More Precisely

Table of Contents

This is the Table of Contents for the Second Edition, which is now available.

Preface

Sets

• 1. Collections of Things
• 2. Sets and Members
• 3. Set Builder Notation
• 4. Subsets
• 5. Small Sets
• 6. Unions of Sets
• 7. Intersections of Sets
• 8. Difference of Sets
• 9. Set Algebra
• 10. Sets of Sets
• 11. Union of a Set of Sets
• 12. Power Sets
• 13. Sets and Selections
• 14. Pure Sets
• 15. Sets and Numbers
• 16. Sums of Sets of Numbers
• 17. Ordered Pairs
• 18. Ordered Tuples
• 19. Cartesian Products

Relations

• 1. Relations
• 2. Some Features of Relations
• 3. Equivalence Relations and Classes
• 4. Closures of Relations
• 5. Recursive Definitions and Ancestrals
• 6. Personal Persistence
• 6.1 The Diachronic Sameness Relation
• 6.2 The Memory Relation
• 6.3 Symmetric then Transitive Closure
• 6.4 The Fission Problem
• 6.5 Transitive then Symmetric Closure
• 7. Closure Under an Operation
• 8. Closure Under Physical Relations
• 9. Order Relations
• 10. Degrees of Perfection
• 11. Parts of Sets
• 12. Functions
• 13. Some Examples of Functions
• 14. Isomorphisms
• 15. Functions and Sums
• 16. Sequences and Operations on Sequences
• 17. Cardinality
• 18. Sets and Classes

Machines

• 1. Machines
• 2. Finite State Machines
• 2.1 Rules for Machines
• 2.2 The Careers of Machines
• 2.3 Utilities of States and Careers
• 3. The Game of Life
• 3.1 A Universe Made from Machines
• 3.2 The Causal Law in the Game of Life
• 3.3 Regularities in the Causal Flow
• 3.4 Constructing the Game of Life from Pure Sets
• 4. Turing Machines
• 5. Lifelike Worlds

Semantics

• 1. Extensional Semantics
• 1.1 Words and Referents
• 1.2 A Sample Vocabulary and Model
• 1.3 Sentences and Truth-Conditions
• 2. Simple Modal Semantics
• 2.1 Possible Worlds
• 2.2. A Sample Modal Structure
• 2.3 Sentences and Truth at Possible Worlds
• 2.4 Modalities
• 2.5 Intensions
• 2.6 Propositions
• 3. Modal Semantics with Counterparts
• 3.1 The Counterpart Relation
• 3.2 A Sample Model for Counterpart Theoretic Semantics
• 3.3 Truth-Conditions for Non-Modal Statements
• 3.4 Truth-Conditions for Modal Statements

Probability

• 1. Sample Spaces
• 2. Simple Probability
• 3. Combined Probabilities
• 4. Probability Distributions
• 5. Conditional Probabilities
• 5.1 Restricting the Sample Space
• 5.2 The Definition of Conditional Probability
• 5.3 An Example Involving Marbles
• 5.4 Independent Events
• 6. Bayes Theorem
• 6.1 The First Form of Bayes Theorem
• 6.2 An Example Involving Medical Diagnosis
• 6.3 The Second Form of Bayes Theorem
• 6.4 An Example Involving Envelopes with Prizes
• 7. Degrees of Belief
• 7.1 Sets and Sentences
• 7.2 Subjective Probability Functions
• 8. Bayesian Confirmation Theory
• 8.1 Confirmation and Disconfirmation
• 8.2 Bayesian Conditionalization
• 9. Knowledge and the Flow of Information

Information Theory

• 1. Communication
• 2. Exponents and Logarithms
• 3. The Probabilities of Messages
• 4. Efficient Codes for Communication
• 4.1 A Method for Making Binary Codes
• 4.2 The Weight Moving across a Bridge
• 4.3 The Information Flowing through a Channel
• 4.4 Messages with Variable Probabilities
• 4.5 Compression
• 4.6 Compression using Huffman Codes
• 5. Entropy
• 5.1 Probability and the Flow of Information
• 5.2 Shannon Entropy
• 5.3 Entropy in Aesthetics
• 5.4 Joint Probability
• 5.5 Joint Entropy
• 6. Mutual Information
• 6.1 From Joint Entropy to Mutual Information
• 6.2 From Joint to Conditional Probabilities
• 6.3 Conditional Entropy
• 6.4 From Conditional Entropy to Mutual Information
• 6.5 An Illustration of Entropies and Codes
• 7. Information and Mentality
• 7.1 Mutual Information and Mental Representation
• 7.2 Integrated Information Theory and Consciousness

Decisions and Games

• 1. Act Utilitarianism
• 1.1 Agents and Actions
• 1.2 Actions and their Consequences
• 1.3 Utility and Moral Quality
• 2. Expected Utility
• 3. Game Theory
• 4. Static Games
• 5. The Prisoner's Dilemma
• 6. Philosophical Issues in the Prisoner's Dilemma
• 7. Dominant Strategies
• 8. The Stag Hunt
• 9. Nash Equilibria
• 10. The Iterated Prisoners Dilemma
• 11. The Spatialized Iterated Prisoner's Dilemma
• 12. Public Goods Games
• 13. The Evolution of Cooperation

From the Finite to the Infinite

• 1. Recursively Defined Series
• 2. Limits of Recursively Defined Series
• 2.1 Counting Through All the Numbers
• 2.2 Cantor's Three Number Generating Rules
• 2.3 The Series of Von Neumann Numbers
• 3. Some Examples of Series with Limits
• 3.1 Achilles Runs on Zeno's Racetrack
• 3.2 The Royce Map
• 3.3 The Hilbert Paper
• 3.4 An Enuless Series of Degrees of Perfection
• 4. Infinity
• 4.1 Infinity and Infinite Complexity
• 4.2 The Hilbert Hotel
• 4.3 Operations on Infinite Sequences
• 5. Supertasks
• 5.1 Reading the Borges Book
• 5.2 The Thomson Lamp
• 5.3 Zeus Performs a Super-Computation
• 5.4 Accelerating Turing Machines

Bigger Infinities

• 1. Some Transfinite Ordinal Numbers
• 2. Comparing the Sizes of Sets
• 3. Ordinal and Cardinal Numbers
• 4. Cantor's Diagonal Argument
• 5. Cantor's Power Set Argument
• 5.1 Sketch of the Power Set Argument
• 5.2 The Power Set Argument in Detail
• 5.3 The Beth Numbers
• 6. The Aleph Numbers
• 7. Transfinite Recursion
• 7.1 Rules for the Long Line
• 7.2 The Sequence of Universes
• 7.3 Degrees of Divine Perfection

Further Study

Glossary of Symbols

References

Index