Probabilistic Lattices: With Applications To Psychology

by Louis E Narens

Overview

There are many books on lattice theory in the field, but none interfaces with the foundations of probability. This book does. It also develops new probability theories with rigorous foundations for decision theory and applies them to specific well-known problematic examples. There is only one other book that attempts this. It uses quantum probability theory from physics. The new probability theories developed in this book are different; they are not borrowed from physics but are explicitly designed for decision theory.

Product Details

ISBN-13:	9789814630412
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/04/2015
Series:	Advanced Series On Mathematical Psychology , #5
Pages:	208
Product dimensions:	6.10(w) x 9.10(h) x 0.80(d)

Table of Contents

Acknowledgments vii

1 Introduction 1

1.1 What This Book is About 1

1.2 Level and Kind of Mathematics Used 2

1.3 How I Came to Write This Book 3

1.4 Differentiation from Fuzziness 8

1.5 Toward a Generalized Theory of Probability 9

1.6 Extended Probability Functions 10

1.7 Lattices 15

1.8 Forms of Complementation 18

1.9 Rationality 24

1.10 Logic 25

2 Basic Lattice Theory 27

2.1 Introduction 27

2.2 General Notation, Conventions, and Definitions 28

2.3 Lattices: Definitions and Duality 31

2.4 Distributive and Modular Lattices 37

2.5 Lattices with Negative Operators 43

2.6 Representation Theorems 47

2.7 Miscellaneous Results 50

2.8 Unique Complementation and Distributivity 52

2.9 Orthomodular Lattices 61

2.10 Finite Boolean Lattices 67

3 Pseudo Complemented Distributive Lattices 71

3.1 Pseudo Complementation and Relative Pseudo Complementation 72

3.2 Pseudo Complemented Distributive Algebras of Events 78

3.3 Lattice Representation Theorems 81

3.4 Comparison with Boolean Lattices 85

3.5 The Refutation Lattice of € 87

3.6 Proof of Stone's Representation Theorem for Distributive Lattices 90

4 Probability and Coherence 95

4.1 L-Probabilistic Lattices 96

4.1.1 Infinitesimals 97

4.1.2 Extended probability functions 102

4.1.3 L-probabilistic lattices 103

4.2 Dutch Books 106

4.3 Preference Books 109

4.4 Finite Cancellation Axioms 112

4.5 Additional Theorems and Proofs 116

4.5.1 Preliminary lemma 116

4.5.2 Proof of Scott's Theorem 118

4.5.3 Proofs using the compactness theorem of logic 122

5 Rationality Heuristics, and Human Judgments of Probability 131

5.1 Introduction 131

5.2 Cognitive Heuristics 133

5.3 Support Theory 136

5.4 Rationality 145

5.4.1 Modeling DSEU 147

6 Orthomodulax Modeling of Psychological Paradigms 155

6.1 Orthoprobability Theory 155

6.2 Experimental Psychological Paradigms 157

6.2.1 Formalism and lattice foundation 157

6.2.2 Between-subject paradigms and modeling the individual subject 166

6.3 The Disjunction Effect 168

6.4 Dynamic Version of the Allais Paradox 174

6.5 A Use of Probability in Making Logical Inferences Across Experiments 178

6.6 Boolean Subalgebras and Blocks 179

6.7 Other Non-Boolean Modeling of Decision Making 181

References 183

Index 189