THE LOGIC OF LEIBNIZ

IN ACCORDANCE WITH UNPUBLISHED DOCUMENTS

 

Louis Couturat

 
 
 

"When God calculates...
He creates the world."
Phil., VII, 191.


 



 
 
 
 

TRANSLATED BY

Donald Rutherford

R. Timothy Monroe

 
 
 
 

Translation copyright (c) 1997-2002 by Donald Rutherford and R. Timothy Monroe, all rights reserved

 



 

TABLE OF CONTENTS

TRANSLATORS' INTRODUCTION

ABBREVIATIONS

PREFACE

FOREWORD

CHAPTER I SYLLOGISTIC

1.         Leibniz's View of Classical Logic
2.         Hospinianus's Theory of the Syllogism
3.         The 12 Simple Moods
4.         The 24 Figured Moods
5.         Reduction of Moods to the First Figure
6.         Subalternation and Conversion
7.         Reduction by Regression
8.         The Intensional Point of View
9.         All Moods Deduced from the Five Universal Moods
10.       Remarks on the Table of 24 Moods
11.       The Nine Imperfect Moods
12.       Principles of the Universal Moods
13.       Discussion of These Principles
14.       Extension and Intension
15.       Implicit Quantification of the Predicate
16.       Schemata of Propositions in Extension
17.       Schemata of Syllogisms in Extension
18.       Schemata of Propositions in Intension
19.       Schemata of Syllogisms in Intension
 

CHAPTER II THE COMBINATORY

1.         First Idea of the Combinatory
2.         The Alphabet of Human Thoughts
3.         On the Art of Combinations
4.         Precedents: Ramon Lull, etc.
5.         Criticism of Lull; Other Precursors
6.         Analysis and Classification of Concepts
7.         Finding All the Predicates of a Given Subject
8.         Finding All the Subjects of a Given Predicate
9.         Finding the Particular Predicates or Subjects of a Given Term
10.       Finding the Negative Subjects or Predicates of a Given Term
11.       Finding All the Middle Terms Capable of Proving a Given Conclusion
12.       Leibniz's Later Opinion of On the Art of Combinations; New Projects
 

CHAPTER III THE UNIVERSAL LANGUAGE

1.         Previous Plans
2.         Plan of On the Art of Combinations
3.         Contemporary Plans
4.         Descartes's Ideas on the Subject
5.         Plans of Wilkins and Dalgarno
6.         Leibniz's Opinion of These Plans
7.         Principle of the Characteristic
8.         Plan for a Rational Grammar
9.         Latin as an Intermediary
10.       The Simplification of Grammar
11.       Inflections and Particles
12.       Reduction of the Parts of Speech
13.       Analysis of Particles
14.       Analysis of Inflections
15.       Grammatical Analysis of Relations
16.       Analysis of Concepts by Definition
17.       The Philosophical Language is Not a Calculus
18.       The Characteristic and the Encyclopedia
 

CHAPTER IV THE UNIVERSAL CHARACTERISTIC

1.         The Characteristic Must be Real
2.         Algebra and Infinitesimal Analysis
3.         Utility of a Good Characteristic
4.         Conditions Required of the Characters
5.         Analogy with the Cartesian Method
6.         Superiority over the Cartesian Method
7.         Critique of Descartes's Method of Doubt
8.         Logic Settles Disagreements
9.         Utility of the Logical Calculus
10.       The Calculus Replaces Reasoning
11.       Refutation of Nominalism
12.       Application to Mathematics
13.       Importance of Natural Signs
14.       Various Plans for the Characteristic: Numbers
15.       Geometrical Figures
16.       Mechanisms
17.       The Characteristic Was Never Realized
 

CHAPTER V THE ENCYCLOPEDIA

1.         First Works: Juridical Logic
2.         Philosophy of Law: Recasting Legal Codes
3.         Application of Logic to Moral Questions
4.         Plan for the Literary Quarterly (1668-1669)
5.         First Plans for Encyclopedic Compilations
6.         Plans of 1676: the Philadelphic Society
7.         Plans of 1678-1679
8.         On Secret Things
9.         Plus Ultra
10.       Preliminaries of Plus Ultra
11.       Foundations and Illustrations of the General Science
12.       Measures Taken for Oldenburg and Galloys
13.       New Measures for the Royal Society
14.       Reports Addressed to Louis XIV
15.       Satire of the Republic of Letters
16.       Plan of the Demonstrative Encyclopedia
17.       Rational Sciences
18.       Experimental Sciences
19.       Arts and Crafts
20.       History
21.       The Elements of Philosophy
22.       The Story of Leibniz's "Vow"
23.       Collections of Definitions
24.       Tables of Definitions (1702-1704)
25.       Plan for an Encyclopedia in the New Essays
26.       Plan Presented to Peter the Great
 

CHAPTER VI THE GENERAL SCIENCE

1,         The Art of Judgment and the Art of Invention
2.         Analysis and Synthesis
3.         Development of the Cartesian Method
4.         Analysis of Truths
5.         Demonstration Rests on Definitions
6.         And on Axioms of Identity
7.         Nominal and Real Definitions
8.         Example of the Geometers
9.         Finding the Most Complete Definition
10.       Criterion of a True Idea
11.       Critique of the Ontological Argument
12.       Theory of Clear, Distinct, and Adequate Ideas
13.       Analysis of Ideas
14.       Critique of the Cartesian Method
15.       Demonstration of Axioms
16.       Principle of the Substitution of Equivalents
17.       Every Truth Is Analytic
18.       Truths of Reason and Truths of Fact
19.       Principle of Reason and Principle of Contradiction
20.       Relation of the Two Principles and of the Two Orders of Truths
21.       Necessary and Contingent Truths
22.       Contingency of the Laws of Nature
23.       Theory of Possibles and Compossibles
24.       Principles of Symmetry and Indiscernibles
25.       Principle of the Simplicity of the Laws of Nature
26.       Principle of Continuity or General Order
27.       Mathematical Principles and Metaphysical Principles
28.       The Logic of Probabilities
29.       Mathematical Theory of Games
30.       Principles of the Calculus of Probabilities
31.       Role of the Logic of Probabilities
32.       Degrees of Determination of Problems
33.       Examples from Algebra and Geometry
34.       Analogy of Cryptography
35.       Role of Experience in the Sciences
36.       Theory of First Empirical Truths
37.       Theory of Empirical Induction
38.       Role of Deduction in the Experimental Sciences
39.       Theory of Pappus's Analysis
40.       Theory of Hypotheses
41.       The Experimental Method is Deduction
42.       The Characteristic and the Art of Invention
43.       The Art of Invention and the Calculus of Probabilities
44.       The Art of Invention in the Technical and Practical Sciences
45.       Logic, Mathematics, and Metaphysics
 

CHAPTER VII UNIVERSAL MATHEMATICS

1.         Formal Analogy Between Logic and Mathematics
2.         Algebra Subordinated to the Combinatory
3.         Conception of the Combinatory
4.         Central Division of Mathematics
5.         Algebra Is Not the Universal Method
6.         The Combinatory Is the Art of Invention
7.         The New On the Art of Combinations
8.         Relations of Size
9.         Relations of Identity and Inclusion
10.       Relations of Determination
11.       Congruence, Equality, Similarity
12.       Axioms All These Relations Share
13.       Axioms Only These Relations Share
14.       Mathematics Reduced to Logic
15.       Logic Reduced to Mathematics
16.       Idea of Universal Algebra
 

CHAPTER VIII THE LOGICAL CALCULUS

1.         Logical Multiplication Is Commutative
2.         The System of Characteristic Numbers (1679)
3.         Translation of the Propositions A, E, I, O
4.         Translation of Negation
5.         New Translation of the Propositions A, E, I, O
6.         Conversions and Subalternation
7.         Faults of the System
8.         System Founded on Logical Inclusion
9.         An Illustration of the Universal Calculus: Principles
10.       Rules of Composition and Decomposition
11.       Additions to an Illustration of the Universal Calculus
12.       Attempt at a Calculus of Alternatives
13.       General Investigations of 1686: Principles
14.       Theory of the Syllogism
15.       Other Attempts to Translate the Propositions A, E, I, O
16.       Analogy Between Categorical and Hypothetical Propositions
17.       Fragments of August 1690
18.       Logical Difficulties
19.       Addition from the Extensional Point of View
20.       A Not Inelegant Illustration: Axioms and Definitions
21.       Demonstration of the Principle of the Syllogism
22.       Rules for the Composition of Inclusions
23.       Meaning of Inclusion: Inclusion of Intensions
24.       Meaning of Addition: Addition of Intensions
25.       Theory of Subtraction
26.       Properties of Subtraction
27.       Theory of Compensations
28.       Theorems Relating to Subtraction
29.       Summary and Conclusion
 

CHAPTER IX THE GEOMETRICAL CALCULUS

1.         The Essay of 1679
2.         Correspondence with Huygens
3.         Correspondence with L'Hospital
4.         Works Relating to the Geometrical Calculus
5.         Defects of Algebra and Analytic Geometry
6.         Defects of Synthetic Geometry
7.         Advantages of a Properly Geometrical Analysis
8.         Analysis Situs
9.         Definitions of Situation, Point, and Space
10.       Description of Figures by Points and Lines
11.       Theory of Similarity
12.       Definitions of Straight Line and Plane Using the Idea of Similarity
13.       Other Definitions, Using the Ideas of Section and Motion
14.       Definitions Based on the Idea of Congruence
15.       Definitions of Sphere and Circle
16.       Definition of Straight Line
17.       Critique of Euclid's Definition of Straight Line
18.       Definition of Plane, of Intersections
19.       Another Definition of Straight Line
20.       Critical Remarks
 

CONCLUSION 

APPENDIX I OUTLINE OF CLASSICAL LOGIC

1.         Universal Affirmative Judgment
2.         Four Types of Propositions
3.         Rules of Opposition
4.         Rules of Conversion
5.         Syllogistic Rules Relating to Terms
6.         Syllogistic Rules Relating to Propositions
7.         Rules of the Syllogistic Figures
8.         Reduction of Moods
9.         Regression or Reductio ad Absurdum
10.       Justification of the Fourth Figure

APPENDIX II LEIBNIZ AND HOBBES: THEIR LOGIC, THEIR NOMINALISM

1.         Citation of Hobbes in On the Art of Combinations
2.         Reasoning Is Calculation
3.         Hobbes's Conception of Logical Addition
4.         Nominalism and Syllogism
5.         Hobbes's View of the Mathematical Method
6.         Relations Between Leibniz and Hobbes
7.         Leibniz Opposes Hobbes's Materialism
8.         Leibniz: A Cartesian or an Aristotelian?
9.         Leibniz and Hobbes's Physics
10.       Leibniz and Hobbes's Nominalism
11.       Leibniz's Critique of Nominalism
12.       Leibniz's Apparent Nominalism
13.       Leibniz's Real Nominalism

APPENDIX III SOME OF LEIBNIZ'S MATHEMATICAL DISCOVERIES RELATING TO THE COMBINATORY AND CHARACTERISTIC

1.         Binary Arithmetic
2.         Metaphysical Symbol: Interpretation of Fohi's Characters
3.         Periodicity of Columns in the Number Series
4.         Alleged Periodicity of Prime Numbers
5.         Algebraic Tables or Canons
6.         The Difference Between Arithmetic and Algebra
7.         Notation for Coefficients Using Imaginary Numbers
8.         Elimination in First-Degree Equations
9.         Rule of Determinants. Casting Out Nines
10.       Resolution of First-Degree Equations (Cramer's Rule)
11.       Elimination in Higher-Order Equations
12.       Multiplication of Polynomials; Symbolic Homogeneity
13.       Division of Polynomials
14.       Method of Undetermined Coefficients (De Moivre's Theorem)
15.       Notation of Algebraic Forms
16.       Tables for the Calculus of Forms and the Resolution of Equations
17.       The Binomial Formula: Powers of a Polynomial
18.       Analogy Between the Binomial and Differentials
19.       Theorems of Divisibility Derived From the Combinatory
20.       Demonstration of Fermat's Theorem

APPENDIX IV LEIBNIZ AS FOUNDER OF ACADEMIES

1.         Literary Quarterly (1668)
2.         Trusteeship of Books
3.         First Plan for an Encyclopedic Society
4.         Plan for a German Society
5.         Plan for a Philadelphic Society
6.         Plans of 1676-1679
7.         Plans Submitted to the Dukes of Hanover
8.         Imperial Historical College (1688)
9.         Idea of a Religious Order (1693)
10.       Approaches to the Jesuits
11.       Plan for a Teutophilic Society
12.       Founding of the Academy of Sciences in Berlin (1700)
13.       Scientific Functions of the Academy
14.       Financial Resources of the Academy
15.       Project for Sericulture
16.       Plan for an Academy of Sciences in Dresden (1704)
17.       Plan for an Imperial Academy in Vienna (1712)
18.       Failure: Opposition of the Jesuits
19.       Plans Submitted to Tsar Peter the Great
20.       Leibniz's Patriotism and Cosmopolitanism

APPENDIX V GRASSMANN'S GEOMETRICAL CALCULUS

1.         Principles of Ausdehnungslehre
2.         Fundamental Definitions
3.         Equality and Congruence
4.         Regressive Multiplication
5.         Geometrical and Mechanical Applications
6.         Comparison with Leibniz's Geometrical Calculus
 

NOTES

I.          Thomas Barton
II.         The Ars Magna of Ramon Lull and Athanasius Kircher
III.       Dalgarno's Ars Signorum
IV.       Wilkins's Philosophical Language
V.        On Conditions
VI.       Extract from On the Art of Combinations
VII.      New Method of Learning and Teaching Jurisprudence (1667)
VIII.     Specimen of Demonstrations in Politics (1669)
IX.       Definition of Universal Justice
X.        Definition of Love
XI.       The Usefulness of Albert von Hotten's Cylindrical Grammar
XII.      The Reason for Completing and Emending Alsted's Encyclopedia
XIII.     Judgment on the Writings of Comenius
XIV.    Leibniz as a Librarian
XV.      Leibniz's "Distractions"
XVI.    The Principle of Least Action
XVII.   Mathematical Theory of Games
XVIII. Letter to Eler (10 May 1716)
XIX.    Letter to Lange (5 June 1716)
XX.      Letter to Kestner (1 July 1716)

INDEX


G.W. Leibniz: Texts and Translations
Donald Rutherford, University of California, San Diego / drutherford@ucsd.edu
Last update: March 19, 2002