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I. THE SENTENTIAL CALCULUS 1. Introduction of the Fundamental Logical Connectives 3 2. Equivalence; Dispensability of Fundamental Connec- tives 5 3. Normal Form for Logical Expressions 11 4. Characterization of Logically True Combinations of Sentences 13 5. The Principle of Duality 15 6. The Disjunctive Normal Form for Logical Expressions 17 7. The Totality of Combinations which Can be Formed from Given Elementary Sentences 18 8. Supplementary Remarks on the Problem of Universal Validity and Satisfiability 21 9. Systematic Survey of All the Deductions from Given Axioms 23 10. The Axioms of the Sentential Calculus 27 11. Examples of the Proof of Theorems from the Axioms 30 12. The Consistency of the System of Axioms 38 13. The Independence and Completeness of the System 40 II. THE CALCULUS OF CLASSES (MONADIC PREDICATE CALCULUS) 1. Reinterpretation of the Symbolism of the Sentential Calculus 44 2. The Combination of the Calculus of Classes with the Sentential Calculus 47 3. Systematic Derivation of the Traditional Aristotelian Inferences 48 III. THE RESTRICTED PREDICATE CALCULUS 1. Inadequacy of the Foregoing Calculus 55 2. Methodological Basis of the Predicate Calculus 56 3. Preliminary Orientation on the Use of the Predicate Calculus 61 4. Precise Notation for the Predicate Calculus 65 5. The Axioms of the Predicate Calculus 67 6. The System of Universally Valid Formulas 71 7. The Rule of Substitution; Construction of the Contra- dictory of a Formula 79 8. The Extended Principle of Duality; Normal Forms 81 9. Consistency and Independence of the System of Axioms 87 10. The Completeness of the Axiom System 92 11. Derivation of Consequences from Given Premises; Relation to Universally Valid Formulas 101 12. The Decision Problem 112 IV. THE EXTENDED PREDICATE CALCULUS 1. The Predicate Calculus of Second Order 125 2. Introduction of Predicates of Second Level; Logical Treatment of the Concept of Number 135 3. Representation of the Fundamental Concepts of Set Theory in the Extended Calculus 139 4. The Logical Paradoxes 143 5. The Predicate Calculus of Order co 152 6. Applications of the Calculus of Order w 158 EDITOR'S NOTES 165 BIBLIOGRAPHY 169 INDEX 171Library of Congress Subject Headings for this publication: Logic, Symbolic and mathematical