o f t h e r e s p e c t i v e e v e n t s , so t h a t t h e event valued e l i s t h e e a r l i e s t , and t h e e v e n t v a l u e d en i s t h e l a t e s t . If V d e n o t e s t h e s e t o f a l l outcomes which e n t e r t h e

e v a l u a t i o n , t h e n t h e o b j e c t is t o a s s i g n t o V a v a l u e , say E(V), representing the overall ethical valuation o f outcomes i n V . We may r e q u i r e h e r e t h e f o l l o w i n g postulates. POSTULATE 1. The v a l u e E ( V ) depends o n l y on t h e v a l u e s e ) f o r some funce l , . . . , e n’ s o t h a t E ( V ) = f ( e l , n tion f.

...,

POSTULATE 2 . The f u n c t i o n f i s monotone i n c r e a s i n g w i t h r e s p e c t t o each o f i t s arguments, i . e . f o r every i, yei,l,ei+l,. ,en whenever e . e ’ then f o r a l l el,. 1 iy we have

B.

The r e l a t i o n * > i s c l e a r l y t r a n s i t i v e , b u t i n g e n e r a l

874

CHAPTER 6

i t need n o t be c o n n e c t e d , s o t h a t two d e s c r i p t i o n s may n o t be comparable. We have h e r e t h e f o l l o w i n g t h e orem THEOREM.

If A * ,> B *

f o r every B

* , then A *

*.

= S

For t h e p r o o f suppose f i r s t t h a t A * $ S*. We t h e n show

*

*.

t h a t one can f i n d B which i s b e t t e r t h a n A Indeed, * * * * * * * l e t p E S - A , and l e t B = ( A + p ) Then A $c B * * * and B c S , s i n c e p k S Consequently, A S $

B

*

0

* , while A*

S

*.

fi

(Sf)'

= B

*

*n

( S )'

.*

= @,

which

*

*

means t h a t B" .> A " .

*

*

Suppose now t h a t A $ S , and l e t q e A - S so that * * * * 2 q (S")'. Define B = ( A - q ) Then B C A , hence * * * c * * * * B 0 (S*)'$ A 0 (S ) , while A A S = B n S , * * which a g a i n means t h a t B .> A

.

.

*

*

'*.

In general, i f A # S t h e n t h e symmetric d i f f e r e n c e * * * * * A Y S = (A - S ) U (S - A ) i s n o t empty, and one I

Y

can combine t h e above two c a s e s t o c o n s t r u c t B" which * i s b e t t e r than A according t o the r e l a t i o n This completes t h e p r o o f .

>.

Thus, t h e theorem a s s e r t s t h a t t h e u n i v e r s a l l y b e s t A must simply c o i n c i d e w i t h S

*

.

S i m i l a r l y , we have t h e p r o p e r t y t h a t whenever B

*

*

*

* c

*

+> A

*

f o r every B , t h e n A = ( S ) , s o t h a t t h e worst desc r i p t i o n must c o n s i s t o n l y o f s e n t e n c e s which a r e i n t h e * complement o f S

.

It i s also w o r t h w h i l e t o mention t h e f o l l o w i n g i n t e r -

p r e t a t i o n o f t h e a g g r e g a t i o n problem. Suppose t h a t we e v a l u a t e N e v e n t s a c c o r d i n g t o K c r i t e r i a , and t h a t

875


t h e s e e v a l u a t i o n s ( d i s r e g a r d i n g t h e p e r s o n s who p e r f o r m e v a l u a t i o n s and t h e o c c a s i o n s ) a l l o w u s t o o r d e r t h e e v e n t s w i t h r e s p e c t t o e a c h c r i t e r i o n . L e t a 3 t h e c o n d i t i o n R k ( a , b ) > q h o l d s i f and o n l y i f R . ( a , b ) 7 q . Here q i s a measure J o f t h e " s t r e n g t h " o f t h e concordance r e l a t i o n .

For t h e a g g r e g a t i o n p r o c e s s , we may r e q u i r e t h e f o l l o w ing postulate. POSTULATE. If f o r a l l k , j t h e r e l a t i o n s R k and R

j

are

c o n c o r d a n t , t h e n t h e o v e r a l l r e l a t i o n R must be concordant w i t h every Rk. The c o n s t a n t q i n t h e l a t t e r concordance i s t h e maximum of t h e c o r r e s p o n d i n g c o n s t a n t s o f concordances b e t ween t h e p a i r s o f r e l a t i o n s R k ,

Rj.

This p o s t u l a t e i s a n a t u r a l requirement, corresponding i n some s e n s e t o t h e p o s t u l a t e o f s o v e r e i g n t y o f Arrow. L e t us f i n a l l y c o n s i d e r t h e s i t u a t i o n of e v a l u a t i o n s by a group o f p e r s o n s i n c a s e when n o t a l l c r i t e r i a a r e

878

CHAPTER 6

a c c e p t e d by a l l members o f t h e g r o u p . With r e g a r d s t o a f i x e d p e r s o n p , h i s s t a t e w i t h resp e c t t o t h e a c c e p t a n c e o r r e j e c t i o n o f c r i t e r i a may be d e s c r i b e d by a v e c t o r

where r

i s o o r 1 depending whether person p r e j e c t s kp o r a c c e p t s t h e k-th c r i t e r i o n .

T h u s , t h e g r o u p o f p e r s o n s w i l l b e d e s c r i b e d by t h e matrix

r

9

=

r

21 K 1

r r

12

...

22

K2

...

w h e r e e a c h row c o r r e s p o n d s t o a c r i t e r i o n , a n d e a c h column -- t o a p e r s o n . N a t u r a l l y , when o n e s p e a k s o f a c c e p t a n c e o r r e j e c t i o n of a c r i t e r i o n , one e x p e c t s some d e g r e e o f " s i m i l a r i t y " b e t w e e n members o f t h e g r o u p ( t h i s d o e s n o t mean a g r e e m e n t when t h e s e c r i t e r i a

are a p p l i e d t o any p a r t i c u l a r o b j e c t ) . T h i s c o n d i t i o n means t h a t columns o f t h e m a t r i x Q o u g h t t o h e s i m i l a r i n some s u f f i c i e n t l y h i g h d e g r e e . One o f t h e p o s s i b l e i n d i c e s o f s i m i l a r i t y may h e des c r i b e d as f o l l o w s . L e t (

,

d u c t o f two v e c t o r s , s o t h a t

) d e n o t e t h e s c a l a r pro-

819


...,

Moreover, l e t 1 d e n o t e t h e v e c t o r (l,l, 1). I f now z and z a r e two column v e c t o r s of m a t r i x Q, t h e n i j we may p u t

which i s simply e q u a l t o t h e number o f matching comp o n e n t s i n v e c t o r s z and z i

5

.

A s a c o n s i s t e n c y measure one may now t a k e t h e v a l u e

F o r example, for t h e m a t r i x

C(Q)

=

(3+1+2+3+4+2t1+2+3+1+0+1+3+2t3)

~

=

31

4.6-5 ( t h e m a x i m a l v a l u e o f t h e i n d e x i s 1, when a l l columns

are i d e n t i c a l ) . With r e s p e c t t o a g i v e n c r i t e r i o n , t h e s t a t e o f t h e

group i s d e s c r i b e d by t h e row v e c t o r o f m a t r i x Q. We

880

CHAPTER 6

m a y i n t r o d u c e a p r o c e s s o f change o f t h i s v e c t o r . One

can namely e x p e c t t h a t t h e group w i l l atteinpt t o a d j u s t e i t h e r t h e c h o i c e o f c r i t e r i a , o r even i t c o m p o s i t i o n ( b y a c c e p t i n g new members o r r e j e c t i n g old o n e s ) , s o

as t o have C(Q) a s c l o s e t o 1 as p o s s i b l e . F o r f o r m a l d e s c r i p t i o n , we s h a l l d e f i n e a Markov c h a i n Xt, t = 0,1, , w i t h s t a t e s 0,1, P , where X = r t means t h a t a t t i m e t , t h e v e c t o r c o r r e s p o n d i n g t o a g i v e n f i x e d row ( s a y k - t h ) has r c o o r d i n a t e s e q u a l

...

...,

1 and K-r c o o r d i n a t e s e q u a l 0 .

The c h o i c e o f t r a n s i t i o n p r o b a b i l i t y w i l l be based on t h e f o l l o w i n g i n t u i t i o n . I n c a s e when t h e s t a t e i s 0 o r P , t h e whole group r e j e c t s o r t h e whole group a c c e p t s t h e k - t h c r i t e r i o n . Then t h e r e i s a ( s m a l l ) chance t h a t one p e r s o n w i l l change h i s mind, s o t h a t

P0 , l =

pP ,P-1

= b,

and o f c o u r s e , P o o = 1-a, P p p = 1-b. For 1

c

3

c' f

cs.

T h i s means t h a t i f s has some goods c , t h e n h e a l s o

has a l l goods which he p r e f e r s t o c . T h e main n o t i o n h e r e i s t h a t o f a d m i s s i b i l i t y , or Pare-

t o o p t i m a l i t y . T h i s means t h a t t h e r e i s no a l l o c a t i o n which would be b e t t e r (or a t l e a s t as good) f o r a l l p e r s o n s . F o r m a l l y , an a l l o c a t i o n f i s a d m i s s i b l e , i f t h e r e e x i s t s no a l l o c a t i o n f ' s u c h t h a t

where C' a r e s e t s o f goods a s s i g n e d t o p e r s o n s under S allocation f ' . The main problem o f t h e o r y o f freedom l i e s i n d e t e r m i n i n g t h e c l a s s o f a l l o c a t i o n s which are f e a s i b l e . I n t u i t i -

v e l y , f e a s i b l e a l l o c a t i o n i s such

t h a t may be a t t a i n e d


891

by some a l l o w a b l e means, a n d may a l s o be r e t a i n e d i n

t h e s o c i e t y . N a t u r a l l y , t h e s p e c i a l i s t s may d i f f e r w i d e l y as t o f e a s i b i l i t y o f some a l l o c a t i o n s w h i c h a r e only contemplated. Assuming, h o w e v e r , t h a t o n e may d e t e r m i n e ( p o s s i b l y i n

a f u z z y way) t h e c l a s s o f a l l f e a s i b l e a l l o c a t i o n s , o n e c a n t h e n t r y t o f i n d a n optimum among $hem ( s a y , an admissible a l l o c a t i o n , e t c . ) . Here t h e p r o b l e m a p p e a r s t o be c l o s e l y r e l a t e d t o t h e p r o b l e m o f Arrow o f s o c i a l w e l f a r e f u n c t i o n . However, two f a c t o r s d i s t i n g u i s h i t f r o m t h e A r r o w ' s s e t u p , a n d t h e r e f o r e o f f e r some hope o f a v o i d i n g t h e I m p o s s i b i l i t y Theorem. One o f them i s t h e f a c t t h a t o n e a s s u m e s o n l y p a r t i a l o r d e r i n g o f goods ( s a y , c u l t u r a l g o o d s may b e n o t c o m p a r a b l e t o e d u c a t i o n a l g o o d s , e t c . ) ,

and s e c o n d l y , one l o o k s f o r a s o l u t i o n o n ly i n t h e c l a s s o f a l l f e a s i b l e s o l u t i o n s ( t h e a s s e r t i o n of Arrow Theorem i s known t o be f a l s e i n c a s e when t h e p r e f e r e n c e s of i n d i v i d u a l persons a r e a p p r o p r i a t e l y r e s t r i c t e d ) .

1 4 . 4 . A l i e n a t i o n and dynamics o f s o c i a l change I n t h i s s e c t i o n we s h a l l o u t l i n e t h e main i d e a s o f t h e f o r m a l t h e o r y o f a l i e n a t i o n s u g g e s t e d b y Nowakowska ( 1 9 7 7 a ) , a c c o r d i n g t o which a l i e n a t i o n i s a s t a t e o f b e i n g c o n s c i o u s o f t h e f a c t t h a t o n e i s d e n i e d some goods w h i c h o n e e x p e c t s a n d h a s t h e r i g h t t o have bec a u s e o f t h e work o n e p e r f o r m s , or b e c a u s e o f o n e ' s social status, etc. C o n s i d e r a f r a g m e n t o f t h e s y s t e m from t h e p r e c e d i n g

892

CHAPTER 6

s e c t i o n s , c o n s i s t i n g o f s o c i e t y S, goods C , a l l o c a t i o n s f , and p r e f e r e n c e s 7 F o r s i m p l i c i t y , assume t h a t C / S c o n t a i n s o n l y b i n a r y goods, and c o n s e q u e n t l y , e v e r y

.

a l l o c a t i o n ( a c t u a l or h y p o t h e t i c a l ) assumes o n l y t h e v a l u e s 0 and 1. Two b a s i c a d d i t i o n a l c o n c e p t s w i l l be ( a ) a d m i s s i b i l i t y function a ( t , s , c ) , representing the degree, t o which g i v i n g goods c t o p e r s o n s a t t i m e t i s admissi b l e , and ( b ) t h e f o r c e s F ( s , f , f ' ) and G ( s , f , f ' ) which t h e p e r s o n s i s c a p a b l e , given t h e p r e s e n t a l l o c a t i o n f , o f e x e r t i n g r e s p e c t i v e l y t o w a r d s and a g a i n s t t h e change o f a l l o c a t i o n f t o an a l t e r n a t i v e f ' . The two main p o s t u l a t e s a s s e r t t h e f o l l o w i n g : 1) Every a ( t , s , c ) , sa a f u n c t i o n o f t , h a s a t most one

peak; 2 ) The change from f t o f ' w i l l n o t o c c u r , i f t h e

forces against it are stronger than forces f o r i t , i . e .

lE

G(s,f,f')

s €A

>seBUC

F(s,f,f')

where A , B and C a r e r e s p e c t i v e l y t h e s e t s o f t h o s e p e r s o n s s who p r e f e r f t o f' ( f f ' ) , who a r e i n d i f f e r S e n t between f and f' ( f m S f ' ) , and who p r e f e r f ' t o

>

f (f'

> sf).

One may now d e f i n e t h e a d m i s s i b i l i t y o f an a l l o c a t i o n f , a t t i m e t , as f o l l o w s : A(t,f)

= min{a(t,s,c):

f ( s , c ) = 1)

.

Thus, t h e a d m i s s i b i l i t y o f a n a l l o c a t i o n e q u a l s to t h e


893

a d m i s s i b i l i t y o f t h e l e a s t a d m i s s i b l e of i t s a s s i g n m e n t s

o f goods. The f i r s t p o s t u l a t e i m p l i e s t h a t t h e f u n c t i o n A ( t , f )

has

a l s o t h e p r o p e r t y of s i n g l e - p e a k e d n e s s . T h i s means t h a t a d m i s s i b i l i t y o f any a l l o c a t i o n w i l l e v e n t u a l l y s t a r t

diminishing. Next, an a l l o c a t i o n f i s c a l l e d f a i r ( a t time t ) , i f A ( t , f ) 3 A ( t , f ' ) f o r any a l l o c a t i o n f ' . T h i s means t h a t f i s the allocation w i t h highest admissibility a t t . The n o t i o n o f f a i r n e s s c o v e r s t o some e x t e n t t h e i n t u i t i o n of " s o c i a l j u s t i c e " . An a l l o c a t i o n i s c a l l e d s t a b l e , i f t h e i n e q u a l i t y b e t ween f o r c e s from p o s t u l a t e 2 h o l d s f o r any a l l o c a t i o n f ' . Since A ( f , t ) w i l l eventually begin t o d e c l i n e , ther e w i l l a p p e a r a t l e a s t one a l t e r n a t i v e a l l o c a t i o n f ' which w i l l b e more f a i r t h a n f . T h i s l e a d s t o t h e f o l lowing dilemma: i f one wants t o e n s u r e f a i r n e s s , one has t o make f r e q u e n t changes; i f one wants t o p r e s e r v e s t a b i l i t y , one has t o r e s i g n from f a i r n e s s . Any group of p e r s o n s M c a p a b l e of e n s u r i n g t h e i n e q u a lity

1G(s,f,f') > s t;M

F(s,f,f') SdN

f o r some f ' w i t h A ( t , f ' ) 7 A ( t , f ) , i . e . group c a p a b l e o f b l o c k i n g a s o c i a l l y more f a i r a l t e r n a t i v e f ' , w i l l b e c a l l e d a monopole w i t h r e s p e c t t o a l l o c a t i o n f ' . The n o t i o n o f an e n f o r c i n g monopole ( a s opposed t o b l o c k i n g monopole above) i s d e f i n e d a n a s i m i l a r way. S i n c e t h e f o r c e s F and G a r e , i n g e n e r a l , n o t known

CHAPTER 6

894

e x a c t l y , t h e problem o f c h e c k i n g whether a g i v e n group i s a monopole may be d i f f i c u l t . M o r e o v e r , monopoles

are u s u a l l y fuzzy s u b s e t s of S. Given t h e a b o v e c o n c e p t s , o n e c a n s p e c i f y t h e c o n d i t i o n s f o r a l i e n a t i o n of a person s . T h u s , s u p p o s e t h a t t h e a c t u a l a l l o c a t i o n i s f , and t h e r e e x i s t s f ' s u c h t h a t f ( s , c ) = 0 , f ' ( s , c ) = 1 f o r some goods c ; t h a t i s , p e r s o n s d o e s n o t h a v e a c c e s s t o goods c u n d e r f , b u t he would have s u c h a n a c c e s s u n d e r t h e a l l o c a t i o n f'. If t h e f o l l o w i n g c o n d i t i o n s are met:

1) a ( t , s , c ) i s h i g h a n d i n c r e a s i n g ;

2)

f' >s

3) A ( t , f ' )

f (i.e. s prefers f ' t o f ) ;

>

A(t,f);

4) f' i s b l o c k e d by some monopole M; t h e n o n e s a y s t h a t p e r s o n s i s a l i e n a t e d by monopole M . T h u s , a l i e n a t i o n o c c u r s if t h e r e i s a n a l t e r n a t i v e a l l o c a t i o n f ' , which g i v e s goods c t o s , w h i l e he d o e s n o t h a v e t h e s e goods u n d e r f , a n d d e s e r v e s t h e m ; moreo v e r , t h e a l t e r n a t i v e f ' i s a t t h e same t i m e s o c i a l l y b e t t e r t h a n f , p r e f e r r e d by s , a n d b l o c k e d by a mono-

pole. The p a p e r s of Nowakowska ( 1 9 7 7 a , b ) p r e s e n t a number o f

h y p o t h e s e s a b o u t s t r a t e g i e s o f m o n o p o l e s , and t h e e f f e c t o f a l i e n a t i o n on t h e t e n d e n c i e s t o form counter-monop o l e s which t r y t o i n d u c e changes i n t h e a l l o c a t i o n o f g o o d s . These h y p o t h e s e s may t h e r e f o r e be t r e a t e d as


895

as h y p o t h e s e s about t h e dynamics which induce some s p e c i a l k i n d o f s o c i a l c h a n g e s , namely t h o s e which cons i s t o f a t r a n s f o r m a t i o n of a l l o c a t i o n o f goods i n the society. Under c o n d i t i o n s ( a )

-

( d ) , one may c o n s t r u c t a l s o a

measure o f s t r e n g t h o f a l i e n a t i o n . A p e r s o n may be a l i e n a t e d w i t h r e s p e c t t o v a r i o u s goods,

and by v a r i o u s monopoles. T h i s a l l o w s us t o s p l i t t h e a l i e n a t i o n of a g i v e n p e r s o n i n t o v a r i o u s components, and d e f i n e t h e n o t i o n s , s u c h as scope and d e p t h o f a l i e n a t i o n , depending on t h e r e l a t i v e " l o c a t i o n " o f a p e r s o n w i t h r e s p e c t t o a monopole. When a l a r g e number o f p e r s o n s a r e a l i e n a t e d by t h e

same monopole, and t h e i r communication network i s suf f i c i e n t l y "dense", t h e r e i s a high p r o b a b i l i t y t h a t t h a y w i l l form a new s o c i a l g r o u p , w i t h i t s own g o a l s , program f o r a c t i o n , e t c . , aimed p r i m a r i l y a t r e d u c i n g t h e l e v e l o f a l i e n a t i o n from some goods. I n t h i s s e n s e , a l i e n a t i o n may produce a s o c i a l change i n t h e form o f a t r a n s f o r m a t i o n o f t h e s t r u c t u r e o f t h e s o c i e t y and t r a n s f o r m a t i o n o f t h e p r e f e r e n c e system. Such s o c i a l c h a n g e s , as a r u l e , a r e p r e l i m i n a r y t o major changes i n allocations. One may t h e r e f o r e s a y t h a t , g i v e n a p p r o p r i a t e c o n d i t i o n s , t h e phenomenon of a l i e n a t i o n i s a primary s o u r c e o f s o c i a l c h a n g e s , f i r s t i n t h e " c o n s c i o u s n e s s domain", and t h e n -- i f t h e s t r e n g t h o f t h e new group i s s u f f i cient

--

a l s o i n t h e domain of a l l o c a t i o n of goods.

The main r e s u l t o f t h e s u g g e s t e d model, however, i s t h e t h e o r e t i c a l j u s t i f i c a t i o n of t h e u n a v o i d a b i l i t y

896

CHAPTER 6

o f a l i e n a t i o n , due t o t h e dilemma between s t a b i l i t y and f a i r n e s s o f a l l o c a t i o n s o f goods.

1 4 . 5 . S u b j e c t i v e p e r c e p t i o n of o n e ' s l e v e l o f a l i e n a t i o n I n t h i s s e c t i o n we s h a l l s u g g e s t a c e r t a i n model, des i g n e d t o c a p t u r e c e r t a i n dynamic f e a t u r e s o f t h e p e r c e p t i o n of o n e ' s l e v e l of a l i e n a t i o n . The b a s i c assumption i s t h a t t h e p e r c e i v e d l e v e l o f a l i e n a t i o n changes s e q u e n t i a l l y , as one o b s e r v e s t h e l e v e l s o f a c c e s s t o c e r t a i n goods o f o t h e r p e r s o n s . I n t u i t i v e l y , e v e r y time one e n c o u n t e r s a p e r s o n whose l e v e l o f a c c e s s t o some goods i s e x c e e d i n g o n e ' s own a c c e s s t o t h e same goods, t h e p e r c e i v e d a l i e n a t i o n l e v e l i n c r e a s e s . On t h e o t h e r hand, when one e n c o u n t e r s a p e r s o n who i s , s o t o s p e a k , i n worse c o n d i t i o n s , t h e l e v e l of a l i e n a t i o n d e c r e a s e s . These p e r c e p t i o n s a r e " f i l t e r e d " , o r b i a s e d , depending o f o n e ' s p e r c e i v e d l e v e l of a l i e n a t i o n . The l a t t e r e f f e c t , o f a p s y c h o l o g i c a l c h a r a c t e r , i s r e l a t e d t o t h e f a c t t h a t f o r a person who i s h i g h l y a l i e n a t e d ( w i t h something l i k e a " f e e l i n g of b i t t e r n e s s " ) each encounter w i t h a person w i t h h i g h e r a c c e s s t o some goods t e n d s t o have a somewhat e x a g g e r a t e d e f f e c t . On t h e o t h e r hand, i n s u c h a i t u a t i o n , e n c o u n t e r i n g a p e r s o n who i s worse-off t o be n e g l e c t e d .

tends

Thus, t h e model w i l l a t t e m p t t o c a p t u r e t h e dynamic f e a t u r e s o f p e r c e p t i o n o f o n e ' s own l e v e l of a l i e n a t i o n , as a n i n t e r p l a y o f t h e f o l l o w i n g t h r e e f a c t o r s :

---

t h e a c t u a l perceived l e v e l of alienation; t h e a c t u a l a c c e s s t o t h e goods;


--

891

t h e p e r c e i v e d a c c e s s t o t h e goods by o t h e r s .

14.5.1. The model. We c o n s i d e r p e r c e p t i o n of t h e l e v e l o f a l i e n a t i o n o f an a r b i t r a r y f i x e d s u b j e c t , r e s u l t i n g from h i s p e r c e p t i o n of b e i n g d e p r i v e d of c e r t a i n goods s a y , C . The n a t u r e o f t h e s e goods need n o t be made s p e c i f i c i n t h e g e n e r a l model.

For s i m p l i c i t y , l e t u s assume t h a t t h e l e v e l o f a c c e s s t o the goods C i s r e p r e s e n t a b l e i n form o f a c e r t a i n i n d e x ( i t may be a s i m p l e v a r i a b l e , s u c h as s a l a r y l e v e l , or a composite i n d e x , i n v o l v i n g p e r c e p t i o n o f l e s s t a n g i b l e goods, which may i n t e r v e n e i n a l i e n a t i o n level). N e x t , we assume t h a t from t i m e t o t i m e ( i n t h e model,

a t t i m e s t = 1,2, . . . ) t h e s u b j e c t e n c o u n t e r s o t h e r p e r s o n s , whom he s e e s as e q u a l l y d e s e r v i n g t h e a c c e s s t o t h e goods a s h e , and p e r c e i v e s t h e i r a c t u a l a c c e s s t o t h e s e goods.

...

be t h e s u b j e c t ' s a c t u a l l e v e l s F o r m a l l y , l e t zl, z2, of a c c e s s t o goods C a t t i m e s t = 1,2,..., and l e t xl, x 2 , be h i s p e r c e i v e d l e v e l s o f a c c e s s t o C o f t h e s u c c e s s i v e persons whom he e n c o u n t e r s .

...

...

F i n a l l y , l e t Y1, Y2, be t h e p e r c e i v e d by t h e subj e c t l e v e l s o f a l i e n a t i o n from goods C . B e f o r e f o r m u l a t i n g t h e a s s u m p t i o n s about t h e n a t u r e o f i t i s w o r t h w h i l e t o comment b r i e f l y the process on t h e i n t e r p r e t a t i o n o f t h e t h r e e p r o c e s s e s i n v o l v e d , namely {xn\, { z n l and { y n j .

fYnl,

898

CHAPTER 6

F i r s t l y , i t w i l l be assumed t h a t f z 1 form some f i x e d n s e q u e n c e , r e p r e s e n t i n g t h e s u b j e c t Is e v o l u t i o n w i t h r e s p e c t t o goods C . I f I z n l r e p r e s e n t s t h e s a l a r y l e v e l s , t h e n i t may be assumed t h a t z n grows s t e a d i l y a t a c e r t a i n r a t e , o r remains c o n s t a n t , e t c . S e c o n d l y , xn i s t h e a c c e s s t o goods C o f t h e n - t h successive person encountered (it i s not important t o i n t e r p r e t t h e t e r m " e n c o u n t e r " l i t e r a l l y ) . We imagine t h a t a t t5me t = n , t h e s u b j e c t l e a r n s about some o t h e r p e r s o n , who ( a c c o r d i n g t o h i m ) d e s e r v e s t h e acc e s s t o goods C i n t h e same d e g r e e as he ( e . g . has t h e same q u a l i f i c a t i o n s , e t c . ) , and p e r c e i v e s ( r i g h t l y o r n o t ) h i s a c c e s s t o C as xn

.

>

Now, i f xn z n , t h e s u b j e c t w i l l f e e l l a c k of "fairn e s s " , and t h i s w i l l t e n d t o i n c r e a s e h i s p e r c e p t i o n MOo f a l i e n a t i o n , from Y n t o some h i g h e r v a l u e Y n t l . r e o v e r , t h e i n c r e a s e Y 1 - Y n w i l l t e n d t o be h i g h e r nt i f t h e a c t u a l l e v e l of a l i e n a t i o n Y n i s h i g h e r . On t h e o t h e r hand, i f xn < z n , t h e a l i e n a t i o n l e v e l Yn w i l l d e c r e a s e t o a lower l e v e l Yntl. The d e c r e a s e w i l l be l e s s i n a b s o l u t e v a u l u e , i f t h e s u b j e c t f e e l s h i g h l y a l i e n a t e d , i . e . i f Yn i s h i g h . S p e c i f i c a l l y , t h e model w i l l have t h e f o l l o w i n g form 'n+ 1 = f ( Y n )

+ F(xn, zn, Y n ) .

(14.2)

Here f i s a f u n c t i o n which d e s c r i b e s t h e " s p o n t a n e o u s "

changes i n t h e p e r c e i v e d a l i e n a t i o n l e v e l , o c c u r r i n g d u r i n g the p e r i o d between c o n s e c u t i v e e n c o u n t e r s . One o f t h e p o s s i b l e forms of t h e f u n c t i o n f i s

FORMAL THEORY OF ACDONS

899

where 0 < r 4 1. T h i s c o r r e s p o n d s t o t h e c a s e when t h e p e r c e i v e d l e v e l o f a l i e n a t i o n "wears o f f " i n t h e p e r i o d s between t h e e n c o u n t e r s , i n a b s e n c e of s u p p o r t i n g or i n h i b i t i n g f a c t o r s , w i t h r being t h e decrease r a t e . Next, f u n c t i o n F ( x , z , Y ) r e p r e s e n t s t h e change ( p o s i t i ve o r negative) i n one's perceived a l i e n a t i o n l e v e l , r e s u l t i n g from an e n c o u n t e r w i t h a p e r s o n whose a c c e s s t o goods C i s x , i n s i t u a t i o n when t h e s u b j e c t ' s a c c e s s t o goods C i s z , and h i s a l i e n a t i o n l e v e l i s Y . The gen e r a l a s s u m p t i o n s about t h e f u n c t - o n s f and F are as follows. ASSUMPTION 1. The f u n c t i o n f i s s t r i c t l y i n c r e a s i n g . T h i s a s s u m p t i o n means t h a t t h e h i g h e r i s t h e p e r c e i v e d

l e v e l o f a l i e n a t i o n a f t e r t h e n - t h e n c o u n t e r , t h e highe r w i l l be i t s " r e s i d u a l " l e v e l , p r i o r t o ( n + l l - s t encounter. ASSUMPTION 2 . F o r e v e r y f i x e d Y and z , t h e f u n c t i o n F(x,z,Y) i s s t r i c t l y increasing i n z , with F(x,z,Y) f o r x > z and F ( x , z , Y ) < 0 for x < z .

>

9

T h i s assumption means t h a t e a c h e n c o u n t e r w i t h a p e r -

s o n who i s p e r c e i v e d as " b e t t e r o f f " ( x > z ) g i v e s an i n c r e a s e i n a l e v e l o f a l i e n a t i o n , while each enc-unter w i t h a p e r s o n who i s p e r c e i v e d as "worse-off" ( x < z ) g i v e s a d e c r e a s e of l e v e l of a l i e n a t i o n . The v a l u e F ( x , z , Y ) r e p r e s e n t s a change i n t h e p e r c e i v e d l e v e l o f a l i e n a t i o n r e s u l t i n g from an e n c o u n t e r w i t h a p e r s o n who i s i n t h e same s i t u a t i o n as r e g a r d s t h e goods

900

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as t h e s u b j e c t . T h i s v a l u e ma.y be p o s i t i v e , n e g a t i v e

or zero. ASSUMPTION 3 . The f u n c t i o n F(x,z,Y)

increases with Y

f o r any x and z. The i n t e r p r e t a t i o n o f t h i s assumption i s as f o l l o w s . Firstly, if x

> z (an encounter

with

a

p e r s o n who i s

“ b e t t e r o f f ” g i v e s an i n c r e a s e i n a l i e n a t i o n l e v e l ( s i n c e F > 0 ) , and t h i s i n c r e a s e i s l a r g e r when t h e l e v e l of a l i e n a t i o n i s h i g h e r . On t h e o t h e r hand, i f x c z ( a n e n c o u n t e r w i t h a p e r s o n who i s w o r s e - o f f ) , t h e l e v e l of a l i e n a t i o n d e c r e a s e s , but t h e decrease i s s m a l l e r when t h e l e v e l o f a l i e n a t i o n i s h i g h e r . T h i s c o r r e s p o n d s t o t h e p s y c h o l o g i c a l e f f e c t mentioned

a t t h e b e g i n n i n g , a c c o r d i n g t o which when t h e r e i s a h i g h e r f e e l i n g o f d e p r i v a t i o n e a c h new i n s t a n c e o f p e r c e i v e d i n j u s t i c e or u n f a i r n e s s i s f e l t as more a c u t e . On t h e o t h e r hand, a n e n c o u n t e r w i t h a l e s s e s i n j u s t i c e t e n d s t o be n e g l e c t e d .

14.5.2.

A n a l y s i s . The above model i s t o o g e n e r a l t o p e r m i t a n a l y s i s , and one has t o make some a d d i t i o n a l s i m p l i f y i n g a s s u m p t i o n s . T h u s , one may assume f i r s t l y t h a t z1 = z = = z. T h i s means t h a t i n t h e p e r i o d under c o n s i d e r a t i o n , t h e a c c e s s t o goods C by t h e s u b j e c t do n o t change i n any a p p r e c i a b l e manner.

...

i t may be r e a s o n a b l y assumed t h a t t h e y a r e i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s . T h i s c o r r e s p o n d s t o t h e most i m p o r t a n t c a s e when t h e r e i s some i n h e r e n t v a r i a t i o n i n t h e a c c e s s t o goods C i n t h e p o p u l a t i o n o f p e r s o n s whom t h e subA s r e g a r d s xl, x

2’”’


90 1

j e c t p e r c e i v e s as " e q u a l l y d e s e r v i n g " t h e goods C . Under t h e s e a s s u m p t i o n s , d r o p p i n g for s i m p l i c i t y t h e c o n s t a n t z , t h e model becomes

where G i s some f u n c t i o n (f t F ) . One may now impose c o n d i t i o n s under which t h e sequence Yn i s s t a t i o n a r y , o r t e n d s t o a s t a t i o n a r y d i s t r i b u t i o n , depending on t h e d i s t r i b u t i o n of x and c o n s t a n t z. n The e x i s t e n c e of a s t a t i o n a r y d i s t r i b u t i o n means t h a t t h e p e r s o n ' s p e r c e i v e d l e v e l of a l i e n a t i o n f l u c t u a t e s around some a v e r a g e v a l u e ( d e p e n d i n g on z), w i t h o u t tendency t o grow or d e c r e a s e . I n p r a c t i c a l t e r m s , s u c h a s i t u a t i o n means % h a t ( f r o m p s y c h o l o g i c a l p o i n t o f view) a n a c c e p t a b l e e q u i l i b r i u m has been r e a c h e d : t h e s u b j e c t f e e l s t h a t he i s " l o c a t e d " a t some p l a c e o f t h e s c a l e of a c c e s s t o goods C , and a c c e p t s t h e v a r i a t i o n s o f t h i s l e v e l as normal. He u n d e r s t a n d s t h a t some p e r s o n s who a r e e q u a l l y d e s e r v i n g as h i m s e l f a r e w o r s e - o f f , w h i l e some o t h e r s a r e b e t t e r - o f f , b u t t h e numbers of these persons a r e balanced, so t h a t h i s f e e l i n g of a l i e n a t i o n ( f r o m a c c e s s t o goods C ) i s s t a b l e . On t h e o t h e r hand, i f Y n t e n d s t o i n c r e a s e , we have t h e s i t u a t i o n of a p e r s o n who f e e l s i n c r e a s i n g l y f r u s t r a t e d , b y e n c o u n t e r i n g t o o many p e r s o n s who a r e (undes e r v e d l y ) b e t t e r off t h a n h i m s e l f . I n t h i s c a s e , t h e p e r c e i v e d l e v e l of a l i e n a t i o n i s a self-perpetuating process. It i s c l e a r t h a t t h e model s u g g e s t e d h e r e may be e n r i c h -

902

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ed and m o d i f i e d , s o as t o t a k e i n t o a c c o u n t v a r i o u s other factors.

14.6.

Some h y p o t h e s e s

There i s no need t o p o i n t o u t t h a t t h e t h e o r y of t h e p r e s e n t s e c t i o n g i v e s only a very b r i e f s k e t c h o f t h e t h e o r y o f s o c i a l change, w i t h many d e t a i l s o m i t t e d f o r s i m p l i c i t y and s h o r t n e s s , and many a s p e c t s o f t h e phenomenon l e f t o u t . To i l l u s t r a t e , however, t h e a p p l i c a b i l i t y o f t h e s y s t e m

and i t s r i c h n e s s , some h y p o t h e s e s w i l l b e f o r m u l a t e d c o n c e r n i n g v a r i o u s a s p e c t s o f t h e phenomenon d e s c r i b e d . F i r s t l y , a s o c i a l change was d e f i n e d i n a f u z z y way as

a " s u b s t a n t i a l " change i n one o r more v a r i a b l e s o f t h e system. Whether a g i v e n change i s a c c e p t e d as s u b s t a n t i a l or n o t ( i . e . whether i t may be c l a s s i f i e d as a soci a l change) depends on many f a c t o r s , mainly on t h e pos i t i o n of a c l a s s i f y i n g p e r s o n i n t h e s o c i e t y , and h i s a l i e n a t i o n l e v e l . We have h e r e t h e f o l l o w i n g hypothesis. HYPOTHESIS 1. Any p e r s o n h a s a t h r e s h o l d , above which he t e n d s t o c l a s s i f y changes a s s u b s t a n t i a l . T h i s

threshold, i n general, i s

-- h i g h e r for changes which a f f e c t p o s i t i v e l y t h e judgi n g p e r s o n , and lower i f t h e y a f f e c t him n e g a t i v e l y ;

--

n e g a t i v e l y r e l a t e d t o t h e a l i e n a t i o n l e v e l of t h e person.

903


Another a s p e c t

o f s o c i a l changes i s c o n n e c t e d w i t h

t h e concepts sketched b r i e f l y i n t h e preceding sections. The p o i n t h e r e i s t h a t t h e t h e o r y c o v e r s what may be termed " t o t a l " c h a n g e s , a l o n g one or more o f t h e d i mensions o f t h e system i n t r o d u c e d . One can a l s o c o n s i d e r " l o c a l " changes, r e s t r i c t e d t o one o r more o f t h e g r o u p s , and c o n s i s t i n g o f changes of t h e s t r a t e g i e s ; changes o f g o a l s , changes o f r u l e s , as w e l l as changes i n t h e s t r u c t u r e o f t h e groups t h e m s e l v e s , even i f r e s t r i c t e d t o one group o n l y , can be c o n s i d e r e d as s o c i a l changes i n t h e s e n s e o f t h e p r e c e d i n g s e c t i o n s , which l e a v e i n v a r i a n t t h e r e l e v a n t a s p e c t s o f a l l b u t one group. The s t r a t e g y means h e r e t h e change o f s t r e s s o f t h e

a l l o c a t i o n of e f f o r t s by a group among t h e t h r e e domains o f i t s a c t i v i t y : (A1) i d e o l o g i c a l , (A2) f a c t u a l , and i n t e r n a l . Here one may f o r m u l a t e t h e f o l l o w i n g hypothesis. (A3)

HYPOTHESIS 2 . During t h e t i m e p r e c e d i n g and immediat e l y f o l l o w i n g a s o c i a l change i f form o f t r a n s i t i o n t o

a new a l l o c a t i o n o f goods, t h e group a c t i v i t i e s a r e c o n c e n t r a t e d a t f i r s t m o s t l y on a c t i v i t i e s o f t h e t y p e A1, t h e n on a c t i v i t i e s o f t h e t y p e A2, and t h e n , i n t h e p e r i o d immediately p r e c e d i n g t h e t r a n s i t i o n and f o l l o w i n g i t , on a c t i v i t i e s o f t y p e s A and A1.

3

HYPOTHESIS 3 . A s t h e a d m i s s i b i l i t y o f t h e a c t u a l a l l o c a t i o n d e c r e a s e s (as it must, a c c o r d i n g t o t h e p o s t u l a t e s of t h e t h e o r y ) , t h e groups i n t e r e s t e d i n p r e s e r v i n g i t w i l l c o n c e n t r a t e mainly on a c t i v i t i e s o f t h e t y p e s w h i l e t h e groups i n t e r e s t e d i n changing i t A1 and A

3'

904

CHAPTER 6

w i l l c o n c e n t r a t e mainly on a c t i v i t i e s o f t h e t y p e A

2

and A1. HYPOTHESIS 4. The e f f e c t i v e n e s s a f a c t i v i t y o f t h e t y p e A1 ( i d e o l o g i c a l ) depends p r i m a r i l y on t h e p r o p e r t i e s o f t h e network o f communication i n S , and i n c r e a s e s w i t h t h e i n c r e a s e of t h e number o f c o n n e c t i o n s p e r node. HYPOTHESIS 5 . An i n c r e a s e o f i n t e n s i t y of a c t i v i t y o f t y p e A1 ( i d e o l o g i c a l ) by two or more groups w i t h oppos i n g p r e f e r e n c e p a t t e r n s , l e a d s t o a d e c r e a s e o f fuzzin e s s o f t h e groups s t r u c t u r e ( " p o l a r i z a t i o n o f t h e society"). HYPOTHESIS 6 . Major changes i n t h e s t r u c t u r e and cont e n t o f t h e s e t of r u l e s associated with various groups a r e u s u a l l y f o l l o w e d by major changes i n t h e groups s t r u c t u r e i t s e l f , by l e a d i n g t o an a p p e a r e n c e o f new g r o u p s . A s a l r e a d y mentioned, t h e main d r i v i n g f o r c e s b e h i n d s o c i a l changes a r e induced by a l i e n a t i o n of

members

o f v a r i o u s g r o u p s . Thus, t h e h y p o t h e s e s about a l i e n a t -

i o n ( s e e Nowakowskal977a) a r e r e l e v a n t h e r e , i n t h e c o n t e x t o f s o c i a l changes. I n p a r t i c u l a r , one s h o u l d menion here t h e f o l l o w i n g h y p o t h e s e s . HYPOTHESIS 7 . A b l o c k i n g monopole, hence a monopole i n t e r e s t e d i n p r e v e n t i n g a s o c i a l change a d v e r s e t o

i t s i n t e r e s t s , w i l l t e n d t o s u p p o r t o t h e r s o c i a l change s , i n p a r t i c u l a r t h o s e which l e a d t o i n c r e a s e of t h e t o t a l a d m i s s i b i l i t y o f a l l o c a t i o n s , hence a l s o l e a d i n g t o a decrease i n alienation l e v e l s .


905

HYPOTHESIS 8 . A s o c i a l change i n form o f an a p p e a r e n c e o f a new group s t r u c t u r e , t h r o u g h t h e c r e a t i o n o f a counter-monopole, i s more l i k e l y i n c a s e s when t h e l e v e l o f a l i e n a t i o n i s h i g h , and i t s v a r i a n c e i s s m a l l , t h a n i n the opposite cases. F i n a l l y , t h e f o l l o w i n g h y p o t h e s i s i s r e l e v a n t f o r pred i c t i o n o f dynamics of monopoles, hence a l s o t h e dynamics of t h e s o c i a l s t r u c t u r e . HYPOTHESIS 9 . I f f ( t ) i s t h e p r o b a b i l i t y t h a t a p e r s o n ' s p a r t i c i p a t i o n i n a monopole i s e x a c t l y t , and F ( t ) i s t h e probability t h a t it i s t o r l e s s , then t h e r a t i o f ( t ) / [ l - F ( t ) l i s an i n c r e a s i n g f u n c t i o n o f t . The l a s t h y p o t h e s i s , a s s e r t i n g t h e " a g i n g p r o p e r t y " o f t h e d i s t r i b u t i o n o f p a r t i c i p a t i o n i n a monopole i s j u s t i f i e d by t h e i n t u i t i o n a c c o r d i n g t o which t h e longe r a p e r s o n i s i n a monopole, t h e more chances t h a t he i s s a t i s f i e d w i t h t h e goods he r e c e i v e d , and t h e r e f o r e , t h e more c h a n c e s he w i l l a t t e m p t t o j o i n a n o t h e r monopole. The p r e c e d i n g h y p o t h e s e s concerned t h e r o l e o f a l i e n -

a t i o n i n i n d u c i n g s o c i a l changes. I n connection w i t h t h i s , one can a l s o s t a t e t h e f o l l o w i n g h y p o t h e s e s of a "negative" character. HYPOTHESIS 1 0 . A s o c i a l change i s l e s s l i k e l y t o o c c u r , i f t h e a l i e n a t e d groups: -- r e c e i v e some " g o o d s - s u b s t i t u t e s " , as l o n g as t h e y a r e approved o r v a l u a b l e , o r -- t h e group a c t i v i t y i s d i r e c t e d towards some " a c t i o n substitutes", o r

906

CHAPTER 6

-- t h e g r o u p ’ s p r e s e n t a l l o c a t i o n i s t h r e a t e n e d . HYPOTHESIS 11. If t h e group i s under n e g a t i v e i n f o r m a t i o n p r e s s u r e o r g a n i z e d by competing g r o u p s , i t may dec r e a s e t h e l i k e l i h o o d o f r e a l changes by making a p p a r e n t changes ( e . g . t h r o u g h s e m a n t i c a l l y e q u i v a l e n t p h r a s i n g o f programs, r e a s o n s and r u l e s ) . HYPOTHESIS 1 2 . The t i g h t e r a group i s o r g a n i z e d , and t h e more s u c h a group a p p r o v e s o f i t s r e a s o n s and programs, t h e more r e s i s t a n t i t w i l l be t o new i d e a s , and t h e l e s s p r o n e t o change.

1 4 . 7 . Some o t h e r mechanisms o f s o c i a l change The mechanisms connected w i t h a l i e n a t i o n p r e s e n t e d above a r e g e n e r a l l y t r e a t e d as one o f t h e main c a u s e s o f s o c i a l changes. Another mechanism i s t h a t o f t h e s o c i a l p r e s s s u r e on d e c i s i o n s . These p r e s s u r e s , i n some con-

t e x t s may be towards t h e c h a n g e s , and i n some o t h e r contexts

--

a g a i n s t t h e changes. I n c o n n e c t i o n w i t h

s o c i a l p r e s s u r e s , we s h a l l s u g g e s t a s i m p l e model o f a group d e c i s i o n i n o p p o s i n g or s u p p o r t i n g some i s s u e . The c e n t r a l a s s u m p t i o n w i l l be t h a t t h e p r o b a b i l i t y o f a p o s i t i v e d e c i s i o n by n - t h p e r s o n depends on t h e number o f p o s i t i v e d e c i s i o n s o f t h e p r e c e d i n g n-1 p e r sons.

I n o t h e r words, we imagine t h a t t h e group d e -

c i s i o n i s r e a c h e d s e q u e n t i a l l y , and t h a t e v e r y p e r s o n h a s some knowledge about t h e d e c i s i o n s o f h i s p r e d e c e s -

sors, and a c t s a c c o r d i n g l y . The r e s u l t s o f t h i s model w i l l allow us t o optimally a l l o c a t e t h e resources between v a r i o u s propaganda t a s k s .

907


I n c i d e n t a l l y , i t i s w o r t h n o t i n g t h a t a new d e c i s i o n model o f p r o s o c i a l and a n t i s o c i a l b e h a v i o u r , g i v e n i n C h a p t e r 5 , i s a l s o r e l e v a n t h e r e , namely as a model of p r e s s u r e e x e r t e d by a group, l e a d i n g t o t h e mo d if ic a t i o n o f g r o u p r u l e s , and i n d i r e c t l y , a s o c i a l c h a n g e o f group s t r u c t u r e and g o a l s . T h i s model r e a c h e s d e e p e s t i n t o t h e mechanisms o f g r o u p b e h a v i o u r , by s h o w i n g t h e r e l a t i o n s b e t w e e n a c t i o n s a n d o p i n i o n s , e x p e c t a t i o n s , and t y p e s o f reward and punishment by t h e g r o u p . One o f t h e most i m p o r t a n t f e a t u r e s o f t h i s model i s i t s a b i l i t y o f s u p p l y i n g p r e d i c t i o n s . F i n a l l y , t h e communication network i n system ( 1 4 . 1 ) and t y p e o f s p r e a d of i n f o r m a t i o n i n t h e group, a r e a l s o o n e o f t h e most i m p o r t a n t f a c t o r s d e t e r m i n i n g

t h e group

a n d i t s d y n a m i c s . An o u t l i n e c f s u c h a t h e o r y o f n e t works w i l l be g i v e n i n n e x t s u b s e c t i o n s .

14.7.1. Group p r e s s u r e s a n d g r o u p c h o i c e .

The r e s u l t s

o f t h i s s e c t i o n were f i r s t p u b l i s h e d i n Nowakowska (1982b), and t h e r e a d e r i s r e f e r r e d t h e r e f o r t h e p r o o f s o f t h e t h e o r e m s . The model c o n c e r n s a s i t u a t i o n i n w h i c h a g r o u p , s a y G , o f a f i x e d s i z e N , makes a d e c i s i o n l l f o r l r or " a g a i n s t " a c e r t a i n i s s u e . C l e a r l y , s u c h

a s i t u a t i o n i s a b a s i c i n g r e d i e n t o f more complex s o c i a l s i t u a t i o n s o f i n t e r p l a y b e t w e e n a number o f g r o u p s f i g h t i n g f o r and a g a i n s t some i s s u e .

We s h a l l assume t h a t t h e i s s u e i n q u e s t i o n , s a y I , i s f i x e d , a n d e v e r y member o f t h e g r o u p h a s t o r e s p o n d t o i t p o s i t i v e l y o r n e g a t i v e l y ( v o t e " f o r " or " a g a i n s t f f ) . One may t h i n k here o f s u c h i s s u e s I as j o i n i n g a s t r i k e ,

908

CHAPTER 6

e l e c t i n g a c a n d i d a t e , d o n a t i n g money t o some c a u s e , e t c . Group membership i n f l u e n c e s a p e r s o n l s r e s p o n s e i n a t l e a s t two ways. F i r s t l y , t h e p r i o r p r o b a b i l i t y o f a pos i t i v e r e s p o n s e may be d i f f e r e n t t h a n f o r members i n o t h e r s o c i a l g r o u p s . I n o t h e r w o r d s , g r o u p membership i s some s e n s e c o n d i t i o n s t h i n k i n g and c o n v i c t i o n s , a n d makes members o f t h e g r o u p t h i n k " a l i k e " i n h i g h e r d e g r e e t h a n members o f d i f f e r e n t g r o u p s ( w h i c h d o e s n o t mean t h a t t h e y have t o be o f t h e same o p i n i o n , o f course. ) S e c o n d l y , a p e r s o n ' s d e c i s i o n may be i n f l u e n c e d by h i s knowledge a b o u t t h e d e c i s i o n s o f o t h e r members o f t h e g r o u p , who have a l r e a d y made t h e d e c i s i o n . The model below c o n c e r n s t h i s s e c o n d e f f e c t , w h i c h may be c a l l e d "group p r e s s u r e " . We assume t h e r e f o r e t h a t e a c h member of t h e g r o u p G i s

t o make a p o s i t i v e o r n e g a t i v e d e c i s i o n a b o u t t h e i s s u e I , w i t h no t i e s a l l o w e d . L e t xN b e t h e t o t a l number of p e r s o n s i n g r o u p G who d e c i d e s d r r f o r r r .Then xN may b e c a l l e d t h e g r o u p s u p p o r t f o r I , a n d o u r main o b j e c t i v e w i l l b e t o make i n f e r e n c e and p r e d i c t i o n s a b o u t x

"

as w e l l as t o f i n d ways o f c o n t r o l l i n g E ( x N ) . We c o n c e p t u a l i z e t h e d e c i s i o n p r o c e s s as o c c u r r i n g

s e q u e n t i a l l y ; moreover, w e imagine t h a t a f t e r n-th p e r s o n made h i s d e c i s i o n , t h e ( n + l ) - s t p e r s o n knows how many among t h e f i r s t n made t h e d e c i s i o n i n f a v o u r of I. N a t u r a l l y , such an assumption i s a n i d e a l i z a t i o n : i n r e a l s i t u a t i o n s , a p e r o n s who makes h i s d e c i s i o n d o e s n o t h a v e a c o m p l e t e knowledge a b o u t t h e d e c i s i o n s of

909


o t h e r s . However, he may h av e a f a i r i d e a a b o u t i t , a n d t h i s know l e dg e, v ag u e as i t may b e , c a n i n f l u e n c e t h e p r o b a b i l i t y of h i s p o s i t i v e d e c i s i o n . A s an example, i m a g i n e a v o t i n g a t a c o n v e n t i o n o f some s o r t . A g i v e n v o t e r u s u a l l y has some i d e a ( f r o m t a l k i n g w i t h o t h e r d e l e g a t e s , from s p e e c h e s made, e t c . ) w h e t h e r m a j o r i t y w i l l be r r f o r r lo r " a g a i n s t " , and t h i s knowledge may influence his decision. I n t h e model of t h i s s e c t i o n , t h e a b o v e phenomenon w i l l b e d e s c r i b e d i n form o f d ep en d e n c e o f p r o b a b i l i t y of p o s i t i v e d e c i s i o n of n-th c o n s e c u t i v e p e r s o n on t h e number o f v o t e s ' ' f o r r r among t h e n -1 p e r s o n s who p r e c e ded him.

...,

x l , x2, x N , where x i s t h e number o f v o t e s " f o r r r among t h e f i r s t n n members o f t h e g r o u p . We assume t h a t t h i s p r o c e s s i s a Markov c h a i n , w i t h t h e f o l l o w i n g t r a n s i t i o n p r o b a b i l i t i e s : l e t xo = 0 , an d f o r n = 1,2, We d e a l h e r e w i t h t h e p r o c e s s

...

P(xntl

= ktllx,

= k) = p

P

= k

I xn

= k) = 1

+

k

a(1 -p )--nt1

(14.5)

an d X

where 0

nt1 p

Quantitative Psychology: Some Chosen Problems and New Ideas (Advances in Psychology)

Quantitative Models in Psychology

Quantitative Models in Psychology

Personal Construct Psychology: New Ideas

Personal Construct Psychology: New Ideas

Key Ideas in Psychology

On Some Methodological Problems of Psychology

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Advances in Psychology Volume 40 Current Issues in Theoretical Psychology

Advances in Psychology Volume 66 A History of Great Ideas in Abnormal Psychology

Advances in Psychology Volume 66 A History of Great Ideas in Abnormal Psychology

Advances in Psychology Volume 20 Trends in Mathematical Psychology

Environmental Psychology: New Developments (Psychology Research Progress)

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Gestalt Psychology: An Introduction to New Concepts in Modern Psychology

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Advances in Experimental Social Psychology Vol. 33 (Advances in Experimental Social Psychology)

Advances in Experimental Social Psychology, Volume 23

Advances in Experimental Social Psychology, Volume 4

Quantitative Psychology: Some Chosen Problems and New Ideas (Advances in Psychology)

Quantitative Models in Psychology

Quantitative Models in Psychology

Personal Construct Psychology: New Ideas

Personal Construct Psychology: New Ideas

Key Ideas in Psychology

On Some Methodological Problems of Psychology

Trends in Mathematical Psychology: Advances in Psychology

Advances in Psychology Volume 40 Current Issues in Theoretical Psychology

Advances in Psychology Volume 66 A History of Great Ideas in Abnormal Psychology

Advances in Psychology Volume 66 A History of Great Ideas in Abnormal Psychology

Advances in Psychology Volume 20 Trends in Mathematical Psychology

Environmental Psychology: New Developments (Psychology Research Progress)

Psychology (BPS Textbooks in Psychology)

Energy Psychology (Innovations in Psychology)

Psychology (BPS Textbooks in Psychology)

Schizophrenic Psychology, New Research

Gestalt Psychology: An Introduction to New Concepts in Modern Psychology

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Psychology

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Abnormal Psychology (Psychology Focus)

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