All of the sciences — physical, biological, and social — have a need for quantitative measurement. This influential series, Foundations of Measurement, established the formal basis for measurement, justifying the assignment of numbers to objects in terms of their structural correspondence.
Volume I introduces the distinct mathematical results that serve to formulate numerical representations of qualitative structures.
Volume II extends the subject in the direction of geometrical, threshold, and probabilistic representations, Volume III examines representation as expressed in axiomatization and invariance.
Reprint of the Academic Press, New York and London, 1990 edition.
Author(s): Patrick Suppes, David H. Krantz, R. Duncan Luce, Amos Tversky
Series: Dover Books on Mathematics
Publisher: Dover Publications
Year: 2007
Language: English
Pages: C+xvi+356+B
Cover
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
Back Cover
