논리의 대수 수학.The Algebra of Logic, by Louis Couturat

논리의 대수 수학.The Algebra of Logic, by Louis Couturat
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2 The Two Interpretations of the Logical Calculus . . . . . . . . . 2
0.3 Relation of Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.4 De nition of Equality . . . . . . . . . . . . . . . . . . . . . . . . 4
0.5 Principle of Identity . . . . . . . . . . . . . . . . . . . . . . . . . 5
0.6 Principle of the Syllogism . . . . . . . . . . . . . . . . . . . . . . 6
0.7 Multiplication and Addition . . . . . . . . . . . . . . . . . . . . . 6
0.8 Principles of Simpli cation and Composition . . . . . . . . . . . 8
0.9 The Laws of Tautology and of Absorption . . . . . . . . . . . . . 9
0.10 Theorems on Multiplication and Addition . . . . . . . . . . . . . 10
0.11 The First Formula for Transforming Inclusions into Equalities . . 11
0.12 The Distributive Law . . . . . . . . . . . . . . . . . . . . . . . . 13
0.13 De nition of 0 and 1 . . . . . . . . . . . . . . . . . . . . . . . . . 14
0.14 The Law of Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 16
0.15 De nition of Negation . . . . . . . . . . . . . . . . . . . . . . . . 17
0.16 The Principles of Contradiction and of Excluded Middle . . . . . 19
0.17 Law of Double Negation . . . . . . . . . . . . . . . . . . . . . . . 19
0.18 Second Formulas for Transforming Inclusions into Equalities . . . 20
0.19 The Law of Contraposition . . . . . . . . . . . . . . . . . . . . . 21
0.20 Postulate of Existence . . . . . . . . . . . . . . . . . . . . . . . . 22
0.21 The Development of 0 and of 1 . . . . . . . . . . . . . . . . . . . 23
0.22 Properties of the Constituents . . . . . . . . . . . . . . . . . . . . 23
0.23 Logical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
0.24 The Law of Development . . . . . . . . . . . . . . . . . . . . . . 24
0.25 The Formulas of De Morgan . . . . . . . . . . . . . . . . . . . . . 26
0.26 Disjunctive Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
0.27 Properties of Developed Functions . . . . . . . . . . . . . . . . . 28
0.28 The Limits of a Function . . . . . . . . . . . . . . . . . . . . . . 30
0.29 Formula of Poretsky. . . . . . . . . . . . . . . . . . . . . . . . . . 32
0.30 Schr？der's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 33
0.31 The Resultant of Elimination . . . . . . . . . . . . . . . . . . . . 34
0.32 The Case of Indetermination . . . . . . . . . . . . . . . . . . . . 36
0.33 Sums and Products of Functions . . . . . . . . . . . . . . . . . . 36
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0.34 The Expression of an Inclusion by Means of an Indeterminate . . 39
0.35 The Expression of a Double Inclusion by Means of an Indeterminate 40
0.36 Solution of an Equation Involving One Unknown Quantity . . . . 42
0.37 Elimination of Several Unknown Quantities . . . . . . . . . . . . 45
0.38 Theorem Concerning the Values of a Function . . . . . . . . . . . 47
0.39 Conditions of Impossibility and Indetermination . . . . . . . . . 48
0.40 Solution of Equations Containing Several Unknown Quantities . 49
0.41 The Problem of Boole . . . . . . . . . . . . . . . . . . . . . . . . 51
0.42 The Method of Poretsky . . . . . . . . . . . . . . . . . . . . . . . 52
0.43 The Law of Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 53
0.44 The Law of Consequences . . . . . . . . . . . . . . . . . . . . . . 54
0.45 The Law of Causes . . . . . . . . . . . . . . . . . . . . . . . . . . 56
0.46 Forms of Consequences and Causes . . . . . . . . . . . . . . . . . 59
0.47 Example: Venn's Problem . . . . . . . . . . . . . . . . . . . . . . 60
0.48 The Geometrical Diagrams of Venn . . . . . . . . . . . . . . . . . 62
0.49 The Logical Machine of Jevons . . . . . . . . . . . . . . . . . . . 64