Foundations of Measurement Volume III Representation, Axiomatization, and Invariance

, by Suppes, Patrick; Krantz, David H.; Luce, R. Duncan; Tversky, Amos

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ISBN: 9780486453163 | 0486453162
Cover: Paperback
Copyright: 12/15/2006

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This classic series in the field of quantitative measurement established the formal foundations for measurement, justifying the assignment of numbers to objects in terms of their structural correspondence.Volume Iintroduces the distinct mathematical results that serve to formulate numerical representations of qualitative structures.Volume IIextends the subject in the direction of geometrical, threshold, and probabilistic representations, andVolume IIIexamines representation as expressed in axiomatization and invariance. 1990 edition.

Preface

xiii

Acknowledgments

Overview

Nonadditive Representations (Chapter 19)

(4)

Examples

(1)

Representation and Uniqueness of Positive Operations

(1)

Intensive Structures

(1)

Conjoint Structures and Distributive Operations

(1)

Scale Types (Chapter 20)

(4)

A Classification of Automorphism Groups

(2)

Unit Representations

(1)

Characterization of Homogeneous Concatenation and Conjoint Structures

(1)

Reprise

(1)

Axiomatization (Chapter 21)

(3)

Types of Axiom

(2)

Theorems on Axiomatizability

(1)

Testability of Axioms

(1)

Invariance and Meaningfulness (Chapter 22)

(4)

Types of Invariance

(1)

Applications of Meaningfulness

(2)

Nonadditive Representations

Introduction

(7)

Inessential and Essential Nonadditives

(4)

General Binary Operations

(1)

Overview

(2)

Types of Concatenation Structure

(12)

Concatenation Structures and Their Properties

(2)

Some Numerical Examples

(6)

Archimedean Properties

(4)

Representations of PCSs

(11)

General Definitions

(2)

Uniqueness and Construction of a Representation of a PCS

(1)

Existence of a Representation

(4)

Automorphism Groups of PCSs

(2)

Continuous PCSs

(2)

Completions of Total Orders and PCSs

(8)

Order Isomorphisms onto Real Intervals

(1)

Completions of Total Orders

(3)

Completions of Closed PCSs

(3)

Proofs about Concatenation Structures

(19)

Theorem 1

(1)

Lemmas 1--6, Theorem 2

(4)

Theorem 2

(1)

Construction of PCS Homomorphisms

(2)

Theorem 3

(5)

Theorem 4

(1)

Theorem 5

(1)

Theorem 6

(2)

Corollary to Theorem 7

(1)

Theorem 9

(2)

Connections between Conjoint and Concatenation Structures

(12)

Conjoint Structures: Introduction and General Definitions

(2)

Total Concatenation Structures Induced by Conjoint Structures

(2)

Factorizable Automorphisms

(2)

Total Concatenation Structures Induced by Closed, Idempotent Concatenation Structures

(2)

Intensive Structures Related to PCSs by Doubling Functions

(2)

Operations That Distribute over Conjoint Structures

(2)

Representations of Solvable Conjoint and Concatenation Structures

(2)

Conjoint Structures

(1)

Solvable, Closed, Archimedean Concatenation Structures

(1)

Intensive Concatenation Structures with Doubling Functions

(1)

Proofs

(12)

Theorem 11

(3)

Theorem 12

(1)

Theorem 13

(4)

Theorem 14, Part (iii)

(1)

Theorem 15

(1)

Theorem 18

(3)

Theorem 21

101

(1)

Bisymmetry and Related Properties

101

(7)

General Definitions

101

(2)

Equivalences in Closed, Idempotent, Solvable, Dedekind Complete Structures

103

(1)

Bisymmetry in the 1-Point Unique Case

103

(1)

Exercises

104

(4)

Scale Types

Introduction

108

(4)

Constructibility and Symmetry

108

(3)

Problem in Understanding Scale Types

111

(1)

Homogeneity, Uniqueness, and Scale Type

112

(14)

Stevens' Classification

112

(2)

Decomposing the Classification

114

(1)

Formal Definitions

115

(2)

Relations among Structure, Homogeneity, and Uniqueness

117

(2)

Scale Types of Real Relational Structures

119

(3)

Structures with Homogeneous, Archimedean Ordered Translation Groups

122

(3)

Representations of Dedekind Complete Distributive Triples

125

(1)

Proofs

126

(16)

Theorem 2

126

(1)

Theorem 3

127

(1)

Theorem 4

128

(1)

Theorem 5

128

(9)

Theorem 7

137

(4)

Theorem 8

141

(1)

Homogeneous Concatenation Structures

142

(14)

Nature of Homogeneous Concatenation Structures

142

(1)

Real Unit Concatenation Structures

143

(3)

Characterizations of Homogeneity: PCS

146

(2)

Characterizations of Homogeneity: Solvable, Idempotent Structures

148

(2)

Mixture Spaces of Gambles

150

(2)

The Dual Bilinear Utility Model

152

(4)

Proofs

156

(24)

Theorem 9

156

(2)

Theorem 11

158

(2)

Theorem 24, Chapter 19

160

(3)

Theorem 14

163

(2)

Theorem 15

165

(1)

Theorem 16

166

(1)

Theorem 17

167

(1)

Theorem 18

168

(7)

Theorem 19

175

(5)

Homogeneous Conjoint Structures

180

(4)

Component Homogeneity and Uniqueness

180

(1)

Singular Points in Conjoint Structures

181

(2)

Forcing the Thomsen Condition

183

(1)

Proofs

184

(12)

Theorem 22

184

(1)

Theorem 23

185

(1)

Theorem 24

186

(4)

Theorem 25

190

(2)

Exercises

192

(4)

Axiomatization

Axiom Systems and Representations

196

(8)

Why Do Scientists and Mathematicians Axiomatize?

196

(5)

The Axiomatic-Representational Viewpoint in Measurement

201

(1)

Types of Representing Structures

202

(2)

Elementary Formalization of Theories

204

(14)

Elementary Languages

204

(4)

Models of Elementary Languages

208

(5)

General Theorems about Elementary Logic

213

(2)

Elementary Theories

215

(3)

Definability and Interpretability

218

(7)

Definability

218

(6)

Interpretability

224

(1)

Some Theorems on Axiomatizability

225

(4)

Proofs

229

(2)

Theorem 6

229

(1)

Theorem 7

230

(1)

Theorem 8

230

(1)

Theorem 9

231

(1)

Finite Axiomatizability

231

(15)

Axiomatizable by a Universal Sentence

234

(7)

Proof of Theorem 12

241

(1)

Finite Axiomatizability of Finitary Classes

242

(4)

The Archimedean Axiom

246

(5)

Testability of Axioms

251

(16)

Finite Data Structures

254

(2)

Convergence of Finite to Infinite Data Structures

256

(3)

Testability and Constructibility

259

(2)

Diagnostic versus Global Tests

261

(4)

Exercises

265

(2)

Invariance and Meaningfulness

Introduction

267

(2)

Methods of Defining Meaningful Relations

269

(16)

Definitions in First-Order Theories

271

(2)

Reference and Structure Invariance

273

(3)

An Example: Independence in Probability Theory

276

(1)

Definitions with Particular Representations

277

(1)

Parametrized Numerical Relations

278

(2)

An Example: Hooke's Law

280

(2)

A Necessary Condition for Meaningfulness

282

(2)

Irreducible Structures: Reference Invariance of Numerical Equality

284

(1)

Characterizations of Reference Invariance

285

(5)

Permissible Transformations

285

(2)

The Criterion of Invariance under Permissible Transformations

287

(1)

The Condition of Structure Invariance

287

(3)

Proofs

290

(2)

Theorem 3

290

(1)

Theorem 4

290

(1)

Theorem 5

291

(1)

Definability

292

(2)

Meaningfulness and Statistics

294

(13)

Examples

295

(3)

Meaningful Relations Involving Population Means

298

(1)

Inferences about Population Means

299

(1)

Parametric Models for Populations

299

(2)

Measurement Structures and Parametric Models for Populations

301

(4)

Meaningful Relations in Uniform Structures

305

(2)

Dimensional Invariance

307

(19)

Structures of Physical Quantities

309

(3)

Triples of Scales

312

(3)

Representation and Uniqueness Theorem for Physical Attributes

315

(3)

Physically Similar Systems

318

(5)

Fundamental versus Index Measurement

323

(3)

Proofs

326

(3)

Theorem 6

326

(1)

Theorem 7

327

(2)

Reprise: Uniqueness, Automorphisms, and Constructibility

329

(9)

Alternative Representations

329

(1)

Nonuniqueness and Automorphisms

330

(2)

Invariance under Automorphisms

332

(1)

Constructibility of Representations

333

(3)

Exercises

336

(2)

References

338

(9)

Author Index

347

(4)

Subject Index

351

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Foundations of Measurement Volume III Representation, Axiomatization, and Invariance

Summary

Table of Contents

Supplemental Materials