More Precisely

The Math You Need to do Philosophy

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. Networks of Machines
3.1 Interacting Machines
3.2 Machines in the Network
3.3 Configurations in the Network
3.4 The History of a Network of Machines
3.5 Mechanical Metaphysics
4. The Game of Life
4.1 A Universe Made from Machines
4.2 The Causal Law in the Game of Life
4.3 Regularities in the Causal Flow
5. Turing Machines

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

Utilitarianism

1. Act Utilitarianism
1.1 Agents and Actions
1.2 Actions and their Consequences
1.3 Utility and Moral Quality
2. Expected Utility
3. World Utilitarianism
3.1 Agents and their Careers
3.2 Careers and Compatible Worlds
3.3 Runs and Prefixes
3.4 Runs and the Worlds Open to Them
3.5 The Utilities of Worlds
3.6 Optimal Open Worlds
3.7 Actions and their Moral Qualities

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 Endless 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