Theories in Probability: An Examination of Logical and Qualitative Foundations (Advanced Series on Mathematical Psychology) (Advanced Series on Mathematical Psychology)

Standard probability theory has been an enormously successful contribution to modern science. However, from many perspectives it is too narrow as a general theory of uncertainty, particularly for issues involving subjective uncertainty. This first-of-its-kind book is primarily based on qualitative approaches to probabilistic-like uncertainty, and includes qualitative theories for the standard theory as well as several of its generalizations. One of these generalizations produces a belief function composed of two functions: a probability function that measures the probabilistic strength of an uncertain event, and another function that measures the amount of ambiguity or vagueness of the event. Another unique approach of the book is to change the event space from a boolean algebra, which is closely linked to classical propositional logic, to a different event algebra that is closely linked to a well-studied generalization of classical propositional logic known as intuitionistic logic. Together, these new qualitative theories succeed where the standard probability theory fails by accounting for a number of puzzling empirical findings in the psychology of human probability judgments and decision making. Contents......Page 8 1.1 Introduction......Page 12 1.2 Preliminary Conventions and Definitions......Page 14 2 Kolmogorov Probability Theory......Page 18 3.1 Introduction......Page 22 3.2 Ultrafilters......Page 26 3.3 Ultrapowers of the Reals......Page 29 3.4 Totally Ordered Field Extensions of the Reals......Page 36 4.1 Extensive Measurement of Probability......Page 40 4.2 Probability Representations......Page 47 4.2.1 Scott’s Theorem......Page 50 4.3 R-probability Representations......Page 54 4.4 Weak Probability Representations......Page 61 4.5 Independence......Page 64 4.5.1 Independent Experiments......Page 65 5.2 Subjective Expected Utility......Page 72 5.3 Dual Bilinear and Rank Dependent Utility......Page 77 6.1 Brief Overview......Page 86 6.2 Luce’s Choice Property......Page 87 6.4 Basic Choice Axioms......Page 89 7.1 Introduction......Page 96 7.2 Belief Axioms......Page 100 7.3 Choice Axioms......Page 107 7.4 Belief Support Functions......Page 108 7.4.1 Fair Bets......Page 110 7.4.2 Ellsberg’s Paradox......Page 111 7.5.1 Empirical Phenomena......Page 114 7.5.2 Tversky’s and Koehler’s Theory......Page 115 7.5.3 Rottenstreich’s and Tversky’s Theory......Page 122 7.6 Conclusions......Page 123 8.1 Introduction......Page 126 8.2 Boolean Lattices......Page 127 8.3 Representation Theorems of Stone......Page 129 8.4 u-Complemented Lattices......Page 135 8.4.1 Lattices of Open Sets......Page 136 8.4.2 Properties of lattices of open sets......Page 139 8.5 Intuitionistic Propositional Logic......Page 141 8.6 Non-Boolean Event Spaces......Page 142 9.1 Introduction......Page 144 9.2 Lattices of Scientific Events......Page 145 9.3 Boolean Lattice of Refutations......Page 148 9.4 Probability Theory for Scientific Events......Page 151 9.5 An Example Involving American Law......Page 156 10.1 Introduction......Page 160 10.2.1 Judgmental Heuristics......Page 161 10.2.2 Empirical Basis......Page 162 10.2.3 Summary of the Theory......Page 164 10.3.1 Focal Descriptions and Contexts in the New Formulation......Page 166 10.3.2 Semantical Representations......Page 167 10.3.3 Cognitive Representations......Page 168 10.3.4 Cognitive Model......Page 170 10.3.5 Clear Instances......Page 171 10.3.6 Single Probability Judgments......Page 173 10.4 Frequency Based Judgments......Page 175 10.5 Judgments Based on Representativeness......Page 179 10.6 Probability Judgments for Binary Partitions......Page 183 10.7 Topological Considerations......Page 187 11.1 Brief Summary......Page 190 11.2 First Order Languages......Page 191 11.3 Models......Page 193 11.4 Lo ́s’s Theorem......Page 197 11.5 Compactness Theorem of Logic......Page 201 11.6 Lowenheim-Skolem Theorem......Page 202 11.7 Axiomatizability and Uniqueness of Probability Representations......Page 207 11.8.1 Classical Results of Vitali, Hausdor , Banach, and Ulam......Page 213 11.8.2 Implications for -additivity......Page 216 References......Page 218 Index......Page 224 Contents 8 1 Elementary Concepts 12 1.1 Introduction 12 1.2 Preliminary Conventions and Definitions 14 2 Kolmogorov Probability Theory 18 3 Infinitesimals 22 3.1 Introduction 22 3.2 Ultrafilters 26 3.3 Ultrapowers of the Reals 29 3.4 Totally Ordered Field Extensions of the Reals 36 4 Qualitative Probability 40 4.1 Extensive Measurement of Probability 40 4.2 Probability Representations 47 4.2.1 Scott’s Theorem 50 4.3 R-probability Representations 54 4.4 Weak Probability Representations 61 4.5 Independence 64 4.5.1 Independent Experiments 65 5 Qualitative Utility 72 5.1 Brief Overview 72 5.2 Subjective Expected Utility 72 5.3 Dual Bilinear and Rank Dependent Utility 77 6 Axioms for Choice Proportions 86 6.1 Brief Overview 86 6.2 Luce’s Choice Property 87 6.3 BTL Model of Choice 89 6.4 Basic Choice Axioms 89 7 Conditional Belief 96 7.1 Introduction 96 7.2 Belief Axioms 100 7.3 Choice Axioms 107 7.4 Belief Support Functions 108 7.4.1 Fair Bets 110 7.4.2 Ellsberg’s Paradox 111 7.5 Application to Support Theory 114 7.5.1 Empirical Phenomena 114 7.5.2 Tversky’s and Koehler’s Theory 115 7.5.3 Rottenstreich’s and Tversky’s Theory 122 7.6 Conclusions 123 8 Lattices 126 8.1 Introduction 126 8.2 Boolean Lattices 127 8.3 Representation Theorems of Stone 129 8.4 u-Complemented Lattices 135 8.4.1 Lattices of Open Sets 136 8.4.2 Properties of lattices of open sets 139 8.5 Intuitionistic Propositional Logic 141 8.6 Non-Boolean Event Spaces 142 9 Belief for Scientific Events and Refutations 144 9.1 Introduction 144 9.2 Lattices of Scientific Events 145 9.3 Boolean Lattice of Refutations 148 9.4 Probability Theory for Scientific Events 151 9.5 An Example Involving American Law 156 10 The Psychology of Human Probability Judgments 160 10.1 Introduction 160 10.2 Support Theory 161 10.2.1 Judgmental Heuristics 161 10.2.2 Empirical Basis 162 10.2.3 Summary of the Theory 164 10.2.4 Focal Descriptions and Contexts in Support Theory 166 10.3 Concepts Used in the New Formulation 166 10.3.1 Focal Descriptions and Contexts in the New Formulation 166 10.3.2 Semantical Representations 167 10.3.3 Cognitive Representations 168 10.3.4 Cognitive Model 170 10.3.5 Clear Instances 171 10.3.6 Single Probability Judgments 173 10.3.7 Multiple Probability Judgments 175 10.4 Frequency Based Judgments 175 10.5 Judgments Based on Representativeness 179 10.6 Probability Judgments for Binary Partitions 183 10.7 Topological Considerations 187 11 Metamathematical Considerations 190 11.1 Brief Summary 190 11.2 First Order Languages 191 11.3 Models 193 11.4 Lo ́s’s Theorem 197 11.5 Compactness Theorem of Logic 201 11.6 Lowenheim-Skolem Theorem 202 11.7 Axiomatizability and Uniqueness of Probability Representations 207 11.8 Axiom of Choice and Probability Theory 213 11.8.1 Classical Results of Vitali, Hausdor , Banach, and Ulam 213 11.8.2 Implications for -additivity 216 References 218 Index 224 "Standard probability theory has been an enormously successful contribution to modern science. However, from many perspectives it is too narrow as a general theory of uncertainty, particularly for issues involving subjective uncertainty. This book is primarily based on qualitative approaches to probabilistic-like uncertainty, and includes qualitative theories for the standard theory as well as several of its generalizations." "One of these generalizations produces a belief function composed of two functions: a probability function that measures the probabilistic strength of an uncertain event, and another function that measures the amount of ambiguity or vagueness of the event. Another unique approach of the book is to change the event space from a boolean algebra, which is closely linked to classical propositional logic, to a different event algebra that is closely linked to a well-studied generalization of classical propositional logic known as intuitionistic logic. Together, these new qualitative theories succeed where the standard probability theory fails by accounting for a number of puzzling empirical findings in the psychology of human probability judgments and decision making."--Jacket