Foundations of Measurement Volume II Geometrical, Threshold, and Probabilistic Representations

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

• ISBN13: 9780486453156
• ISBN10: 0486453154
• 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. 1989 edition.

Author Biography

David H. Krantz is affiliated with Columbia University; R. Duncan Luce with the University of California, Irvine, and Patrick Suppes with Stanford University. Amos Tversky is deceased.

Table of Contents

Preface	xiii
Acknowledgments	xv
Overview
Geometry Unit	1	(8)
Geometrical Representations (Chapter 12)	3	(1)
Axiomatic Synthetic Geometry (Chapter 13)	4	(3)
Proximity Measurement (Chapter 14)	7	(2)
Color and Force Measurement (Chapter 15)	9	(1)
Threshold and Error Unit	9	(4)
Representations with Thresholds (Chapter 16)	10	(1)
Representations of Choice Probabilities (Chapter 17)	11	(2)
Geometrical Representations
Introduction	13	(1)
Vector Representations	14	(37)
Vector Spaces	16	(5)
Analytic Affine Geometry	21	(3)
Analytic Projective Geometry	24	(7)
Analytic Euclidean Geometry	31	(4)
Meaningfulness in Analytic Geometry	35	(7)
Minkowski Geometry	42	(4)
General Projective Metrics	46	(5)
Metric Representations	51	(29)
General Metrics with Geodesies	52	(5)
Elementary Spaces and the Helmholtz--Lie Problem	57	(2)
Riemannian Metrics	59	(12)
Other Metrics	71	(6)
Exercises	77	(3)
Axiomatic Geometry and Applications
Introduction	80	(3)
Order on the Line	83	(6)
Betweenness: Affine Order	83	(2)
Separation: Projective Order	85	(4)
Proofs	89	(4)
Projective Planes	93	(9)
Projective Spaces	102	(2)
Affine and Absolute Spaces	104	(10)
Ordered Geometric Spaces	105	(2)
Affine Spaces	107	(2)
Absolute Spaces	109	(1)
Euclidean Spaces	110	(1)
Hyperbolic Spaces	111	(3)
Elliptic Spaces	114	(4)
Double Elliptic Spaces	115	(2)
Single Elliptic Spaces	117	(1)
Classical Space-Time	118	(3)
Space-Time of Special Relativity	121	(10)
Other Axiomatic Approaches	127	(4)
Perceptual Spaces	131	(28)
Historical Survey through the Nineteenth Century	131	(3)
General Considerations Concerning Perceptual Spaces	134	(4)
Experimental Work before Luneburg's Theory	138	(1)
Luneburg Theory of Binocular Vision	139	(6)
Experiments Relevant to Luneburg's Theory	145	(5)
Other Studies	150	(3)
Exercises	153	(6)
Proximity Measurement
Introduction	159	(4)
Metrics with Additive Segments	163	(6)
Collinearity	163	(2)
Constructive Methods	165	(3)
Representation and Uniqueness Theorems	168	(1)
Proofs	169	(6)
Theorem 2	169	(1)
Reduction to Extensive Measurement	170	(1)
Theorem 3	171	(1)
Theorem 4	172	(3)
Multidimensional Representations	175	(13)
Decomposability	178	(1)
Intradimensional Subtractivity	179	(2)
Interdimensional Additivity	181	(3)
The Additive-Difference Model	184	(1)
Additive-Difference Metrics	185	(3)
Proofs	188	(12)
Theorem 5	188	(1)
Theorem 6	189	(2)
Theorem 7	191	(1)
Theorem 9	192	(2)
Preliminary Lemma	194	(3)
Theorem 10	197	(3)
Experimental Tests of Multidimensional Representations	200	(7)
Relative Curvature	201	(1)
Translation Invariance	202	(3)
The Triangle Inequality	205	(2)
Feature Representations	207	(15)
The Contrast Model	209	(6)
Empirical Applications	215	(3)
Comparing Alternative Representations	218	(4)
Proofs	222	(4)
Theorem 11	222	(2)
Exercises	224	(2)
Color and Force Measurement
Introduction	226	(3)
Grassmann Structures	229	(8)
Formulation of the Axioms	229	(5)
Representation and Uniqueness Theorems	234	(1)
Discussion of Proofs of Theorems 3 and 4	235	(2)
Proofs	237	(3)
Theorem 3	237	(2)
Theorem 4	239	(1)
Color Interpretations	240	(15)
Metameric Color Matching	240	(3)
Tristimulus Colorimetry	243	(7)
Four Ways to Misunderstand Color Measurement	250	(2)
Asymmetric Color Matching	252	(3)
The Dimensional Structure of Color and Force	255	(16)
Color Codes and Metamer Codes	256	(4)
Photopigments	260	(3)
Force Measurement and Dynamical Theory	263	(5)
Color Theory in a Measurement Framework	268	(3)
The Konig and Hurvich-Jameson Color Theories	271	(20)
Representations of 2-Chromatic Reduction Structures	271	(6)
The Konig Theory and Alternatives	277	(2)
Codes Based on Color Attributes	279	(1)
The Cancellation Procedure	280	(3)
Representation and Uniqueness Theorems	283	(3)
Tests and Extensions of Quantitative Opponent-Colors Theory	286	(5)
Proofs	291	(8)
Theorem 6	291	(3)
Theorem 9	294	(1)
Theorem 10	295	(1)
Exercises	296	(3)
Representations with Thresholds
Introduction	299	(4)
Three Approaches to Nontransitive Data	300	(1)
Idea of Thresholds	301	(1)
Overview	302	(1)
Ordinal Theory	303	(23)
Upper, Lower, and Two-Sided Thresholds	304	(2)
Induced Quasiorders: Interval Orders and Semiorders	306	(5)
Compatible Relations	311	(2)
Biorders: A Generalization of Interval Orders	313	(1)
Tight Representations	314	(4)
Constant-Threshold Representations	318	(4)
Interval and Indifference Graphs	322	(4)
Proofs	326	(5)
Theorem 2	326	(1)
Lemma 1	326	(1)
Theorem 6	327	(1)
Theorem 9	327	(1)
Theorem 10	328	(1)
Theorem 11	329	(1)
Theorems 14 and 15	330	(1)
Ordinal Theory for Families of Orders	331	(10)
Finite Families of Interval Orders and Semiorders	332	(4)
Order Relations Induced by Binary Probabilities	336	(3)
Dimension of Partial Orders	339	(2)
Proofs	341	(3)
Theorem 16	341	(1)
Theorem 17	342	(1)
Theorem 18	343	(1)
Theorem 19	344	(1)
Semiordered Additive Structures	344	(10)
Possible Approaches to Semiordered Probability Structures	345	(4)
Probability with Approximate Standard Families	349	(3)
Axiomatization of Semiordered Probability Structures	352	(1)
Weber's Law and Semiorders	353	(1)
Proof of Theorem 24	354	(5)
Random-Variable Representations	359	(24)
Weak Representations of Additive Conjoint and Extensive Structures	364	(2)
Variability as Measured by Moments	366	(2)
Qualitative Primitives for Moments	368	(3)
Axiom System for Qualitative Moments	371	(3)
Representation Theorem and Proof	374	(5)
Exercises	379	(4)
Representation of Choice Probabilities
Introduction	383	(5)
Empirical Interpretations	384	(1)
Probabilistic Representations	385	(3)
Ordinal Representations for Pair Comparisons	388	(13)
Stochastic Transitivity	388	(2)
Difference Structures	390	(1)
Local Difference Structures	391	(2)
Additive Difference Structures	393	(4)
Intransitive Preferences	397	(4)
Proofs	401	(9)
Theorem 2	401	(4)
Theorem 3	405	(2)
Theorem 4	407	(3)
Constant Representations for Multiple Choice	410	(9)
Simple Scalability	410	(3)
The Strict-Utility Model	413	(6)
Proofs	419	(2)
Theorem 5	419	(1)
Theorem 7	420	(1)
Random Variable Representations	421	(11)
The Random-Utility Model	421	(3)
The Independent Double-Exponential Model	424	(3)
Error Tradeoff	427	(5)
Proofs	432	(4)
Theorem 9	432	(1)
Theorem 12	433	(2)
Theorem 13	435	(1)
Markovian Elimination Processes	436	(14)
The General Model	436	(1)
Elimination by Aspects	437	(7)
Preference Trees	444	(6)
Proofs	450	(7)
Theorem 15	450	(2)
Theorem 16	452	(3)
Theorem 17	455	(2)
Exercises	457	(2)
References	459	(22)
Author Index	481	(6)
Subject Index	487