Philosophical Devices, Proofs, Probabilities, Possibilities, and Sets

Cover......Page 1
Contents......Page 8
List of Boxes......Page 14
Preface......Page 16
Introduction......Page 18
Part I: SETS AND NUMBERS......Page 22
1.1 Sets......Page 24
1.2 Membership and the Axiom of Extensionality......Page 25
1.3 Unions, Intersections, and the Empty Set......Page 26
1.5 Members versus Subsets......Page 27
1.6 Power Sets......Page 29
1.7 The Axiom of Comprehension......Page 30
1.8 Russell’ s Set......Page 31
1.9 Russell’ s Paradox......Page 32
1.10 Barbers and Sets......Page 33
1.11 Alternatives to Naive Set Theory......Page 34
Exercises......Page 36
2.1 Some Infinite Sets......Page 38
2.2 Different Kinds of Numbers......Page 39
2.3 Two Senses of ‘ More’......Page 41
2.4 Denumerability......Page 43
2.5 More Denumerable Sets......Page 45
2.6 The Non-Denumerability of the Real Numbers......Page 46
2.7 The Abundance of the Real Numbers......Page 48
Exercises......Page 49
3.2 The Numerical Size of Sets......Page 51
3.3 The Reals and the Power Set of the Natural Numbers......Page 53
3.4 The Continuum Hypothesis......Page 56
3.5 An Infinity of Infinities......Page 57
3.6 The Generalized Continuum Hypothesis......Page 59
Exercises......Page 61
Part II: ANALYTICITY, A PRIORICITY, AND NECESSITY......Page 64
4.2 Analytic and Synthetic......Page 66
4.3 A Priori and A Posteriori......Page 67
4.4 Synthetic A Prioris......Page 68
4.5 How is Synthetic A Priori Knowledge Possible?......Page 70
4.6 Pure and Applied Geometry......Page 71
Exercises......Page 77
5.1 Necessity and Contingency......Page 79
5.2 A Posteriori Necessities......Page 80
5.3 A Priori Contingencies......Page 81
5.4 Possibility and Necessity......Page 82
5.5 Possible Worlds......Page 83
5.6 Necessity and Possibility in terms of Worlds......Page 84
5.7 Constraints on Possible Worlds......Page 85
5.8 Essential Properties......Page 87
5.9 The Nature of Necessity......Page 88
5.10 Different Kinds of Possibility......Page 89
Exercises......Page 91
6.1 Two Readings of Statements of Necessity......Page 93
6.2 Scope Distinctions......Page 94
6.3 Julius and the Inventor of the Zip......Page 95
6.4 Rigid Designators......Page 96
6.5 The Causal Theory of Reference......Page 97
6.6 Rigidity and the Causal Theory......Page 98
6.7 De Dicto and De Re......Page 99
6.8 Necessary and A Priori Again......Page 101
6.9 A Limit to Scepticism about A Posteriori Necessity......Page 102
Exercises......Page 106
Part III: THE NATURE AND USES OF PROBABILITY......Page 108
7.2 Kolmogorov’s Axioms......Page 110
7.3 Some Consequences......Page 111
7.4 Joint Probabilities......Page 112
7.5 Subjective and Objective Probabilities......Page 115
7.6 Subjective Probability......Page 116
7.7 Action, Utility, and Subjective Probability......Page 117
7.8 Dutch Books......Page 119
7.9 Objective Probability......Page 120
Exercises......Page 123
8.1 The Principal Principle......Page 125
8.2 Conditional Probability......Page 127
8.3 Updating Degrees of Belief—Conditionalization......Page 128
8.4 Bayes’ Theorem......Page 130
8.5 Conditional Probabilities and Conditional Statements......Page 131
8.6 Material Conditionals......Page 132
8.7 Indicative and Subjunctive Conditionals......Page 135
8.8 Rational and Metaphysical Changes......Page 136
Exercises......Page 138
9.1 Probabilistic Independence......Page 140
9.2 Probabilistic Dependence......Page 141
9.3 Correlation......Page 142
9.4 Causation and Correlation......Page 143
9.5 Screening Off......Page 144
9.6 Spurious Correlations......Page 145
9.7 Randomized Experiments......Page 146
9.8 Survey Research......Page 148
9.9 Simpson’s Paradox......Page 149
Exercises......Page 152
Part IV: LOGICS AND THEORIES......Page 156
10.1 Validity......Page 158
10.2 Logic and Metalogic......Page 159
10.4 Truth-Functional Connectives......Page 160
10.5 Syntax and Semantics......Page 163
10.6 Syntactic Consequence......Page 164
10.7 Semantic Consequence......Page 165
Exercises......Page 169
11.1 Soundness and Completeness......Page 170
11.2 Proving Soundness and Completeness......Page 171
11.3 Reflections on Circularity......Page 172
11.4 Predicate Logic......Page 174
11.5 Predicate Syntax......Page 175
11.6 Predicate Semantics......Page 177
11.8 Predicate Logic—Undecidability......Page 178
11.9 Second-Order Logic......Page 180
11.10 The Incompleteness of Second-Order Logic......Page 182
Further Reading......Page 184
12.1 Theories......Page 185
12.2 Syntax and Semantics for Theories......Page 186
12.3 Theoretical Completeness......Page 187
12.4 Completeness for Theories versus Completeness for Logics......Page 189
12.5 Gödel’s Theorem Stated......Page 190
12.6 A Sketch of Gödel’s Proof......Page 191
12.7 The Inescapability of Gödel’s Theorem......Page 193
12.8 Meta-Theorizing......Page 195
Further Reading......Page 198
Solutions to Exercises......Page 200
C......Page 210
L......Page 211
R......Page 212
W......Page 213