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Softcover ISBN:  9781470470562 
Product Code:  CHEL/69.S 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 
eBook ISBN:  9781470470609 
Product Code:  CHEL/69.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
Softcover ISBN:  9781470470562 
eBook ISBN:  9781470470609 
Product Code:  CHEL/69.S.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $107.20 $81.20 

Book DetailsAMS Chelsea PublishingVolume: 69; 1950; 172 pp
David Hilbert was particularly interested in the foundations of mathematics. Among many other things, he is famous for his attempt to axiomatize mathematics. This now classic text is his treatment of symbolic logic. It lays the groundwork for his later work with Bernays.
This translation is based on the second German edition, and has been modified according to the criticisms of Church and Quine. In particular, the authors' original formulation of Gödel's completeness proof for the predicate calculus has been updated.
In the first half of the twentieth century, an important debate on the foundations of mathematics took place. Principles of Mathematical Logic represents one of Hilbert's important contributions to that debate. Although symbolic logic has grown considerably in the subsequent decades, this book remains a classic.
ReadershipGraduate students and research mathematicians interested in logic and foundations.

Table of Contents

Front Cover

EDITOR'S PREFACE

PREFACE TO THE FIRST (GERMAN) EDITION

PREFACE TO THE SECOND (GERMAN) EDITION

TABLE OF CONTENTS

INTRODUCTION

CHAPTER I: THE SENTENTIAL CALCULUS

§ 1. Introduction of the Fundamental Logical Connectives

§ 2. Equivalence ; Dispensability of Fundamental Connectives

§ 3. Normal Form for Logical Expressions

§ 4. Characterization of Logically True Combinations of Sentences

§ 5. The Principle of Duality

§ 6. The Disjunctive Normal Form for Logical Expressions

§ 7. The Totality of Combinations Which Can Be Formed from Given Elementary Sentences

§ 8. Supplementary Remarks on the Problem of Universal Validity and Satisfiability

§ 9. Systematic Survey of All the Deductions from Given Axioms

§ 10. The Axioms of the Sentential Calculus

§ 11. Examples of the Proof of Theorems from The Axioms

§ 12. The Consistency of the System of Axioms

§ 13. The Independence and Completeness of the System

CHAPTER II: THE CALCULUS OF CLASSES (MONADIC PREDICATE CALCULUS)

§ 1 . Reinterpretation of the Symbolism of the Sentential Calculus

§ 2. The Combination of the Calculus of Classes with the Sentential Calculus

§ 3. Systematic Derivation of the Traditional Aristotelian Inferences

CHAPTER III: THE RESTRICTED PREDICATE CALCULUS

§ I. The Inadequacy of the Foregoing Calculus

§ 2. Methodological Basis of the Predicate Calculus

§ 3 . Preliminary Orientation on the Use of the Predicate Calculus

§ 4. Precise Notation for the Predicate Calculus

§ 5. The Axioms of the Predicate Calculus

§ 6. The System of Universally Valid Formulas

§ 7. The Rule of Substitution; Construction of the Contradictory of a Formula

§ 8. The Extended Principle of Duality; Normal Forms

§ 9. Consistency and Independence of the System of Axioms

§ 10. The Completeness of the Axiom System

§ 11. Derivation of Consequences from Given Premises; Relation to Universally Valid Formulas

§ 12. The Decision Problem

CHAPTER IV: THE EXTENDED PREDICATE CALCULUS

§ 1. The Predicate Calculus of Second Order

§ 2. Introduction of Predicates of Second Level; Logical Treatment of the Number Concept

§ 3. Representation of the Fundamental Concepts of Set Theory in the Extended Calculus

§ 4. The Logical Paradoxes

§ 5. The Predicate Calculus of Order ω

§ 6. Applications of the Calculus of Order ω

EDITOR'S NOTES

BIBLIOGRAPHY

INDEX

Back Cover


Additional Material

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This book is unmistakably a mathematician's book, but it goes far beyond the limits of mathematics and makes available to everyone interested in logic one of the most permanent results of the study of the foundations of mathematics, the satisfactory symbolic treatment of the basic relations that play a part in deductive reasoning.
Cambridge University Press


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David Hilbert was particularly interested in the foundations of mathematics. Among many other things, he is famous for his attempt to axiomatize mathematics. This now classic text is his treatment of symbolic logic. It lays the groundwork for his later work with Bernays.
This translation is based on the second German edition, and has been modified according to the criticisms of Church and Quine. In particular, the authors' original formulation of Gödel's completeness proof for the predicate calculus has been updated.
In the first half of the twentieth century, an important debate on the foundations of mathematics took place. Principles of Mathematical Logic represents one of Hilbert's important contributions to that debate. Although symbolic logic has grown considerably in the subsequent decades, this book remains a classic.
Graduate students and research mathematicians interested in logic and foundations.

Front Cover

EDITOR'S PREFACE

PREFACE TO THE FIRST (GERMAN) EDITION

PREFACE TO THE SECOND (GERMAN) EDITION

TABLE OF CONTENTS

INTRODUCTION

CHAPTER I: THE SENTENTIAL CALCULUS

§ 1. Introduction of the Fundamental Logical Connectives

§ 2. Equivalence ; Dispensability of Fundamental Connectives

§ 3. Normal Form for Logical Expressions

§ 4. Characterization of Logically True Combinations of Sentences

§ 5. The Principle of Duality

§ 6. The Disjunctive Normal Form for Logical Expressions

§ 7. The Totality of Combinations Which Can Be Formed from Given Elementary Sentences

§ 8. Supplementary Remarks on the Problem of Universal Validity and Satisfiability

§ 9. Systematic Survey of All the Deductions from Given Axioms

§ 10. The Axioms of the Sentential Calculus

§ 11. Examples of the Proof of Theorems from The Axioms

§ 12. The Consistency of the System of Axioms

§ 13. The Independence and Completeness of the System

CHAPTER II: THE CALCULUS OF CLASSES (MONADIC PREDICATE CALCULUS)

§ 1 . Reinterpretation of the Symbolism of the Sentential Calculus

§ 2. The Combination of the Calculus of Classes with the Sentential Calculus

§ 3. Systematic Derivation of the Traditional Aristotelian Inferences

CHAPTER III: THE RESTRICTED PREDICATE CALCULUS

§ I. The Inadequacy of the Foregoing Calculus

§ 2. Methodological Basis of the Predicate Calculus

§ 3 . Preliminary Orientation on the Use of the Predicate Calculus

§ 4. Precise Notation for the Predicate Calculus

§ 5. The Axioms of the Predicate Calculus

§ 6. The System of Universally Valid Formulas

§ 7. The Rule of Substitution; Construction of the Contradictory of a Formula

§ 8. The Extended Principle of Duality; Normal Forms

§ 9. Consistency and Independence of the System of Axioms

§ 10. The Completeness of the Axiom System

§ 11. Derivation of Consequences from Given Premises; Relation to Universally Valid Formulas

§ 12. The Decision Problem

CHAPTER IV: THE EXTENDED PREDICATE CALCULUS

§ 1. The Predicate Calculus of Second Order

§ 2. Introduction of Predicates of Second Level; Logical Treatment of the Number Concept

§ 3. Representation of the Fundamental Concepts of Set Theory in the Extended Calculus

§ 4. The Logical Paradoxes

§ 5. The Predicate Calculus of Order ω

§ 6. Applications of the Calculus of Order ω

EDITOR'S NOTES

BIBLIOGRAPHY

INDEX

Back Cover

This book is unmistakably a mathematician's book, but it goes far beyond the limits of mathematics and makes available to everyone interested in logic one of the most permanent results of the study of the foundations of mathematics, the satisfactory symbolic treatment of the basic relations that play a part in deductive reasoning.
Cambridge University Press