Lectures on Inductive Logic 1st Edition by Jon Williamson 0199666474 9780199666478

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ISBN 10: 0199666474

ISBN 13: 9780199666478

Author: Jon Williamson

Logic is a field studied mainly by researchers and students of philosophy, mathematics and computing. Inductive logic seeks to determine the extent to which the premisses of an argument entail its conclusion, aiming to provide a theory of how one should reason in the face of uncertainty. It has applications to decision making and artificial intelligence, as well as how scientists should reason when not in possession of the full facts. In this book, Jon Williamson embarks on a quest to find a general, reasonable, applicable inductive logic (GRAIL), all the while examining why pioneers such as Ludwig Wittgenstein and Rudolf Carnap did not entirely succeed in this task. Along the way he presents a general framework for the field, and reaches a new inductive logic, which builds upon recent developments in Bayesian epistemology (a theory about how strongly one should believe the various propositions that one can express). The book explores this logic in detail, discusses some key criticisms, and considers how it might be justified. Is this truly the GRAIL? Although the book presents new research, this material is well suited to being delivered as a series of lectures to students of philosophy, mathematics, or computing and doubles as an introduction to the field of inductive logic

Lectures on Inductive Logic 1st Table of contents:

1. Classical Inductive Logic

1.1 From Deductive to Inductive Logic

1.2 Patterns of Partial Entailment and Support

1.2.1 The Fundamental Inductive Pattern

1.2.2 Diminishing Returns

1.2.3 Examining a Possible Ground

1.2.4 Analogy

1.3 Why Inductive Logic?

1.3.1 Decision Making

1.3.2 Artificial Intelligence

1.3.3 The GRAIL Quest

1.4 Learning from Experience

1.5 Inductive Entailment and Logical Entailment

2. Logic and Probability

2.1 Propositional Logic

2.2 Predicate Logic

2.3 Probability over Logical Languages

2.3.1 Axioms of Probability

2.3.2 Properties of Probability

2.3.3 Truth Tables and Probability

2.3.4 Conditional Probability and Inductive Logic

2.4 Entropy, Divergence and Score

2.5 Interpretations of Probability

2.6 *Probability over Fields of Sets

2.6.1 Fields of Sets

2.6.2 Axioms of Probability

2.6.3 The Valuation Space

3. Combining Probability and Logic

3.1 Entailment

3.2 Support and Consistency

3.3 The Languages of Inductive Logic

3.4 Inductive Qualities

3.5 Probabilistic Logics

3.6 * More Examples of Inductive Logics

4. Carnap’s Programme

4.1 Conditionalizing on a Blank Slate

4.2 Pure and Applied Inductive Logic

4.3 Conditionalization

4.4 The Permutation Postulate

4.5 The Principle of Indifference

4.6 Which Value in the Continuum?

4.7 Which Continuum of Inductive Methods?

4.8 Capturing Logical Entailment

4.9 Summary

5. From Objective Bayesian Epistemology to Inductive Logic

5.1 Objective Bayesian Epistemology

5.2 * Objective versus Subjective Bayesian Epistemology

5.3 Objective Bayesian Inductive Logic

5.4 * Language Invariance

5.5 * Finitely Generated Evidence Sets

5.6 Updating, Expansion and Revision

5.6.1 Maxent and Conditionalization

5.6.2 Maxent and KL-updating

5.7 Summary

6. Logical Entailment

6.1 Truth Tables with Probabilities

6.2 Logical Irrelevance Revisited

6.3 Context and Chance Constraints

6.4 Constraints on Conditional Probabilities

6.5 Revision Under Constraints

6.6 Lottery and Preface Paradoxes Revisited

6.7 The Fundamental Inductive Pattern Revisited

6.7.1 A Surprising Consequence

6.7.2 An Otherwise Surprising Consequence

6.7.3 A Plausible Consequence

6.8 * Inferences in Predicate Inductive Logic

7. Inductive Entailment

7.1 Syntactic Relevance

7.2 The Calibration Norm

7.3 Extended Example

7.4 Is this Application of Confidence Intervals Legitimate?

7.5 Uniqueness of the Interval

7.6 Loss of Information

7.7 Generalization

8. Criticisms of Inductive Logic

8.1 Language Invariance Revisited

8.2 Goodman’s New Problem of Induction

8.3 The Principle of Indifference Revisited

8.4 Universal Hypotheses

8.5 Summary

9. Justification

9.1 Two Problems of Induction

9.2 Two Principles of Rationality

9.3 Minimal Worst-Case Expected Loss

9.4 * Robustness of the Minimax Theorem

9.4.1 Key Assumptions

9.4.2 Rationality Principles

Normalization.

Loss and score.

Worst-case expected loss.

Quantifiers.

Summary and further developments.

10. Conclusion

10.1 Have we Found the GRAIL?

10.2 Open Questions

10.2.1 Knowledge Engineering

10.2.2 Other Questions

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