Foundations of Measurement Volume III: Representation, Axiomatization, and Invariance (Volume 3) (Dover Books on Mathematics)

All of the sciences — physical, biological, and social — have a need for quantitative measurement. This influential series, Foundations of Measurement, established the formal basis for measurement, justifying the assignment of numbers to objects in terms of their structural correspondence. Volume I introduces the distinct mathematical results that serve to formulate numerical representations of qualitative structures. Volume II extends the subject in the direction of geometrical, threshold, and probabilistic representations, Volume III examines representation as expressed in axiomatization and invariance. Reprint of the Academic Press, New York and London, 1990 edition. Cover 1 Frontispiece by Ruth Weisberg 3 Foundations of Measurement, VOLUME III: Representation, Axiomatization, and Invariance 4 Copyright © 1990 by David H. Krantz, R. Duncan Luce, Patrick Suppes,and Barbara Tversky 5 ISBN 0-486-45316-2 5 Table of Contents 8 Preface 14 Acknowledgments 16 Chapter 18 Overview 18 18.1 NONADDITIVE REPRESENTATIONS (CHAPTER 19) 20 18.1.1 Examples 21 18.1.2 Representation and Uniqueness of Positive Operations 21 18.1.3 Intensive Structures 22 18.1.4 Conjoint Structures and Distributive Operations 23 18.2 SCALE TYPES (CHAPTER 20) 24 18.2.1 A Classification of Automorphism Groups 24 18.2.2 Unit Representations 26 18.2.3 Characterization of Homogeneous Concatenation and Conjoint Structures 27 18.2.4 Reprise 28 18.3 AXIOMATIZATION (CHAPTER 21) 28 18.3.1 Types of Axioms 28 18.3.2 Theorems on Axiomatizability 30 18.3.3 Testability of Axioms 31 18.4 INVARIANCE AND MEANINGFULNESS (CHAPTER 22) 31 18.4.1 Types of Invariance 32 18.4.2 Applications of Meaningfulness 33 Chapter 19 Nonadditive Representations 35 19.1 INTRODUCTION 35 19.1.1 Inessential and Essential Nonadditivities 35 19.1.2 General Binary Operations 39 19.1.3 Overview 40 19.2 TYPES OF CONCATENATION STRUCTURE 42 19.2.1 Concatenation Structures and Their Properties 42 19.2.2 Some Numerical Examples 44 19.2.3 Archimedean Properties 50 19.3 REPRESENTATIONS OF PCSs 54 19.3.1 General Definitions 54 19.3.2 Uniqueness and Construction of a Representation of a PCS 56 19.3.3 Existence of a Representation 57 19.3.4 Automorphism Groups of PCSs 61 19.3.5 Continuous PCSs 63 19.4 COMPLETIONS OF TOTAL ORDERS AND PCSs 65 19.4.1 Order Isomorphisms onto Real Intervals 66 19.4.2 Completions of Total Orders 67 19.4.3 Completions of Closed PCSs 70 19.5 PROOFS ABOUT CONCATENATION STRUCTURES 73 19.5.1 Theorem 1 (p. 37) 73 19.5.2 Lemmas 1-6, Theorem 2 74 19.5.3 Theorem 2 (p. 39) 78 19.5.4 Construction of PCS Homomorphisms 79 19.5.5 Theorem 3 (p. 41) 81 19.5.6 Theorem 4 (p. 45) 86 19.5.7 Theorem 5 (p. 46) 87 19.5.8 Theorem 6 (p. 47) 88 19.5.9 Corollary to Theorem 7 (p. 50) 90 19.5.10 Theorem 9 (p. 54) 90 19.6 CONNECTIONS BETWEEN CONJOINT AND CONCATENATION STRUCTURES 92 19.6.1 Conjoint Structures: Introduction and General Definitions 92 19.6.2 Total Concatenation Structures Induced by Conjoint Structures 94 19.6.3 Factorizable Automorphisms 96 19.6.4 Total Concatenation Structures Induced by Closed, Idempotent Concatenation Structures 98 19.6.5 Intensive Structures Related to PCSs by Doubling Functions 100 19.6.6 Operations That Distribute over Conjoint Structures 102 19.7 REPRESENTATIONS OF SOLVABLE CONJOINT AND CONCATENATION STRUCTURES 104 19.7.1 Conjoint Structures 104 19.7.2 Solvable, Closed, Archimedean Concatenation Structures 105 19.8 PROOFS 106 19.8.1 Theorem 11 (p. 78) 106 19.8.2 Theorem 12 (p. 80) 109 19.8.3 Theorem 13 (p. 81) 110 19.8.4 Theorem 14, Part (iii) (p. 81) 114 19.8.5 Theorem 15 (p. 82) 114 19.8.6 Theorem 18 (p. 86) 115 19.8.7 Theorem 21 (p. 88) 117 19.9 BISYMMETRY AND RELATED PROPERTIES 118 19.9.1 General Definitions 118 19.9.2 Equivalences in Closed, Idempotent, Solvable, Dedekind Complete Structures 120 19.9.3 Bisymmetry in the 1-Point Unique Case 120 EXERCISES 121 Chapter 20 Scale Types 125 20.1 INTRODUCTION 125 20.1.1 Constructibility and Symmetry 125 20.1.2 Problem in Understanding Scale Types 128 20.2 HOMOGENEITY, UNIQUENESS, AND SCALE TYPE 129 20.2.1 Stevens' Classification 129 20.2.2 Decomposing the Classification 131 20.2.3 Formal Definitions 132 20.2.4 Relations among Structure, Homogeneity, and Uniqueness 134 20.2.5 Scale Types of Real Relational Structures 136 20.2.6 Structures with Homogeneous, Archimedean Ordered Translation Groups 139 20.2.7 Representations of Dedekind Complete Distributive Triples 142 20.3 PROOFS 143 20.3.1 Theorem 2 (p. 117) 143 20.3.2 Theorem 3 (p. 118) 144 20.3.3 Theorem 4 (p. 118) 145 20.3.4 Theorem 5 (p. 120) 145 20.3.5 Theorem 7 (p. 124) 154 20.3.6 Theorem 8 (p. 125) 158 20.4 HOMOGENEOUS CONCATENATION STRUCTURES 159 20.4.1 Nature of Homogeneous Concatenation Structures 159 20.4.2 Real Unit Concatenation Structures 160 20.4.3 Characterization of Homogeneity: PCS 163 20.4.4 Characterizations of Homogeneity: Solvable, Idempotent Structures 165 20.4.5 Mixture Spaces of Gambles 167 20.4.6 The Dual Bilinear Utility Model 169 20.5 PROOFS 173 20.5.1 Theorem 9 (p. 142) 173 20.5.2 Theorem 11 (p. 144) 175 20.5.3 Theorem 24, Chapter 19 (p. 103) 177 20.5.4 Theorem 14 (p. 147) 180 20.5.5 Theorem 15 (p. 147) 182 20.5.6 Theorem 16 (p. 148) 183 20.5.7 Theorem 17 (p. 148) 184 20.5.8 Theorem 18 (p. 150) 188 20.5.9 Theorem 19 (p. 153) 192 20.6 HOMOGENEOUS CONJOINT STRUCTURES 197 20.6.1 Component Homogeneity and Uniqueness 197 20.6.2 Singular Points in Conjoint Structures 198 20.6.3 Forcing the Thomsen Condition 200 20.7 PROOFS 201 20.7.1 Theorem 22 (p. 181) 201 20.7.2 Theorem 23 (p. 182) 202 20.7.3 Theorem 24 (p. 182) 203 20.7.4 Theorem 25 (p. 183) 207 EXERCISES 209 Chapter 21 Axiomatization 212 21.1 AXIOM SYSTEMS AND REPRESENTATIONS 213 21.1.1 Why Do Scientists and Mathematicians Axiomatize? 213 21.1.2 The Axiomatic-Representational Viewpoint in Measurement 218 21.1.3 Types of Representing Structures 219 21.2 ELEMENTARY FORMALIZATION OF THEORIES 221 21.2.1 Elementary Languages 221 21.2.2 Models of Elementary Languages 225 21.2.3 General Theorems about Elementary Logic 230 21.2.4 Elementary Theories 232 21.3 DEFINABILITY AND INTERPRETABILITY 235 21.3.1 Definability 235 21.3.2 Interpretability 241 21.4 SOME THEOREMS ON AXIOMATIZABILITY 242 21.5 PROOFS 246 21.5.1 Theorem 6 (p. 226) 246 21.5.2 Theorem 7 (p. 227) 247 21.5.3 Theorem 8 (p. 228) 247 21.5.4 Theorem 9 (p. 229) 248 21.6 FINITE AXIOMATIZABILITY 248 21.6.1 Axiomatizable by a Universal Sentence 251 21.6.2 Proof of Theorem 12 (p. 237) 258 21.6.3 Finite Axiomatizability of Finitary Classes 259 21.7 THE ARCHIMEDEAN AXIOM 263 21.8 TESTABILITY OF AXIOMS 268 21.8.1 Finite Data Structures 271 21.8.2 Convergence of Finite to Infinite Data Structures 273 21.8.3 Testability and Constructability 276 21.8.4 Diagnostic versus Global Tests 278 EXERCISES 282 Chapter 22 Invariance and Meaning fulness 284 22.1 INTRODUCTION 284 22.2 METHODS OF DEFINING MEANINGFUL RELATIONS 286 22.2.1 Definitions in First-Order Theories 288 22.2.2 Reference and Structure Invariance 290 22.2.3 An Example: Independence in Probability Theory 293 22.2.4 Definitions with Particular Representations 294 22.2.5 Parametrized Numerical Relations 295 22.2.6 An Example: Hooke's Law 297 22.2.7 A Necessary Condition for Meaningfulness 299 22.2.8 Irreducible Structures: Reference Invariance of Numerical Equality 301 22.3 CHARACTERIZATIONS OF REFERENCE INVARIANCE 302 22.3.1 Permissible Transformations 302 22.3.2 The Criterion of Invariance under Permissible Transformations 304 22.3.3 The Condition of Structure Invariance 304 22.4 PROOFS 307 22.4.1 Theorem 3 (p. 287) 307 22.4.2 Theorem 4 (p. 287) 307 22.4.3 Theorem 5 (p. 288) 308 22.5 DEFINABILITY 309 22.6 MEANINGFULNESS AND STATISTICS 311 22.6.1 Examples 312 22.6.2 Meaningful Relations Involving Population Means 315 22.6.3. Inferences about Population Means 316 22.6.4 Parametric Models for Populations 316 22.6.5 Measurement Structures and Parametric Models for Populations 318 22.6.6 Meaningful Relations in Uniform Structures 322 22.7 DIMENSIONAL INVARIANCE 324 22.7.1 Structures of Physical Quantities 326 22.7.2 Triples of Scales 329 22.7.3 Representation and Uniqueness Theorem for Physical Attributes 332 22.7.4 Physically Similar Systems 335 22.7.5 Fundamental versus Index Measurement 340 22.8 PROOFS 343 22.8.1 Theorem 6 (p. 315) 343 22.8.2 Theorem 7 (p. 315) 344 22.9 REPRISE: UNIQUENESS, AUTOMORPHISMS, AND CONSTRUCTABILITY 346 22.9.1 Alternative Representations 346 22.9.2 Nonuniqueness and Automorphisms 347 22.9.3 Invariance under Automorphisms 349 22.9.4 Constructability of Representations 350 EXERCISES 353 References 355 Author Index 364 Subject Index 368 Back Cover 374 A classic series in the field of quantitative measurement, Volume I introduces the distinct mathematical results that serve to formulate numerical representations of qualitative structures. Volume II extends the subject in the direction of geometrical, threshold, and probabilistic representations, and Volume III examines representation as expressed in axiomatization and invariance. 1990 edition.
Third volume in the three books (https://www.goodreads.com/series/82607-foundations-of-measurement) Foundations of Measurement series. Table of Contents Preface Acknowledgments 1. Overview 2. Nonadditive representations 3. Scale types 4. Axiomatization 5. Invariance and meaningfulness References