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
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