Additive and Polynomial Representations deals with major representation theorems in which the qualitative structure is reflected as some polynomial function of one or more numerical functions defined on the basic entities. Examples are additive expressions of a single measure (such as the probability of disjoint events being the sum of their probabilities), and additive expressions of two measures (such as the logarithm of momentum being the sum of log mass and log velocity terms). The book describes the three basic procedures of fundamental measurement as the mathematical pivot, as the utilization of constructive methods, and as a series of isomorphism theorems leading to consistent numerical solutions. The text also explains the counting of units in relation to an empirical relational structure which contains a concatenation operation. The book notes some special variants which arise in connection with relativity and thermodynamics. The text cites examples from physics and psychology for which additive conjoint measurement provides a possible method of fundamental measurement. The book will greatly benefit mathematicians, econometricians, and academicians in advanced mathematics or physics.

Contenu

Preface

Mathematical Background

Selecting Among the Chapters

Acknowledgments

Notational Conventions

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
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 Hölder'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
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
3.10.1 Extensive Multiples
3.10.2 Standard Sequences
3.11 Proofs
3.11.1 Theorem 8
3.11.2 Preliminary Lemmas
3.11.3 Theorem 9
3.12 Conditionally Connected Extensive Structures
3.12.1 Thermodynamic Motivation
3.12.2 Formulation of the Axioms
3.12.3 The Axiom System and Representation Theorem
3.12.4 Statistical Entropy
3.13 Proofs
3.13.1 Preliminary Lemmas
3.13.2 A Group-Theoretic Result
3.13.3 Theorem 10
3.13.4 Theorem 11
3.14 Extensive Measurement in the Social Sciences
3.14.1 The Measurement of Risk
3.14.2 Proof of Theorem 13
3.15 Limitations of Extensive Measurement
Exercises
Difference Measurement
4.1 Introduction
4.1.1 Direct Comparison of Intervals
4.1.2 Indirect Comparison of Intervals
4.1.3 Axiomatization of Difference Measurement
4.2 Positive-Difference Structures
4.3 Proof of Theorem 1
4.4 Algebraic-Difference Structures
4.4.1 Axiom System and Representation Theorem
4.4.2 Alternative Numerical Representations
4.4.3 Difference-and-Ratio Structures
4.4.4 Strict Inequalities and Approximate Standard Sequences
4.5 Proofs
4.5.1 Preliminary Lemmas
4.5.2 Theorem 2
4.5.3 Theorem 3
4.6 Cross-Modality Ordering
4.7 Proof of Theorem 4
4.8 Finite, Equally Spaced Difference Structures
4.9 Proofs
4.9.1 Preliminary Lemma
4.9.2 Theorem 5
4.10 Absolute-Difference Structures
4.11 Proofs
4.11.1 Preliminary Lemmas
4.11.2 Theorem 6
4.12 Strongly Conditional Difference Structures
4.13 Proofs
4.13.1 Preliminary Lemmas
4.13.2 Theorem 7
Exercises
Probability Representations
5.1 Introduction
5.2 A Representation by Unconditional Probability
5.2.1 Necessary Conditions: Qualitative Probability
5.2.2 The Nonsufficiency of Qualitative Probability
5.2.3 Sufficient Conditions
5.2.4 Preference Axioms for Qualitative Probability
5.3 Proofs
5.3.1 Preliminary Lemmas
5.3.2 Theorem 2
5.4 Modifications of the Axiom System
5.4.1 QM-Algebra of Sets
5.4.2 Countable Additivity
5.4.3 Finite Probability Structures with Equivalent Atoms
5.5 Proofs
5.5.1 Structure of QM-Algebras of Sets
5.5.2 Theorem 4
5.5.3 Theorem 6
5.6 A Representation by Conditional Probability
5.6.1 Necessary Conditions: Qualitative Conditional Probability
5.6.2 Sufficient Conditions
5.6.3 Further Discussion of Definition 8 and Theorem 7
5.6.4 A Nonadditive Conditional Representation
5.7 Proofs
5.7.1 Preliminary Lemmas
5.7.2 An Additive Unconditional Representation
5.7.3 Theorem 7
5.7.4 Theorem 8
5.8 Independent Events
5.9 Proof of Theorem 10
Exercises
Additive Conjoint Measurement
6.1 Several Notions of Independence
6.1.1 Independent Realization of the Components
6.1.2 Decomposable Structures
6.1.3 Additive Independence
6.1.4 Independent Relations
6.2 Additive Representation of Two Components
6.2.1 Cancellation Axioms
6.2.2 Archimedean Axiom
6.2.3 Sufficient Conditions
6.2.4 Representation Theorem and Method of Proof
6.2.5 Historical Note
6.3 Proofs
6.3.1 Independence of the Axioms of Definition 7
6.3.2 Theorem 1
6.3.3 Preliminary Lemmas for Bounded Symmetric Structures
6.3.4 Theorem 2
6.4 Empirical Examples
6.4.1 Examples from Physics …