Foundations of Measurement Volume III Representation, Axiomatization, and Invariance

by Suppes, Patrick; Krantz, David H.; Luce, R. Duncan; Tversky, Amos

• ISBN13: 9780486453163
• ISBN10: 0486453162
• Format: Paperback
• Copyright: 2006-12-15
• Publisher: Dover Publications

Summary

This classic series in the field of quantitative measurement established the formal foundations for measurement, justifying the assignment of numbers to objects in terms of their structural correspondence.Volume Iintroduces the distinct mathematical results that serve to formulate numerical representations of qualitative structures.Volume IIextends the subject in the direction of geometrical, threshold, and probabilistic representations, andVolume IIIexamines representation as expressed in axiomatization and invariance. 1990 edition.

Table of Contents

Preface	xiii
Acknowledgments	xv
Overview
Nonadditive Representations (Chapter 19)	3	(4)
Examples	4	(1)
Representation and Uniqueness of Positive Operations	4	(1)
Intensive Structures	5	(1)
Conjoint Structures and Distributive Operations	6	(1)
Scale Types (Chapter 20)	7	(4)
A Classification of Automorphism Groups	7	(2)
Unit Representations	9	(1)
Characterization of Homogeneous Concatenation and Conjoint Structures	10	(1)
Reprise	11	(1)
Axiomatization (Chapter 21)	11	(3)
Types of Axiom	11	(2)
Theorems on Axiomatizability	13	(1)
Testability of Axioms	14	(1)
Invariance and Meaningfulness (Chapter 22)	14	(4)
Types of Invariance	15	(1)
Applications of Meaningfulness	16	(2)
Nonadditive Representations
Introduction	18	(7)
Inessential and Essential Nonadditives	18	(4)
General Binary Operations	22	(1)
Overview	23	(2)
Types of Concatenation Structure	25	(12)
Concatenation Structures and Their Properties	25	(2)
Some Numerical Examples	27	(6)
Archimedean Properties	33	(4)
Representations of PCSs	37	(11)
General Definitions	37	(2)
Uniqueness and Construction of a Representation of a PCS	39	(1)
Existence of a Representation	40	(4)
Automorphism Groups of PCSs	44	(2)
Continuous PCSs	46	(2)
Completions of Total Orders and PCSs	48	(8)
Order Isomorphisms onto Real Intervals	49	(1)
Completions of Total Orders	50	(3)
Completions of Closed PCSs	53	(3)
Proofs about Concatenation Structures	56	(19)
Theorem 1	56	(1)
Lemmas 1--6, Theorem 2	57	(4)
Theorem 2	61	(1)
Construction of PCS Homomorphisms	62	(2)
Theorem 3	64	(5)
Theorem 4	69	(1)
Theorem 5	70	(1)
Theorem 6	71	(2)
Corollary to Theorem 7	73	(1)
Theorem 9	73	(2)
Connections between Conjoint and Concatenation Structures	75	(12)
Conjoint Structures: Introduction and General Definitions	75	(2)
Total Concatenation Structures Induced by Conjoint Structures	77	(2)
Factorizable Automorphisms	79	(2)
Total Concatenation Structures Induced by Closed, Idempotent Concatenation Structures	81	(2)
Intensive Structures Related to PCSs by Doubling Functions	83	(2)
Operations That Distribute over Conjoint Structures	85	(2)
Representations of Solvable Conjoint and Concatenation Structures	87	(2)
Conjoint Structures	87	(1)
Solvable, Closed, Archimedean Concatenation Structures	88	(1)
Intensive Concatenation Structures with Doubling Functions	89	(1)
Proofs	89	(12)
Theorem 11	89	(3)
Theorem 12	92	(1)
Theorem 13	93	(4)
Theorem 14, Part (iii)	97	(1)
Theorem 15	97	(1)
Theorem 18	98	(3)
Theorem 21	101	(1)
Bisymmetry and Related Properties	101	(7)
General Definitions	101	(2)
Equivalences in Closed, Idempotent, Solvable, Dedekind Complete Structures	103	(1)
Bisymmetry in the 1-Point Unique Case	103	(1)
Exercises	104	(4)
Scale Types
Introduction	108	(4)
Constructibility and Symmetry	108	(3)
Problem in Understanding Scale Types	111	(1)
Homogeneity, Uniqueness, and Scale Type	112	(14)
Stevens' Classification	112	(2)
Decomposing the Classification	114	(1)
Formal Definitions	115	(2)
Relations among Structure, Homogeneity, and Uniqueness	117	(2)
Scale Types of Real Relational Structures	119	(3)
Structures with Homogeneous, Archimedean Ordered Translation Groups	122	(3)
Representations of Dedekind Complete Distributive Triples	125	(1)
Proofs	126	(16)
Theorem 2	126	(1)
Theorem 3	127	(1)
Theorem 4	128	(1)
Theorem 5	128	(9)
Theorem 7	137	(4)
Theorem 8	141	(1)
Homogeneous Concatenation Structures	142	(14)
Nature of Homogeneous Concatenation Structures	142	(1)
Real Unit Concatenation Structures	143	(3)
Characterizations of Homogeneity: PCS	146	(2)
Characterizations of Homogeneity: Solvable, Idempotent Structures	148	(2)
Mixture Spaces of Gambles	150	(2)
The Dual Bilinear Utility Model	152	(4)
Proofs	156	(24)
Theorem 9	156	(2)
Theorem 11	158	(2)
Theorem 24, Chapter 19	160	(3)
Theorem 14	163	(2)
Theorem 15	165	(1)
Theorem 16	166	(1)
Theorem 17	167	(1)
Theorem 18	168	(7)
Theorem 19	175	(5)
Homogeneous Conjoint Structures	180	(4)
Component Homogeneity and Uniqueness	180	(1)
Singular Points in Conjoint Structures	181	(2)
Forcing the Thomsen Condition	183	(1)
Proofs	184	(12)
Theorem 22	184	(1)
Theorem 23	185	(1)
Theorem 24	186	(4)
Theorem 25	190	(2)
Exercises	192	(4)
Axiomatization
Axiom Systems and Representations	196	(8)
Why Do Scientists and Mathematicians Axiomatize?	196	(5)
The Axiomatic-Representational Viewpoint in Measurement	201	(1)
Types of Representing Structures	202	(2)
Elementary Formalization of Theories	204	(14)
Elementary Languages	204	(4)
Models of Elementary Languages	208	(5)
General Theorems about Elementary Logic	213	(2)
Elementary Theories	215	(3)
Definability and Interpretability	218	(7)
Definability	218	(6)
Interpretability	224	(1)
Some Theorems on Axiomatizability	225	(4)
Proofs	229	(2)
Theorem 6	229	(1)
Theorem 7	230	(1)
Theorem 8	230	(1)
Theorem 9	231	(1)
Finite Axiomatizability	231	(15)
Axiomatizable by a Universal Sentence	234	(7)
Proof of Theorem 12	241	(1)
Finite Axiomatizability of Finitary Classes	242	(4)
The Archimedean Axiom	246	(5)
Testability of Axioms	251	(16)
Finite Data Structures	254	(2)
Convergence of Finite to Infinite Data Structures	256	(3)
Testability and Constructibility	259	(2)
Diagnostic versus Global Tests	261	(4)
Exercises	265	(2)
Invariance and Meaningfulness
Introduction	267	(2)
Methods of Defining Meaningful Relations	269	(16)
Definitions in First-Order Theories	271	(2)
Reference and Structure Invariance	273	(3)
An Example: Independence in Probability Theory	276	(1)
Definitions with Particular Representations	277	(1)
Parametrized Numerical Relations	278	(2)
An Example: Hooke's Law	280	(2)
A Necessary Condition for Meaningfulness	282	(2)
Irreducible Structures: Reference Invariance of Numerical Equality	284	(1)
Characterizations of Reference Invariance	285	(5)
Permissible Transformations	285	(2)
The Criterion of Invariance under Permissible Transformations	287	(1)
The Condition of Structure Invariance	287	(3)
Proofs	290	(2)
Theorem 3	290	(1)
Theorem 4	290	(1)
Theorem 5	291	(1)
Definability	292	(2)
Meaningfulness and Statistics	294	(13)
Examples	295	(3)
Meaningful Relations Involving Population Means	298	(1)
Inferences about Population Means	299	(1)
Parametric Models for Populations	299	(2)
Measurement Structures and Parametric Models for Populations	301	(4)
Meaningful Relations in Uniform Structures	305	(2)
Dimensional Invariance	307	(19)
Structures of Physical Quantities	309	(3)
Triples of Scales	312	(3)
Representation and Uniqueness Theorem for Physical Attributes	315	(3)
Physically Similar Systems	318	(5)
Fundamental versus Index Measurement	323	(3)
Proofs	326	(3)
Theorem 6	326	(1)
Theorem 7	327	(2)
Reprise: Uniqueness, Automorphisms, and Constructibility	329	(9)
Alternative Representations	329	(1)
Nonuniqueness and Automorphisms	330	(2)
Invariance under Automorphisms	332	(1)
Constructibility of Representations	333	(3)
Exercises	336	(2)
References	338	(9)
Author Index	347	(4)
Subject Index	351