### Chapter I. Definition of Pure Mathematics

- § 1. Definition of pure mathematics
- § 2. The principles of mathematics are no longer controversial
- § 3. Pure mathematics uses only a few notions, and these are logical constants
- § 4. All pure mathematics follows formally from twenty premisses
- § 5. Asserts formal implications
- § 6. And employs variables
- § 7. Which may have any value without exception
- § 8. Mathematics deals with types of relations
- § 9. Applied mathematics is defined by the occurrence of constants which are not logical.
- § 10. Relation of mathematics to logic.

### Chapter II. Symbolic Logic

### Chapter III. Implication and Formal Implication

- § 37. Meaning of implication
- § 38. Asserted and unasserted propositions
- § 39. Inference does not require two premisses
- § 40. Formal implication is to be interpreted extensionally
- § 41. The variable in formal implication has an unrestricted field
- § 42. A formal implication is a single propositional function, not a relation of two
- § 43. Assertions
- § 44. Conditions that a term in an implication may be varied
- § 45. Formal implication involved in rules of inference

### Chapter IV. Proper Names, Adjectives and Verbs

- § 46. Proper names, adjectives and verbs distinguished
- § 47. Terms
- § 48. Things and concepts
- § 49. Concepts as such and as terms
- § 50. Conceptual diversity
- § 51. Meaning and the subject-predicate logic
- § 52. Verbs and truth
- § 53. All verbs, except perhaps
*is*, express relations - § 54. Relations per se and relating relations
- § 55. Relations are not particularized by their terms

### Chapter V. Denoting

- § 56. Definition of denoting
- § 57. Connection with subject-predicate propositions
- § 58. Denoting concepts obtained from predicates
- § 59. Extensional account of
*all*,*every*,*any*,*a*and*some* - § 60. Intensional account of the same
- § 61. Illustrations
- § 62. The difference between
*all*,*every*, etc. lies in the objects denoted, not in the way of denoting them. - § 63. The notion of
*the*and definition - § 64. The notion of
*the*and identity - § 65. Summary

### Chapter VI. Classes

- § 66. Combination of intensional and extensional standpoints required
- § 67. Meaning of
*class* - § 68. Intensional and extensional genesis of classes
- § 69. Distinctions overlooked by Peano
- § 70. The class as one and as many
- § 71. The notion of
*and* - § 72.
*All men*is not analyzable into*all*and*men* - § 73. There are null class-concepts, but there is no null class
- § 74. The class as one, except when it has one term, is distinct from the class as many
- § 75.
*Every*,*any*,*a*and*some*each denote one object, but an ambiguous one - § 76. The relation of a term to its class
- § 77. The relation of inclusion between classes
- § 78. The contradiction
- § 79. Summary

### Chapter VII. Propositional Functions.

- § 80. Indefinability of
*such that* - § 81. Where a fixed relation to a fixed term is asserted, a propositional function can be analysed into a variable subject and a constant assertion
- § 82. But this analysis is impossible in other cases
- § 83. Variation of the concept in a proposition
- § 84. Relation of propositional functions to classes
- § 85. A propositional function is in general not analysable into a constant and a variable element

- § 80. Indefinability of
### Chapter VIII. The Variable.

- § 86. Nature of the variable
- § 87. Relation of the variable to
*any* - § 88. Formal and restricted variables
- § 89. Formal implication presupposes
*any* - § 90. Duality of
*any*and*some* - § 91. The class-concept
*propositional function*is indefinable - § 92. Other classes can be defined by means of
*such that* - § 93. Analysis of the variable

### Chapter IX. Relations

### Chapter X. The Contradiction

- § 100. Consequences of the contradiction
- § 101. Various statements of the contradiction
- § 102. An analogous generalized argument
- § 103. Various statements of the contradiction
- § 104. The contradiction arises from treating as one a class which is only many
- § 105. Other primâ facie possible solutions appear inadequate
- § 106. Summary of Part I

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.