20. Distribution of Quantifiers

Kapitel in diesem Buch

1. Frontmatter i
2. PREFACE, 1981 iii
3. PREFACE TO THE REVISED EDITION vii
4. PREFACE viii
5. CONTENTS xi
6. Introduction 1
7. CHAPTER ONE. Statelfzents
8. 1. Conjunction, Alternation, and Denial 11
9. 2. The Conditional 14
10. 3. Iterated Composition 18
11. 4. Use versus Mention 23
12. 5. Statements about Statements 27
13. 6. Quasi-Quotation 33
14. 7. Parentheses and Dots 37
15. 8. Reduction to Three Primitives 42
16. 9. Reduction to One Primitive 45
17. 10. Tautology 50
18. 11. Selected Tautologous Forms 55
19. CHAPTER TWO. Quantification
20. 12. The Quantifier 65
21. 13. Formulae 71
22. 14. Bondage, Freedom, Closure 76
23. 15. Axioms of Quantification 80
24. 16. Theorems 85
25. 17. Metatheorems 89
26. 18. Substitutivity of the Biconditional 96
27. 19. Existential Quantification 101
28. 20. Distribution of Quantifiers 105
29. 21. Alphabetic Variance 109
30. CHAPTER THREE. Terms
31. 22. Class and Member 119
32. 23. Logical Formulae 123
33. 24. Abstraction 128
34. 25. Identity 134
35. 26. Abstraction Resumed 140
36. 27. Descriptions and Names 146
37. CHAPTER FOUR. Extended Theory of Classes
38. 28. Stratification 155
39. 29. Further Axioms of Membership 160
40. 30. Substitutivity of Identity 167
41. 31. Substitution for Variables 170
42. 32. Further Consequences 175
43. 33. Logical Product, Sum, Complement 179
44. 34. Inclusion 185
45. 35. Unit Classes 189
46. CHAPTER FIVE. Relations
47. 36. Pairs and Relations 197
48. 37. Abstraction of Relations 202
49. 38. Converse, Image, Relative Product 208
50. 39. The Ancestral 215
51. 40. Functions 221
52. 41. Abstraction of Functions 225
53. 42. Identity and Membership as Relations 229
54. CHAPTER SIX. Number
55. 43. Zero, One, Successor 237
56. 44. Natural Numbers 241
57. 45. Counter Sets 246
58. 46. Finite and Infinite 250
59. 47. Powers of Relations 253
60. 48. Arithmetical Sum, Product, Power 259
61. 49. Familiar Identities of Arithmetic 262
62. 50. Ratios 266
63. 51. Real Numbers 271
64. 52. Further Extensions 275
65. CHAPTER SEVEN. Syntax
66. 53. Formality 283
67. 54. The Syntactical Primitive 287
68. 55. Protosyntax 291
69. 56. Formula and Matrix Defined 295
70. 57. Axioms of Quantification Defined 299
71. 58. Theorem Defined 302
72. 59. Protosyntax Self-Applied 306
73. 60. Incompleteness 312
74. APPENDIX. Theorem versus Metatheorem 319
75. List of Definitions 323
76. List of Theorems and Metatheorems 325
77. Bibliographical References 331
78. Index of Proper Names 339
79. Index of Subjects 341