Foundations of Measurement Volume I Additive and Polynomial Representations

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

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ISBN: 9780486453149 | 0486453146
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. 1971 edition.

Preface

xvii

Mathematical Background

xix

Selecting Among the Chapters

Acknowledgments

xxi

Notational Conventions

xxiii

1. Introduction

1.1 THREE BASIC PROCEDURES OF FUNDAMENTAL MEASUREMENT

1.1.1 Ordinal Measurement

1.1.2 Counting of Units

1.1.3 Solving Inequalities

1.2 THE PROBLEM OF FOUNDATIONS

1.2.1 Qualitative Assumptions: Axioms

1.2.2 Homomorphisms of Relational Structures: Representation Theorems

1.2.3 Uniqueness Theorems

1.2.4 Measurement Axioms as Empirical Laws

1.2.5 Other Aspects of the Problem of Foundations

1.3 ILLUSTRATIONS OF MEASUREMENT STRUCTURES

1.3.1 Finite Weak Orders

1.3.2 Finite, Equally Spaced, Additive Conjoint Structures

1.4 CHOOSING AN AXIOM SYSTEM

1.4.1 Necessary Axioms

1.4.2 Nonnecessary Axioms

1.4.3 Necessary and Sufficient Axiom Systems

1.4.4 Archimedean Axioms

1.4.5 Consistency, Completeness, and Independence

1.5 EMPIRICAL TESTING OF A THEORY OF MEASUREMENT

1.5.1 Error of Measurement

1.5.2 Selection of Objects in Tests of Axioms

1.6 ROLES OF THEORIES OF MEASUREMENT IN THE SCIENCES

1.7 PLAN OF THE BOOK

EXERCISES

2. Construction of Numerical Functions

2.1 REAL-VALUED FUNCTIONS ON SIMPLY ORDERED SETS

2.2 ADDITIVE FUNCTIONS ON ORDERED ALGEBRAIC STRUCTURES

2.2.1 Archimedean Ordered Semigroups

2.2.2 Proof of Theorem 4 (Outline)

2.2.3 Preliminary Lemmas

2.2.4 Proof of Theorems 4 and 4' (Details)

2.2.5 Archimedean Ordered Groups

2.2.6 Note on Holder's Theorem

2.2.7 Archimedean Ordered Semirings

2.3 FINITE SETS OF HOMOGENEOUS LINEAR INEQUALITIES

2.3.1 Intuitive Explanation of the Solution Criterion

2.3.2 Vector Formulation and Preliminary Lemmas

2.3.3 Proof of Theorem 7

2.3.4 Topological Proof of Theorem 7

EXERCISES

3. Extensive Measurement

3.1 INTRODUCTION

3.2 NECESSARY AND SUFFICIENT CONDITIONS

3.2.1 Closed Extensive Structures

3.2.2 The Periodic Case

3.3 PROOFS

3.3.1 Consistency and Independence of the Axioms of Definition 1

3.3.2 Preliminary Lemmas

3.3.3 Theorem 1

3.4 SUFFICIENT CONDITIONS WHEN THE CONCATENATION OPERATION IS NOT CLOSED

3.4.1 Formulation of the Non-Archimedean Axioms

3.4.2 Formulation of the Archimedean Axiom

3.4.3 The Axiom System and Representation Theorem

3.5 PROOFS

3.5.1 Consistency and Independence of the Axioms of Definition 3

3.5.2 Preliminary Lemmas

3.5.3 Theorem 3

3.6 EMPIRICAL INTERPRETATIONS IN PHYSICS

3.6.1 Length

3.6.2 Mass

3.6.3 Time Duration

3.6.4 Resistance

3.6.5 Velocity

3.7 ESSENTIAL MAXIMA IN EXTENSIVE STRUCTURES

3.7.1 Nonadditive Representations

3.7.2 Simultaneous Axiomatization of Length and Velocity

3.8 PROOFS

3.8.1 Consistency and Independence of the Axioms of Definition 5

3.8.2 Theorem 6

3.8.3 Theorem 7

3.9 ALTERNATIVE NUMERICAL REPRESENTATIONS

3.10 CONSTRUCTIVE METHODS

102

3.10.1 Extensive Multiples

103

3.10.2 Standard Sequences

105

3.11 PROOFS

106

3.11.1 Theorem 8

106

3.11.2 Preliminary Lemmas

107

3.11.3 Theorem 9

109

3.12 CONDITIONALLY CONNECTED EXTENSIVE STRUCTURES

111

3.12.1 Thermodynamic Motivation

111

3.12.2 Formulation of the Axioms

113

3.12.3 The Axiom System and Representation Theorem

114

3.12.4 Statistical Entropy

116

3.13 PROOFS

117

3.13.1 Preliminary Lemmas

117

3.13.2 A Group-Theoretic Result

119

3.13.3 Theorem 10

120

3.13.4 Theorem 11

121

3.14 EXTENSIVE MEASUREMENT IN THE SOCIAL SCIENCES

123

3.14.1 The Measurement of Risk

124

3.14.2 Proof of Theorem 13

128

3.15 LIMITATIONS OF EXTENSIVE MEASUREMENT

130

EXERCISES

132

4. Difference Measurement

4.1 INTRODUCTION

136

4.1.1 Direct Comparison of Intervals

137

4.1.2 Indirect Comparison of Intervals

141

4.1.3 Axiomatization of Difference Measurement

143

4.2 POSITIVE-DIFFERENCE STRUCTURES

145

4.3 PROOF OF THEOREM 1

148

4.4 ALGEBRAIC-DIFFERENCE STRUCTURES

150

4.4.1 Axiom System and Representation Theorem

151

4.4.2 Alternative Numerical Representations

152

4.4.3 Difference-and-Ratio Structures

152

4.4.4 Strict Inequalities and Approximate Standard Sequences

155

4.5 PROOFS

157

4.5.1 Preliminary Lemmas

157

4.5.2 Theorem 2

158

4.5.3 Theorem 3

158

4.6 CROSS-MODALITY ORDERING

164

4.7 PROOF OF THEOREM 4

166

4.8 FINITE, EQUALLY SPACED DIFFERENCE STRUCTURES

167

4.9 PROOFS

168

4.9.1 Preliminary Lemma

168

4.9.2 Theorem 5

169

4.10 ABSOLUTE-DIFFERENCE STRUCTURES

170

4.11 PROOFS

174

4.11.1 Preliminary Lemmas

174

4.11.2 Theorem 6

176

4.12 STRONGLY CONDITIONAL DIFFERENCE STRUCTURES

177

4.13 PROOFS

184

4.13.1 Preliminary Lemmas

184

4.13.2 Theorem 7

188

EXERCISES

195

5. Probability Representations

5.1 INTRODUCTION

199

5.2 A REPRESENTATION BY UNCONDITIONAL PROBABILITY

202

5.2.1 Necessary Conditions: Qualitative Probability

202

5.2.2 The Nonsufficiency of Qualitative Probability

205

5.2.3 Sufficient Conditions

206

5.2.4 Preference Axioms for Qualitative Probability

208

5.3 PROOFS

211

5.3.1 Preliminary Lemmas

211

5.3.2 Theorem 2

212

5.4 MODIFICATIONS OF THE AXIOM SYSTEM

214

5.4.1 QM-Algebra of Sets

214

5.4.2 Countable Additivity

215

5.4.3 Finite Probability Structures with Equivalent Atoms

216

5.5 PROOFS

217

5.5.1 Structure of QM-Algebras of Sets

217

5.5.2 Theorem 4

218

5.5.3 Theorem 6

220

5.6 A REPRESENTATION BY CONDITIONAL PROBABILITY

220

5.6.1 Necessary Conditions: Qualitative Conditional Probability

222

5.6.2 Sufficient Conditions

224

5.6.3 Further Discussion of Definition 8 and Theorem 7

225

5.6.4 A Nonadditive Conditional Representation

227

5.7 PROOFS

228

5.7.1 Preliminary Lemmas

228

5.7.2 An Additive Unconditional Representation

232

5.7.3 Theorem 7

233

5.7.4 Theorem 8

236

5.8 INDEPENDENT EVENTS

238

5.9 PROOF OF THEOREM 10

241

EXERCISES

243

6. Additive Conjoint Measurement

6.1 SEVERAL NOTIONS OF INDEPENDENCE

245

6.1.1 Independent Realization of the Components

246

6.1.2 Decomposable Structures

247

6.1.3 Additive Independence

247

6.1.4 Independent Relations

248

6.2 ADDITIVE REPRESENTATION OF TWO COMPONENTS

250

6.2.1 Cancellation Axioms

250

6.2.2 Archimedean Axiom

253

6.2.3 Sufficient Conditions

254

6.2.4 Representation Theorem and Method of Proof

257

6.2.5 Historical Note

259

6.3 PROOFS

261

6.3.1 Independence of the Axioms of Definition 7

261

6.3.2 Theorem 1

262

6.3.3 Preliminary Lemmas for Bounded Symmetric Structures

262

6.3.4 Theorem 2

264

6.4 EMPIRICAL EXAMPLES

267

6.4.1 Examples from Physics

267

6.4.2 Examples from the Behavioral Sciences

268

6.5 MODIFICATIONS OF THE THEORY

271

6.5.1 Omission of the Archimedean Property

271

6.5.2 Alternative Numerical Representations

273

6.5.3 Transforming a Nonadditive Representation into an Additive One

273

6.5.4 Subtractive Structures

274

6.5.5 Need for Conjoint Measurement on B subset of A1 x A2

275

6.5.6 Symmetries of Independent and Dependent Variables

276

6.5.7 Alternative Factorial Decompositions

278

6.6 PROOFS

279

6.6.1 Preliminary Lemmas

279

6.6.2 Theorem 3

280

6.6.3 Theorem 4

281

6.6.4 Theorem 6

282

6.7 INDIFFERENCE CURVES AND UNIFORM FAMILIES OF FUNCTIONS

283

6.7.1 A Curve Through Every Point

285

6.7.2 A Finite Number of Curves

286

6.8 PROOFS

288

6.8.1 Theorem 7

288

6.8.2 Theorem 8

289

6.8.3 Preliminary Lemmas About Uniform Families

290

6.8.4 Theorem 9

292

6.9 BISYMMETRIC STRUCTURES

293

6.9.1 Sufficient Conditions

293

6.9.2 A Finite, Equally Spaced Case

297

6.10 PROOFS

298

6.10.1 Theorem 10

298

6.10.2 Theorem 11

300

6.11 ADDITIVE REPRESENTATION OF n COMPONENTS

301

6.11.1 The General Case

301

6.11.2 The Case of Identical Components

303

6.12 PROOFS

306

6.12.1 Preliminary Lemma

306

6.12.2 Theorem 13

307

6.12.3 Theorem 14

309

6.12.4 Theorem 15

310

6.13 CONCLUDING REMARKS

311

EXERCISES

312

7. Polynomial Conjoint Measurement

7.1 INTRODUCTION

316

7.2 DECOMPOSABLE STRUCTURES

317

7.2.1 Necessary and Sufficient Conditions

318

7.2.2 Proof of Theorem 1

320

7.3 POLYNOMIAL MODELS

321

7.3.1 Examples

321

7.3.2 Decomposability and Equivalence of Polynomial Models

325

7.3.3 Simple Polynomials

327

7.4 DIAGNOSTIC ORDINAL PROPERTIES

329

7.4.1 Sign Dependence

329

7.4.2 Proofs of Theorems 2 and 3

335

7.4.3 Joint-Independence Conditions

339

7.4.4 Cancellation Conditions

340

7.4.5 Diagnosis for Simple Polynomials in Three Variables

345

7.5 SUFFICIENT CONDITIONS FOR THREE-VARIABLE SIMPLE POLYNOMIALS

347

7.5.1 Representation and Uniqueness Theorems

348

7.5.2 Heuristic Proofs

350

7.5.3 Generalizations to Four or More Variables

353

7.6 PROOFS

356

7.6.1 A Preliminary Result

356

7.6.2 Theorem 4

357

7.6.3 Theorem 5

361

7.6.4 Theorem 6

364

EXERCISES

366

8. Conditional Expected Utility

8.1 INTRODUCTION

369

8.2 A FORMULATION OF THE PROBLEM

372

8.2.1 The Primitive Notions

372

8.2.2 A Restriction on 2

375

8.2.3 The Desired Representation Theorem

376

8.2.4 Necessary Conditions

376

8.2.5 Nonnecessary Conditions

379

8.2.6 The Axiom System and Representation Theorem

380

8.3 PROOFS

382

8.3.1 Preliminary Lemmas

382

8.3.2 Theorem 1

385

8.4 TOPICS IN UTILITY AND SUBJECTIVE PROBABILITY

391

8.4.1 Utility of Consequences

391

8.4.2 Relations Between Additive and Expected Utility

393

8.4.3 The Consistency Principle for the Utility of Money

395

8.4.4 Expected Utility and Risk

398

8.4.5 Relations Between Subjective and Objective Probability

400

8.4.6 A Method for Estimating Subjective Probabilities

400

8.5 PROOFS

401

8.5.1 Theorem 3

401

8.5.2 Theorem 4

403

8.5.3 Theorem 5

404

8.5.4 Theorem 6

405

8.5.5 Theorem 7

406

8.6 OTHER FORMULATIONS OF RISKY AND UNCERTAIN DECISIONS

406

8.6.1 Mixture Sets and Gambles

407

8.6.2 Propositions as Primitives

411

8.6.3 Statistical Decision Theory

412

8.6.4 Comparision of Statistical and Conditional Decision Theories in the Finite Case

414

8.7 CONCLUDING REMARKS

417

8.7.1 Prescriptive Versus Descriptive Interpretations

417

8.7.2 Open Problems

420

EXERCISES

420

9. Measurement Inequalities

9.1 INTRODUCTION

423

9.2 FINITE LINEAR STRUCTURES

427

9.2.1 Additivity

428

9.2.2 Probability

432

9.3 PROOF OF THEOREM 1

433

9.4 APPLICATIONS

434

9.4.1 Scaling Considerations

434

9.4.2 Empirical Examples

436

9.5 POLYNOMIAL STRUCTURES

447

9.6 PROOFS

450

9.6.1 Theorem 4

450

9.6.2 Theorem 5

451

9.6.3 Theorem 6

451

EXERCISES

452

10. Dimensional Analysis and Numerical Laws

10.1 INTRODUCTION

454

10.2 THE ALGEBRA OF PHYSICAL QUANTITIES

459

10.2.1 The Axiom System

459

10.2.2 General Theorems

462

10.3 THE PI THEOREM OF DIMENSIONAL ANALYSIS

464

10.3.1 Similarities

464

10.3.2 Dimensionally Invariant Functions

466

10.4 PROOFS

467

10.4.1 Preliminary Lemmas

467

10.4.2 Theorems 1 and 2

469

10.4.3 Theorem 3

470

10.4.4 Theorem 4

470

10.5 EXAMPLES OF DIMENSIONAL ANALYSIS

471

10.5.1 The Simple Pendulum

472

10.5.2 Errors of Commission and Omission

474

10.5.3 Dimensional Analysis as an Aid in Obtaining Exact Solutions

479

10.5.4 Conclusion

480

10.6 BINARY LAWS AND UNIVERSAL CONSTANTS

480

10.7 TRINARY LAWS AND DERIVED MEASURES

483

10.7.1 Laws of Similitude

484

10.7.2 Laws of Exchange

488

10.7.3 Compatibility of the Trinary Laws

490

10.7.4 Some Relations Among Extensive, Difference, and Conjoint Structures

491

10.8 PROOFS

493

10.8.1 Preliminary Lemma

493

10.8.2 Theorem 5

494

10.8.3 Theorem 6

496

10.8.4 Theorem 7

498

10.9 EMBEDDING PHYSICAL ATTRIBUTES IN A STRUCTURE OF PHYSICAL QUANTITIES

499

10.9.1 Assumptions About Physical Attributes

499

10.9.2 Fundamental, Derived, and Quasi-Derived Attributes

502

10.10 WHY ARE NUMERICAL LAWS DIMENSIONALLY INVARIANT?

503

10.10.1 Three Points of View

503

10.10.2 Physically Similar Systems

506

10.10.3 Relations to Causey's Theory

512

10.11 PROOFS

513

10.11.1 Theorem 12

513

10.11.2 Theorem 13

515

10.12 INTERVAL SCALES IN DIMENSIONAL ANALYSIS

515

10.13 PROOFS

520

10.13.1 Preliminary Lemma

520

10.13.2 Theorem 14

521

10.13.3 Theorem 15

522

10.13.4 Theorem 16

523

10.14 PHYSICAL QUANTITIES IN MECHANICS AND GENERALIZATIONS OF DIMENSIONAL INVARIANCE

523

10.14.1 Generalized Galilean Invariance

526

10.14.2 Lorentz Invariance and Relativistic Mechanics

532

10.15 CONCLUDING REMARKS

535

EXERCISES

536

DIMENSIONS AND UNITS OF PHYSICAL QUANTITIES

539

Answers and Hints to Selected Exercises

545

References

551

Author Index

565

Subject Index

570

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Foundations of Measurement Volume I Additive and Polynomial Representations

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Author Biography

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