Foundations of Measurement Volume III: Representation, Axiomatization, and Invariance

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Frontispiece by Ruth Weisberg

Foundations of Measurement, VOLUME III: Representation, Axiomatization, and Invariance

Copyright © 1990 by David H. Krantz, R. Duncan Luce, Patrick Suppes,and Barbara Tversky
     ISBN 0-486-45316-2

Table of Contents

Preface

Acknowledgments

Chapter 18  Overview

     18.1 NONADDITIVE REPRESENTATIONS (CHAPTER 19)

          18.1.1 Examples

          18.1.2 Representation and Uniqueness of Positive Operations

          18.1.3 Intensive Structures

          18.1.4 Conjoint Structures and Distributive Operations

     18.2 SCALE TYPES (CHAPTER 20)

          18.2.1 A Classification of Automorphism Groups

          18.2.2 Unit Representations

          18.2.3 Characterization of Homogeneous Concatenation and Conjoint Structures

          18.2.4 Reprise

     18.3 AXIOMATIZATION (CHAPTER 21)

          18.3.1 Types of Axioms

          18.3.2 Theorems on Axiomatizability

          18.3.3 Testability of Axioms

     18.4 INVARIANCE AND MEANINGFULNESS (CHAPTER 22)

          18.4.1 Types of Invariance

          18.4.2 Applications of Meaningfulness

Chapter 19  Nonadditive Representations

     19.1 INTRODUCTION

          19.1.1 Inessential and Essential Nonadditivities

          19.1.2 General Binary Operations

          19.1.3 Overview

     19.2 TYPES OF CONCATENATION STRUCTURE

          19.2.1 Concatenation Structures and Their Properties

          19.2.2 Some Numerical Examples

          19.2.3 Archimedean Properties

     19.3 REPRESENTATIONS OF PCSs

          19.3.1 General Definitions

          19.3.2 Uniqueness and Construction of a Representation of a PCS

          19.3.3 Existence of a Representation

          19.3.4 Automorphism Groups of PCSs

          19.3.5 Continuous PCSs

     19.4 COMPLETIONS OF TOTAL ORDERS AND PCSs

          19.4.1 Order Isomorphisms onto Real Intervals

          19.4.2 Completions of Total Orders

          19.4.3 Completions of Closed PCSs

     19.5 PROOFS ABOUT CONCATENATION STRUCTURES

          19.5.1 Theorem 1 (p. 37)

          19.5.2 Lemmas 1-6, Theorem 2

          19.5.3 Theorem 2 (p. 39)

          19.5.4 Construction of PCS Homomorphisms

          19.5.5 Theorem 3 (p. 41)

          19.5.6 Theorem 4 (p. 45)

          19.5.7 Theorem 5 (p. 46)

          19.5.8 Theorem 6 (p. 47)

          19.5.9 Corollary to Theorem 7 (p. 50)

          19.5.10 Theorem 9 (p. 54)

     19.6 CONNECTIONS BETWEEN CONJOINT AND CONCATENATION STRUCTURES

          19.6.1 Conjoint Structures: Introduction and General Definitions

          19.6.2 Total Concatenation Structures Induced by Conjoint Structures

          19.6.3 Factorizable Automorphisms

          19.6.4 Total Concatenation Structures Induced by Closed, Idempotent Concatenation Structures

          19.6.5 Intensive Structures Related to PCSs by Doubling Functions

          19.6.6 Operations That Distribute over Conjoint Structures

     19.7 REPRESENTATIONS OF SOLVABLE CONJOINT AND CONCATENATION STRUCTURES

          19.7.1 Conjoint Structures

          19.7.2 Solvable, Closed, Archimedean Concatenation Structures

     19.8 PROOFS

          19.8.1 Theorem 11 (p. 78)

          19.8.2 Theorem 12 (p. 80)

          19.8.3 Theorem 13 (p. 81)

          19.8.4 Theorem 14, Part (iii) (p. 81)

          19.8.5 Theorem 15 (p. 82)

          19.8.6 Theorem 18 (p. 86)

          19.8.7 Theorem 21 (p. 88)

     19.9 BISYMMETRY AND RELATED PROPERTIES

          19.9.1 General Definitions

          19.9.2 Equivalences in Closed, Idempotent, Solvable, Dedekind Complete Structures

          19.9.3 Bisymmetry in the 1-Point Unique Case

     EXERCISES

Chapter 20  Scale Types

     20.1 INTRODUCTION

          20.1.1 Constructibility and Symmetry

          20.1.2 Problem in Understanding Scale Types

     20.2 HOMOGENEITY, UNIQUENESS, AND SCALE TYPE

          20.2.1 Stevens\' Classification

          20.2.2 Decomposing the Classification

          20.2.3 Formal Definitions

          20.2.4 Relations among Structure, Homogeneity, and Uniqueness

          20.2.5 Scale Types of Real Relational Structures

          20.2.6 Structures with Homogeneous, Archimedean Ordered Translation Groups

          20.2.7 Representations of Dedekind Complete Distributive Triples

     20.3 PROOFS

          20.3.1 Theorem 2 (p. 117)

          20.3.2 Theorem 3 (p. 118)

          20.3.3 Theorem 4 (p. 118)

          20.3.4 Theorem 5 (p. 120)

          20.3.5 Theorem 7 (p. 124)

          20.3.6 Theorem 8 (p. 125)

     20.4 HOMOGENEOUS CONCATENATION STRUCTURES

          20.4.1 Nature of Homogeneous Concatenation Structures

          20.4.2 Real Unit Concatenation Structures

          20.4.3 Characterization of Homogeneity: PCS

          20.4.4 Characterizations of Homogeneity: Solvable, Idempotent Structures

          20.4.5 Mixture Spaces of Gambles

          20.4.6 The Dual Bilinear Utility Model

     20.5 PROOFS

          20.5.1 Theorem 9 (p. 142)

          20.5.2 Theorem 11 (p. 144)

          20.5.3 Theorem 24, Chapter 19 (p. 103)

          20.5.4 Theorem 14 (p. 147)

          20.5.5 Theorem 15 (p. 147)

          20.5.6 Theorem 16 (p. 148)

          20.5.7 Theorem 17 (p. 148)

          20.5.8 Theorem 18 (p. 150)

          20.5.9 Theorem 19 (p. 153)

     20.6 HOMOGENEOUS CONJOINT STRUCTURES

          20.6.1 Component Homogeneity and Uniqueness

          20.6.2 Singular Points in Conjoint Structures

          20.6.3 Forcing the Thomsen Condition

     20.7 PROOFS

          20.7.1 Theorem 22 (p. 181)

          20.7.2 Theorem 23 (p. 182)

          20.7.3 Theorem 24 (p. 182)

          20.7.4 Theorem 25 (p. 183)

     EXERCISES

Chapter 21  Axiomatization

     21.1 AXIOM SYSTEMS AND REPRESENTATIONS

          21.1.1 Why Do Scientists and Mathematicians Axiomatize?

          21.1.2 The Axiomatic-Representational Viewpoint in Measurement

          21.1.3 Types of Representing Structures

     21.2 ELEMENTARY FORMALIZATION OF THEORIES

          21.2.1 Elementary Languages

          21.2.2 Models of Elementary Languages

          21.2.3 General Theorems about Elementary Logic

          21.2.4 Elementary Theories

     21.3 DEFINABILITY AND INTERPRETABILITY

          21.3.1 Definability

          21.3.2 Interpretability

     21.4 SOME THEOREMS ON AXIOMATIZABILITY

     21.5 PROOFS

          21.5.1 Theorem 6 (p. 226)

          21.5.2 Theorem 7 (p. 227)

          21.5.3 Theorem 8 (p. 228)

          21.5.4 Theorem 9 (p. 229)

     21.6 FINITE AXIOMATIZABILITY

          21.6.1 Axiomatizable by a Universal Sentence

          21.6.2 Proof of Theorem 12 (p. 237)

          21.6.3 Finite Axiomatizability of Finitary Classes

     21.7 THE ARCHIMEDEAN AXIOM

     21.8 TESTABILITY OF AXIOMS

          21.8.1 Finite Data Structures

          21.8.2 Convergence of Finite to Infinite Data Structures

          21.8.3 Testability and Constructability

          21.8.4 Diagnostic versus Global Tests

     EXERCISES

Chapter 22  Invariance and Meaning fulness

     22.1 INTRODUCTION

     22.2 METHODS OF DEFINING MEANINGFUL RELATIONS

          22.2.1 Definitions in First-Order Theories

          22.2.2 Reference and Structure Invariance

          22.2.3 An Example: Independence in Probability Theory

          22.2.4 Definitions with Particular Representations

          22.2.5 Parametrized Numerical Relations

          22.2.6 An Example: Hooke\'s Law

          22.2.7 A Necessary Condition for Meaningfulness

          22.2.8 Irreducible Structures: Reference Invariance of Numerical Equality

     22.3 CHARACTERIZATIONS OF REFERENCE INVARIANCE

          22.3.1 Permissible Transformations

          22.3.2 The Criterion of Invariance under Permissible Transformations

          22.3.3 The Condition of Structure Invariance

     22.4 PROOFS

          22.4.1 Theorem 3 (p. 287)

          22.4.2 Theorem 4 (p. 287)

          22.4.3 Theorem 5 (p. 288)

     22.5 DEFINABILITY

     22.6 MEANINGFULNESS AND STATISTICS

          22.6.1 Examples

          22.6.2 Meaningful Relations Involving Population Means

          22.6.3. Inferences about Population Means

          22.6.4 Parametric Models for Populations

          22.6.5 Measurement Structures and Parametric Models for Populations

          22.6.6 Meaningful Relations in Uniform Structures

     22.7 DIMENSIONAL INVARIANCE

          22.7.1 Structures of Physical Quantities

          22.7.2 Triples of Scales

          22.7.3 Representation and Uniqueness Theorem for Physical Attributes

          22.7.4 Physically Similar Systems

          22.7.5 Fundamental versus Index Measurement

     22.8 PROOFS

          22.8.1 Theorem 6 (p. 315)

          22.8.2 Theorem 7 (p. 315)

     22.9 REPRISE: UNIQUENESS, AUTOMORPHISMS, AND CONSTRUCTABILITY

          22.9.1 Alternative Representations

          22.9.2 Nonuniqueness and Automorphisms

          22.9.3 Invariance under Automorphisms

          22.9.4 Constructability of Representations

     EXERCISES

References

Author Index

Subject Index

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