Book contents
- Frontmatter
- Contents
- Preface
- Note to the Reader
- CHAPTER ONE Confidence
- CHAPTER TWO Evidence
- CHAPTER THREE The Bayesian Challenge
- CHAPTER FOUR Rational Belief
- CHAPTER FIVE The Bayesian Canon
- CHAPTER SIX Decision Theory as Epistemology
- APPENDIX 1 Principles and Definitions
- APPENDIX 2 Proofs
- APPENDIX 3 Probabilism – Some Elementary Theorems
- Bibliography
- Index
APPENDIX 3 - Probabilism – Some Elementary Theorems
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Note to the Reader
- CHAPTER ONE Confidence
- CHAPTER TWO Evidence
- CHAPTER THREE The Bayesian Challenge
- CHAPTER FOUR Rational Belief
- CHAPTER FIVE The Bayesian Canon
- CHAPTER SIX Decision Theory as Epistemology
- APPENDIX 1 Principles and Definitions
- APPENDIX 2 Proofs
- APPENDIX 3 Probabilism – Some Elementary Theorems
- Bibliography
- Index
Summary
There is a small number of useful elementary theorems about how con(–) works to which I have appealed, both in the body of the book and in appendix 2, without first proving that these principles follow from the supposition that con(–) satisfies the Kolmogorov axioms of probability. This appendix is designed to provide, for the sake of those readers who are somewhat unfamiliar with the workings of the probability calculus, the missing proofs.
Recall that con(–) satisfies the Kolmogorov axioms of probability if and only if
A. con(–) assigns a real number to every hypothesis P and hypothesis Q in such a way that
(i) con(P) ≥ 0;
(ii) if P is a tautology, then con(P) = 1; and
(iii) if P and Q are mutually exclusive, then con(P v Q) = con(P) + con(Q).
The theorems that will be proved are
T1. con(P) = 1 – con(~P);
T2. if P entails Q, then con(Q) ≥ con(P);
T3. if P and Q are logically equivalent, con(P) = con(Q);
T4. con(P & Q) = con(P) – con(P & ~Q); and
T5. con(P v Q) = con(P) + con(Q) – con(P & Q).
- Type
- Chapter
- Information
- Decision Theory as Philosophy , pp. 212 - 214Publisher: Cambridge University PressPrint publication year: 1996