Philosophical Devices
Proofs, Probabilities, Possibilities, and Sets
1st Edition
Published on 1. October 2012
213 pages
978-0-19-165624-8 (ISBN)
Description
Philosophical Devices introduces the technical ideas that are taken for granted in contemporary philosophical writing. It offers simple explanations and covers a wealth of material that is normally available only to specialists. This original, distinctive book will appeal to anyone who is curious about the technical infrastructure of philosophy.
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10/2012
1st Edition
Oxford University Press
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Content
- Cover
- Contents
- List of Boxes
- Preface
- Introduction
- Part I: SETS AND NUMBERS
- 1 Naive Sets and Russell's Paradox
- 1.1 Sets
- 1.2 Membership and the Axiom of Extensionality
- 1.3 Unions, Intersections, and the Empty Set
- 1.4 Subsets
- 1.5 Members versus Subsets
- 1.6 Power Sets
- 1.7 The Axiom of Comprehension
- 1.8 Russell' s Set
- 1.9 Russell' s Paradox
- 1.10 Barbers and Sets
- 1.11 Alternatives to Naive Set Theory
- Further Reading
- Exercises
- 2 Infinite Sets
- 2.1 Some Infinite Sets
- 2.2 Different Kinds of Numbers
- 2.3 Two Senses of ' More'
- 2.4 Denumerability
- 2.5 More Denumerable Sets
- 2.6 The Non-Denumerability of the Real Numbers
- 2.7 The Abundance of the Real Numbers
- Further Reading
- Exercises
- 3 Orders of Infinity
- 3.1 Some Harder Stuff
- 3.2 The Numerical Size of Sets
- 3.3 The Reals and the Power Set of the Natural Numbers
- 3.4 The Continuum Hypothesis
- 3.5 An Infinity of Infinities
- 3.6 The Generalized Continuum Hypothesis
- Further Reading
- Exercises
- Part II: ANALYTICITY, A PRIORICITY, AND NECESSITY
- 4 Kinds of Truths
- 4.1 Three Distinctions among Truths
- 4.2 Analytic and Synthetic
- 4.3 A Priori and A Posteriori
- 4.4 Synthetic A Prioris
- 4.5 How is Synthetic A Priori Knowledge Possible?
- 4.6 Pure and Applied Geometry
- Further Reading
- Exercises
- 5 Possible Worlds
- 5.1 Necessity and Contingency
- 5.2 A Posteriori Necessities
- 5.3 A Priori Contingencies
- 5.4 Possibility and Necessity
- 5.5 Possible Worlds
- 5.6 Necessity and Possibility in terms of Worlds
- 5.7 Constraints on Possible Worlds
- 5.8 Essential Properties
- 5.9 The Nature of Necessity
- 5.10 Different Kinds of Possibility
- Further Reading
- Exercises
- 6 Naming and Necessity
- 6.1 Two Readings of Statements of Necessity
- 6.2 Scope Distinctions
- 6.3 Julius and the Inventor of the Zip
- 6.4 Rigid Designators
- 6.5 The Causal Theory of Reference
- 6.6 Rigidity and the Causal Theory
- 6.7 De Dicto and De Re
- 6.8 Necessary and A Priori Again
- 6.9 A Limit to Scepticism about A Posteriori Necessity
- Further Reading
- Exercises
- Part III: THE NATURE AND USES OF PROBABILITY
- 7 Kinds of Probability
- 7.1 Probabilities of Propositions
- 7.2 Kolmogorov's Axioms
- 7.3 Some Consequences
- 7.4 Joint Probabilities
- 7.5 Subjective and Objective Probabilities
- 7.6 Subjective Probability
- 7.7 Action, Utility, and Subjective Probability
- 7.8 Dutch Books
- 7.9 Objective Probability
- Further Reading
- Exercises
- 8 Constraints on Credence
- 8.1 The Principal Principle
- 8.2 Conditional Probability
- 8.3 Updating Degrees of Belief-Conditionalization
- 8.4 Bayes' Theorem
- 8.5 Conditional Probabilities and Conditional Statements
- 8.6 Material Conditionals
- 8.7 Indicative and Subjunctive Conditionals
- 8.8 Rational and Metaphysical Changes
- Further Reading
- Exercises
- 9 Correlations and Causes
- 9.1 Probabilistic Independence
- 9.2 Probabilistic Dependence
- 9.3 Correlation
- 9.4 Causation and Correlation
- 9.5 Screening Off
- 9.6 Spurious Correlations
- 9.7 Randomized Experiments
- 9.8 Survey Research
- 9.9 Simpson's Paradox
- Further Reading
- Exercises
- Part IV: LOGICS AND THEORIES
- 10 Syntax and Semantics
- 10.1 Validity
- 10.2 Logic and Metalogic
- 10.3 Different Kinds of Logic
- 10.4 Truth-Functional Connectives
- 10.5 Syntax and Semantics
- 10.6 Syntactic Consequence
- 10.7 Semantic Consequence
- Further Reading
- Exercises
- 11 Soundness and Completeness
- 11.1 Soundness and Completeness
- 11.2 Proving Soundness and Completeness
- 11.3 Reflections on Circularity
- 11.4 Predicate Logic
- 11.5 Predicate Syntax
- 11.6 Predicate Semantics
- 11.7 Predicate Logic-Soundness and Completeness
- 11.8 Predicate Logic-Undecidability
- 11.9 Second-Order Logic
- 11.10 The Incompleteness of Second-Order Logic
- Further Reading
- 12 Theories and Gödel's Theorem
- 12.1 Theories
- 12.2 Syntax and Semantics for Theories
- 12.3 Theoretical Completeness
- 12.4 Completeness for Theories versus Completeness for Logics
- 12.5 Gödel's Theorem Stated
- 12.6 A Sketch of Gödel's Proof
- 12.7 The Inescapability of Gödel's Theorem
- 12.8 Meta-Theorizing
- Further Reading
- Solutions to Exercises
- Index
- A
- B
- C
- D
- E
- F
- G
- H
- I
- J
- K
- L
- M
- N
- O
- P
- Q
- R
- S
- T
- U
- V
- W
