# Additive and Polynomial Representations

## 1st Edition

**Authors:**David H. Krantz R Duncan Luce Patrick Suppes

**eBook ISBN:**9781483258300

**Imprint:**Academic Press

**Published Date:**28th October 1971

**Page Count:**608

## Description

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.

## Table of Contents

Preface

Mathematical Background

Selecting Among the Chapters

Acknowledgments

Notational Conventions

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 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

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

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

4. 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

5. 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

6. 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

Exercises

7. 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

Exercises

8. 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 𝓓

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

Exercises

9. 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

Exercises

10. 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 Quantities

Answers and Hints to Selected Exercises

References

Author Index

Subject Index

## Details

- No. of pages:
- 608

- Language:
- English

- Copyright:
- © Academic Press 1971

- Published:
- 28th October 1971

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483258300