Additive and Polynomial Representations

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.

¿PrefaceMathematical BackgroundSelecting Among the ChaptersAcknowledgmentsNotational Conventions1. 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 Exercises2. 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 Exercises3. 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 Exercises4. 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 Exercises5. 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 Exercises6. 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 6.4.2 Examples from the Behavioral Sciences 6.5 Modifications of the Theory 6.5.1 Omission of the Archimedean Property 6.5.2 Alternative Numerical Representations 6.5.3 Transforming a Nonadditive Representation into an Additive One 6.5.4 Subtractive Structures 6.5.5 Need for Conjoint Measurement on B ¿ A1 × A2 6.5.6 Symmetries of Independent and Dependent Variables 6.5.7 Alternative Factorial Decompositions 6.6 Proofs 6.6.1 Preliminary Lemmas 6.6.2 Theorem 3 6.6.3 Theorem 4 6.6.4 Theorem 6 6.7 Indifference Curves and Uniform Families of Functions 6.7.1 A Curve Through Every Point 6.7.2 A Finite Number of Curves 6.8 Proofs 6.8.1 Theorem 7 6.8.2 Theorem 8 6.8.3 Preliminary Lemmas About Uniform Families 6.8.4 Theorem 9 6.9 Bisymmetric Structures 6.9.1 Sufficient Conditions 6.9.2 A Finite, Equally Spaced Case 6.10 Proofs 6.10.1 Theorem 10 6.10.2 Theorem 11 6.11 Additive Representation of n Components 6.11.1 The General Case 6.11.2 The Case of Identical Components 6.12 Proofs 6.12.1 Preliminary Lemma 6.12.2 Theorem 13 6.12.3 Theorem 14 6.12.4 Theorem 15 6.13 Concluding Remarks Exercises7. Polynomial Conjoint Measurement 7.1 Introduction 7.2 Decomposable Structures 7.2.1 Necessary and Sufficient Conditions 7.2.2 Proof of Theorem 1 7.3 Polynomial Models 7.3.1 Examples 7.3.2 Decomposability and Equivalence of Polynomial Models 7.3.3 Simple Polynomials 7.4 Diagnostic Ordinal Properties 7.4.1 Sign Dependence 7.4.2 Proofs of Theorems 2 and 3 7.4.3 Joint-Independence Conditions 7.4.4 Cancellation Conditions 7.4.5 Diagnosis for Simple Polynomials in Three Variables 7.5 Sufficient Conditions for Three-Variable Simple Polynomials 7.5.1 Representation and Uniqueness Theorems 7.5.2 Heuristic Proofs 7.5.3 Generalizations to Four or More Variables 7.6 Proofs 7.6.1 A Preliminary Result 7.6.2 Theorem 4 7.6.3 Theorem 5 7.6.4 Theorem 6 Exercises8. Conditional Expected Utility 8.1 Introduction 8.2 A Formulation of the Problem 8.2.1 The Primitive Notions 8.2.2 A Restriction on ?? 8.2.3 The Desired Representation Theorem 8.2.4 Necessary Conditions 8.2.5 Nonnecessary Conditions 8.2.6 The Axiom System and Representation Theorem 8.3 Proofs 8.3.1 Preliminary Lemmas 8.3.2 Theorem 1 8.4 Topics in Utility and Subjective Probability 8.4.1 Utility of Consequences 8.4.2 Relations Between Additive and Expected Utility 8.4.3 The Consistency Principle for the Utility of Money 8.4.4 Expected Utility and Risk 8.4.5 Relations Between Subjective and Objective Probability 8.4.6 A Method for Estimating Subjective Probabilities 8.5 Proofs 8.5.1 Theorem 3 8.5.2 Theorem 4 8.5.3 Theorem 5 8.5.4 Theorem 6 8.5.5 Theorem 7 8.6 Other Formulations of Risky and Uncertain Decisions 8.6.1 Mixture Sets and Gambles 8.6.2 Propositions as Primitives 8.6.3 Statistical Decision Theory 8.6.4 Comparision of Statistical and Conditional Decision Theories in the Finite Case 8.7 Concluding Remarks 8.7.1 Prescriptive Versus Descriptive Interpretations 8.7.2 Open Problems Exercises9. Measurement Inequalities 9.1 Introduction 9.2 Finite Linear Structures 9.2.1 Additivity 9.2.2 Probability 9.3 Proof of Theorem 1 9.4 Applications 9.4.1 Scaling Considerations 9.4.2 Empirical Examples 9.5 Polynomial Structures 9.6 Proofs 9.6.1 Theorem 4 9.6.2 Theorem 5 9.6.3 Theorem 6 Exercises10. Dimensional Analysis and Numerical Laws 10.1 Introduction 10.2 The Algebra of Physical Quantities 10.2.1 The Axiom System 10.2.2 General Theorems 10.3 The PI Theorem of Dimensional Analysis 10.3.1 Similarities 10.3.2 Dimensionally Invariant Functions 10.4 Proofs 10.4.1 Preliminary Lemmas 10.4.2 Theorems 1 and 2 10.4.3 Theorem 3 10.4.4 Theorem 4 10.5 Examples of Dimensional Analysis 10.5.1 The Simple Pendulum 10.5.2 Errors of Commission and Omission 10.5.3 Dimensional Analysis as an Aid in Obtaining Exact Solutions 10.5.4 Conclusion 10.6 Binary Laws and Universal Constants 10.7 Trinary Laws and Derived Measures 10.7.1 Laws of Similitude 10.7.2 Laws of Exchange 10.7.3 Compatibility of the Trinary Laws 10.7.4 Some Relations Among Extensive, Difference, and Conjoint Structures 10.8 Proofs 10.8.1 Preliminary Lemma 10.8.2 Theorem 5 10.8.3 Theorem 6 10.8.4 Theorem 7 10.9 Embedding Physical Attributes in a Structure of Physical Quantities 10.9.1 Assumptions About Physical Attributes 10.9.2 Fundamental, Derived, and Quasi-Derived Attributes 10.10 Why are Numerical Laws Dimensionally Invariant? 10.10.1 Three Points of View 10.10.2 Physically Similar Systems 10.10.3 Relations to Causey's Theory 10.11 Proofs 10.11.1 Theorem 12 10.11.2 Theorem 13 10.12 Interval Scales in Dimensional Analysis 10.13 Proofs 10.13.1 Preliminary Lemma 10.13.2 Theorem 14 10.13.3 Theorem 15 10.13.4 Theorem 16 10.14 Physical Quantities in Mechanics and Generalizations of Dimensional Invariance 10.14.1 Generalized Galilean Invariance 10.14.2 Lorentz Invariance and Relativistic Mechanics 10.15 Concluding Remarks Exercises Dimensions and Units of Physical QuantitiesAnswers and Hints to Selected ExercisesReferencesAuthor IndexSubject Index