Foundations of Measurement Volume I Additive and Polynomial Representations

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

• ISBN13: 9780486453149
• ISBN10: 0486453146
• Format: Paperback
• Copyright: 2006-12-15
• Publisher: Dover Publications

Summary

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.

Author Biography

David H. Krantz is affiliated with Columbia University; R. Duncan Luce with the University of California, Irvine; and Patrick Suppes with Stanford University. Amos Tversky is deceased.

Table of Contents

Preface	xvii
Mathematical Background	xix
Selecting Among the Chapters	xx
Acknowledgments	xxi
Notational Conventions	xxiii
1. Introduction
1.1 THREE BASIC PROCEDURES OF FUNDAMENTAL MEASUREMENT	1
1.1.1 Ordinal Measurement	2
1.1.2 Counting of Units	3
1.1.3 Solving Inequalities	5
1.2 THE PROBLEM OF FOUNDATIONS	6
1.2.1 Qualitative Assumptions: Axioms	6
1.2.2 Homomorphisms of Relational Structures: Representation Theorems	8
1.2.3 Uniqueness Theorems	9
1.2.4 Measurement Axioms as Empirical Laws	13
1.2.5 Other Aspects of the Problem of Foundations	13
1.3 ILLUSTRATIONS OF MEASUREMENT STRUCTURES	13
1.3.1 Finite Weak Orders	14
1.3.2 Finite, Equally Spaced, Additive Conjoint Structures	17
1.4 CHOOSING AN AXIOM SYSTEM	21
1.4.1 Necessary Axioms	21
1.4.2 Nonnecessary Axioms	23
1.4.3 Necessary and Sufficient Axiom Systems	24
1.4.4 Archimedean Axioms	25
1.4.5 Consistency, Completeness, and Independence	26
1.5 EMPIRICAL TESTING OF A THEORY OF MEASUREMENT	26
1.5.1 Error of Measurement	27
1.5.2 Selection of Objects in Tests of Axioms	28
1.6 ROLES OF THEORIES OF MEASUREMENT IN THE SCIENCES	31
1.7 PLAN OF THE BOOK	33
EXERCISES	35
2. Construction of Numerical Functions
2.1 REAL-VALUED FUNCTIONS ON SIMPLY ORDERED SETS	38
2.2 ADDITIVE FUNCTIONS ON ORDERED ALGEBRAIC STRUCTURES	43
2.2.1 Archimedean Ordered Semigroups	44
2.2.2 Proof of Theorem 4 (Outline)	46
2.2.3 Preliminary Lemmas	47
2.2.4 Proof of Theorems 4 and 4' (Details)	48
2.2.5 Archimedean Ordered Groups	53
2.2.6 Note on Holder's Theorem	53
2.2.7 Archimedean Ordered Semirings	54
2.3 FINITE SETS OF HOMOGENEOUS LINEAR INEQUALITIES	59
2.3.1 Intuitive Explanation of the Solution Criterion	59
2.3.2 Vector Formulation and Preliminary Lemmas	61
2.3.3 Proof of Theorem 7	66
2.3.4 Topological Proof of Theorem 7	67
EXERCISES	69
3. Extensive Measurement
3.1 INTRODUCTION	71
3.2 NECESSARY AND SUFFICIENT CONDITIONS	72
3.2.1 Closed Extensive Structures	72
3.2.2 The Periodic Case	75
3.3 PROOFS	77
3.3.1 Consistency and Independence of the Axioms of Definition 1	77
3.3.2 Preliminary Lemmas	77
3.3.3 Theorem 1	80
3.4 SUFFICIENT CONDITIONS WHEN THE CONCATENATION OPERATION IS NOT CLOSED	81
3.4.1 Formulation of the Non-Archimedean Axioms	82
3.4.2 Formulation of the Archimedean Axiom	83
3.4.3 The Axiom System and Representation Theorem	84
3.5 PROOFS	85
3.5.1 Consistency and Independence of the Axioms of Definition 3	85
3.5.2 Preliminary Lemmas	86
3.5.3 Theorem 3	87
3.6 EMPIRICAL INTERPRETATIONS IN PHYSICS	87
3.6.1 Length	87
3.6.2 Mass	89
3.6.3 Time Duration	89
3.6.4 Resistance	90
3.6.5 Velocity	91
3.7 ESSENTIAL MAXIMA IN EXTENSIVE STRUCTURES	92
3.7.1 Nonadditive Representations	92
3.7.2 Simultaneous Axiomatization of Length and Velocity	94
3.8 PROOFS	96
3.8.1 Consistency and Independence of the Axioms of Definition 5	96
3.8.2 Theorem 6	96
3.8.3 Theorem 7	98
3.9 ALTERNATIVE NUMERICAL REPRESENTATIONS	99
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