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Overview 1
The Context
Mathematics and Its Applications 9
Axiomatic Method and Mathematics 9
Prepositional Calculus as an Axiomatic System 12
The Euclidean Geometry Paradigm 13
Formal Science and Applied Mathematics 15
Notes on the Literature 17
The Evolution of Natural Scientists and Their Theories 18
Background 18
"Prediction = Science"? 21
External Aspects of Science 24
Conclusion Theory 29
Law and Learning 33
The Law 33
Learning Theory 35
Interpreting Probability
Introduction to Probability 41
The Axiomatic Theory 42
Interpreting the Axiomatic Theory 47
Enumerative Probability 49
Frequency Probability 50
Propensity Probability-"Tendency to Happen" 51
The Fair Betting Utility Interpretation of Probability 54
Probability as Personal Price 54
Probability as Personal Degree of Belief 58
Conditional Subjective Probability 59
Attitudes Toward Chance 61
Indeterminismand Chance 61
The Location of Probability-Mind or Matter? 62
Equally Likely 63
Frequency and the Law of Large Numbers 67
The Single Instance 69
Locations of Some Kinds of Probabilities 70
Statistical Models of Induction
A Framework for Statistical Evidence 75
Introduction 75
What Is an Experiment? 76
E[subscript v](E, y) Does Not Exist 77
The Fitted Parameter Model 78
Interlaboratory Experimentation 79
Conclusion 83
A Critique of Bayesian Inference 84
Randomness Needs Explaining 84
How to Adjust Your Belief 86
The Economic Approach to Group Belief 88
Objective Bayesian Statistics 94
The Long-Run Consequence of Behavior 97
Decision Theory 97
The Accept-Reject Theory 98
Frequency Interpretation of Confidence Intervals 99
A Critique of p-Values 102
The Context 102
Simple Hypotheses 104
Composite Hypotheses 108
Test of a Direction 110
Asymptotic Considerations 112
Intuitive Test Statistics 113
Discussion 115
The Nature of Statistical Evidence 116
Introduction 116
Birnbaum's Theorem 117
Counterexamples 120
The Science of Statistics 122
Meanings of "Science" 122
Concepts of Statistics 124
Is Statistics a Science? 124
Comparison of Evidential Theories 126
Evidential Models and Their Criteria 126
Two Interpretations of the Mathematics of Testing Are Needed 126
Evidential Meaning 127
Comparison of Bayesian, Likelihood, and "Show-Me" Evidence 127
An Example-Probability Evidence and the Law 130
Deductive Logic 133
The Prepositional Calculus 133
Truth Tables 135
Deductive Arguments and Conditional Statements 136
Critique of Classical Logic 139
Many-Valued Logic 140
Notes on the Literature 142
References 143
Author Index 149
Subject Index 151
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