CSLI, 1996 Cloth: 978-1-57586-067-1 | Paper: 978-1-57586-066-4 | eISBN: 978-1-57586-758-8 Library of Congress Classification QA10.A34 1998 Dewey Decimal Classification 511.3
ABOUT THIS BOOK | TOC
ABOUT THIS BOOK
This book is meant to be a primer, that is, an introduction, to probability logic, a subject that appears to be in its infancy. Probability logic is a subject envisioned by Hans Reichenbach and largely created by Adams. It treats conditionals as bearers of conditional probabilities and discusses an appropriate sense of validity for arguments such conditionals, as well as ordinary statements as premisses.
This is a clear well-written text on the subject of probability logic, suitable for advanced undergraduates or graduates, but also of interest to professional philosophers. There are well-thought-out exercises, and a number of advanced topics treated in appendices, while some are brought up in exercises and some are alluded to only in footnotes. By this means, it is hoped that the reader will at least be made aware of most of the important ramifications of the subject and its tie-ins with current research, and will have some indications concerning recent and relevant literature.
TABLE OF CONTENTS
Contents
Foreword
Brian,
Skyrms
Preface
1
Deduction and Probability: What Probability Logic Is About
1.1
Deduction and Certainty
1.2
Inference and Probability Change: Nonmonotonicity
1.3
Conditionals
1.4
Decision and Action: The Advantage of Being Right
1.5
Summary and Limitations of This Work
1.6
Glossary
2
Probability and Logic
2.1
Logical Symbolism and Basic Concepts
2.2
Fundamental Connections between Logic and Probability
2.3
Probability Functions and Algebra
2.4
Deductive Theory of Probability
2.5
Probabilistic Independence, Symmetry, and Randomness
2.6
Uncertainty and Its Laws
2.7
Glossary
3
Deduction and Probability Part I: Statics
3.1
Introduction
3.2
Two Hauptsätze of the Theory of Uncertainty Accumulation
3.3
Reinforcing Conclusions by Introducing Redundancy among Premises: Degrees of Essentialness
3.4
Remarks on Defaults and Other Assumptions That Limit Conclusion Uncertainties
3.5
Glossary
4
Conditional Probabilities and Conditionalization
4.1
Introduction: The Dependence of Probabilities on States of Knowledge
4.2
The Formalism of Conditional Probability
4.3
The Chain Rule: Probabilistic Dependence and Independence
4.4
Ersatz Formulas and Truth Values
4.5
Probability Change and Bayes' Principle
4.6
On Induction
4.7
Induction and Symmetry
4.8
Critique of Bayesian Epistemology
4.9
Glossary
5
Deduction and Probability Part II: Dynamics
5.1
Introduction
5.2
Deductive Dynamics, General
5.3
Nonmonotonicity and Scope: The Two-premise Case
5.4
What Follows from a Contradiction
5.5
More General Inferences
5.6
Defaults: The Logic of Conversation Again
5.7
Posterior Premise Uncertainty
5.8
Glossary
6
Probability Conditionals: Basics
6.1
Introduction
6.2
The Paradoxes of Material Implication
6.3
Intuitive Analysis of Other Inference Patterns
6.4
Probabilistic Validity
6.5
Order-of-Magnitude Orderings and Distributions
6.6
Enthymemes and Other Defaults
6.7
Problems of Inferential Dynamics
6.8
Glossary
7*
Formal Theory of Probability Conditionals: Derivations and Related Matters
7.1
Aspects of Derivation: Basic Definitions and Equivalences
7.2
Rules of Conditional Inference
7.3
Derived Rules of Inference: Shortcut Derivations
7.4
Quasi-conjunction
7.5
An Ersatz Truth-table Test for Probabilistic Validity
7.6
Validity and Completeness, with Sketches of Their Proofs
7.7
Other Aspects of Metatheory
7.8
P-tautology, Equivalence, and Consistency
7.9
On Probabilistic Certainty Formulas
7.10
Glossary
8*
Truth, Triviality, and Controversy
8.1
Problems, and Some History
8.2
Truth-functionality
8.3
Truth-conditionality: Stalnaker's Theory
8.4
The Problem of Probability
8.5
Triviality Results
8.6
The Controversy about Truth: Theories of Truth
8.7
Glossary
9
Practical Reason
9.1
Introduction
9.2
Practical Inference: A Qualitative Representation
9.3
A Pragmatic Principle
9.4
Decision Matrices
9.5
First Generalization: Degrees of Confidence and Utility
9.6
Critical Degrees of Confidence
9.7
Measuring Confidence and Utility
9.8
Decision Making in the Nonindependence Case
9.9
Unorthodox Reflections on Practical Probabilities
9.10
Glossary
A1
Coherent Degrees of Confidence: Axiomatics and Rational Betting Odds
1.1
Introduction
1.2
The Axiomatic Approach
1.3
Rational Betting Odds and Dutch Books
1.4
Dynamical Dutch Books
1.5
Critique
A2
Infinitesimal Probabilities and Popper Functions
2.1
Introduction
2.2
Two Objections to the Stipulation
2.3
Intuitive Infinitesimals and Non-standard Probabilities
2.4
Popper Functions
A3
David Lewis's Triviality Results
3.1
Introduction
3.2
The Results
3.3
A Way Out
3.4
Discussion
A4
The Problem of Embedded Conditionals
4.1
Introduction
4.2
Some Partial Theories
A5
A Geometrical Argument for the ‘Statistical Reasonableness’ of Contraposition
A6
Counterfactual Conditionals
A7
Probabilistic Predicate Logic
7.1
Introduction
7.2
Language. Worlds, and Probabilities
7.3
Probabilistic Validity
7.4
Dynamics and Scope of Universal Statements: Confirmation
A8
Probabilistic Identity Logic
8.1
Generalities
8.2
Complications: Connections with Frege's Puzzle
A9
Approximate Generalizations
9.1
Introduction
9.2
Approximate Monadic Generalizations
9.3
Multi-variate Approximate Generalizations
9.4
Generalizations and Conditionals: Degrees of Truth and Probability
This book is meant to be a primer, that is, an introduction, to probability logic, a subject that appears to be in its infancy. Probability logic is a subject envisioned by Hans Reichenbach and largely created by Adams. It treats conditionals as bearers of conditional probabilities and discusses an appropriate sense of validity for arguments such conditionals, as well as ordinary statements as premisses.
This is a clear well-written text on the subject of probability logic, suitable for advanced undergraduates or graduates, but also of interest to professional philosophers. There are well-thought-out exercises, and a number of advanced topics treated in appendices, while some are brought up in exercises and some are alluded to only in footnotes. By this means, it is hoped that the reader will at least be made aware of most of the important ramifications of the subject and its tie-ins with current research, and will have some indications concerning recent and relevant literature.
TABLE OF CONTENTS
Contents
Foreword
Brian,
Skyrms
Preface
1
Deduction and Probability: What Probability Logic Is About
1.1
Deduction and Certainty
1.2
Inference and Probability Change: Nonmonotonicity
1.3
Conditionals
1.4
Decision and Action: The Advantage of Being Right
1.5
Summary and Limitations of This Work
1.6
Glossary
2
Probability and Logic
2.1
Logical Symbolism and Basic Concepts
2.2
Fundamental Connections between Logic and Probability
2.3
Probability Functions and Algebra
2.4
Deductive Theory of Probability
2.5
Probabilistic Independence, Symmetry, and Randomness
2.6
Uncertainty and Its Laws
2.7
Glossary
3
Deduction and Probability Part I: Statics
3.1
Introduction
3.2
Two Hauptsätze of the Theory of Uncertainty Accumulation
3.3
Reinforcing Conclusions by Introducing Redundancy among Premises: Degrees of Essentialness
3.4
Remarks on Defaults and Other Assumptions That Limit Conclusion Uncertainties
3.5
Glossary
4
Conditional Probabilities and Conditionalization
4.1
Introduction: The Dependence of Probabilities on States of Knowledge
4.2
The Formalism of Conditional Probability
4.3
The Chain Rule: Probabilistic Dependence and Independence
4.4
Ersatz Formulas and Truth Values
4.5
Probability Change and Bayes' Principle
4.6
On Induction
4.7
Induction and Symmetry
4.8
Critique of Bayesian Epistemology
4.9
Glossary
5
Deduction and Probability Part II: Dynamics
5.1
Introduction
5.2
Deductive Dynamics, General
5.3
Nonmonotonicity and Scope: The Two-premise Case
5.4
What Follows from a Contradiction
5.5
More General Inferences
5.6
Defaults: The Logic of Conversation Again
5.7
Posterior Premise Uncertainty
5.8
Glossary
6
Probability Conditionals: Basics
6.1
Introduction
6.2
The Paradoxes of Material Implication
6.3
Intuitive Analysis of Other Inference Patterns
6.4
Probabilistic Validity
6.5
Order-of-Magnitude Orderings and Distributions
6.6
Enthymemes and Other Defaults
6.7
Problems of Inferential Dynamics
6.8
Glossary
7*
Formal Theory of Probability Conditionals: Derivations and Related Matters
7.1
Aspects of Derivation: Basic Definitions and Equivalences
7.2
Rules of Conditional Inference
7.3
Derived Rules of Inference: Shortcut Derivations
7.4
Quasi-conjunction
7.5
An Ersatz Truth-table Test for Probabilistic Validity
7.6
Validity and Completeness, with Sketches of Their Proofs
7.7
Other Aspects of Metatheory
7.8
P-tautology, Equivalence, and Consistency
7.9
On Probabilistic Certainty Formulas
7.10
Glossary
8*
Truth, Triviality, and Controversy
8.1
Problems, and Some History
8.2
Truth-functionality
8.3
Truth-conditionality: Stalnaker's Theory
8.4
The Problem of Probability
8.5
Triviality Results
8.6
The Controversy about Truth: Theories of Truth
8.7
Glossary
9
Practical Reason
9.1
Introduction
9.2
Practical Inference: A Qualitative Representation
9.3
A Pragmatic Principle
9.4
Decision Matrices
9.5
First Generalization: Degrees of Confidence and Utility
9.6
Critical Degrees of Confidence
9.7
Measuring Confidence and Utility
9.8
Decision Making in the Nonindependence Case
9.9
Unorthodox Reflections on Practical Probabilities
9.10
Glossary
A1
Coherent Degrees of Confidence: Axiomatics and Rational Betting Odds
1.1
Introduction
1.2
The Axiomatic Approach
1.3
Rational Betting Odds and Dutch Books
1.4
Dynamical Dutch Books
1.5
Critique
A2
Infinitesimal Probabilities and Popper Functions
2.1
Introduction
2.2
Two Objections to the Stipulation
2.3
Intuitive Infinitesimals and Non-standard Probabilities
2.4
Popper Functions
A3
David Lewis's Triviality Results
3.1
Introduction
3.2
The Results
3.3
A Way Out
3.4
Discussion
A4
The Problem of Embedded Conditionals
4.1
Introduction
4.2
Some Partial Theories
A5
A Geometrical Argument for the ‘Statistical Reasonableness’ of Contraposition
A6
Counterfactual Conditionals
A7
Probabilistic Predicate Logic
7.1
Introduction
7.2
Language. Worlds, and Probabilities
7.3
Probabilistic Validity
7.4
Dynamics and Scope of Universal Statements: Confirmation
A8
Probabilistic Identity Logic
8.1
Generalities
8.2
Complications: Connections with Frege's Puzzle
A9
Approximate Generalizations
9.1
Introduction
9.2
Approximate Monadic Generalizations
9.3
Multi-variate Approximate Generalizations
9.4
Generalizations and Conditionals: Degrees of Truth and Probability