From Peirce to Skolem : A Neglected Chapter

This book is an account of the important influence on the development of mathematical logic of Charles S. Peirce and his student O.H. Mitchell, through the work of Ernst Schröder, Leopold Löwenheim, and Thoralf Skolem. As far as we know, this book is the first work delineating this line of influence on modern mathematical logic.

Introduction

1

(8)

The Early Work of Charles S. Peirce

9

(14)

Overview of the Mathematical Systems of Charles S. Peirce

9

(2)

Peirce's Influence on the Development of Logic

11

(3)

Peirce's Early Approaches to Logic

14

(9)

Peirce's Calculus of Relatives: 1870

23

(28)

Peirce's Algebra of Relations

24

(22)

Inclusion and Equality

27

(2)

Addition

29

(1)

Multiplication

30

(8)

Peirce's First Quantifiers

38

(1)

Involution

39

(3)

Involution and Mixed-Quantifier Forms

42

(2)

Elementary Relatives

44

(2)

Quantification in the Calculus of Relatives in 1870

46

(2)

Summary

48

(3)

Peirce on the Algebra of Logic: 1880

51

(24)

Overview of Peirce's ``On the algebra of logic''

51

(3)

Discussion

54

(19)

The Origins of Logic

54

(2)

Syllogism and Illation

56

(4)

Forms of Propositions

60

(4)

The Algebra of the Copula

64

(6)

The Logic of Nonrelative Terms

70

(3)

Conclusion

73

(2)

Mitchell on a New Algebra of Logic: 1883

75

(20)

Mitchell's Rule of Inference

75

(3)

Single-Variable Monadic Logic

78

(8)

Single-Variable Monadic Propositions

78

(2)

Disjunctive Normal Form

80

(2)

Rules of Inference for Single-Variable Logic

82

(4)

Two-Variable Monadic Logic

86

(3)

Mitchell's Dimension Theory

86

(2)

Contrast to Peirce

88

(1)

Three-Variable Monadic Logic

89

(1)

Peirce on Mitchell

90

(5)

Peirce on the Algebra of Relatives: 1883

95

(18)

Background in Linear Associative Algebras

95

(3)

The Algebra of Relatives

98

(6)

Types of Relatives

98

(2)

Operations on Relatives

100

(4)

Syllogistic in the Relative Calculus

104

(2)

Prenex Predicate Calculus

106

(3)

Summary of Peirce's Accomplishments in 1883

109

(2)

Syntax and Semantics

109

(1)

Quantifiers

110

(1)

Peirce's Appraisal of His Algebra of Binary Relatives

111

(2)

Peirce's Logic of Quantifiers: 1885

113

(30)

On the Derivation of Logic from Algebra

113

(3)

Nonrelative Logic

116

(11)

Embedding Boolean Algebra in Ordinary Algebra

116

(5)

Five Peirce Icons

121

(4)

Truth-functional Interpretations of Propositions

125

(2)

First-Order Logic

127

(5)

Infinite Sums and Products

127

(1)

Mitchell

128

(1)

Formulas and Rules

129

(3)

Second-Order Logic

132

(11)

Schroder's Calculus of Relatives

143

(26)

Die Algebra der Logik: Volume 1

144

(3)

Die Algebra der Logik: Volume 2

147

(2)

Die Algebra der Logik: Volume 3

149

(16)

Peirce's Attack on the General Solutions of Schroder

153

(2)

Lectures VI-X and Dedekind Chain Theory

155

(5)

Lectures XI-XII and Higher Order Logic

160

(5)

Norbert Wiener's Ph.D. Thesis

165

(4)

Lowenheim's Contribution

169

(28)

Overview of Lowenheim's 1915 Paper

171

(1)

Lowenheim's Theorem

172

(19)

Conclusions

191

(4)

Impact of Lowenheim's Paper

195

(2)

Skolem's Recasting

197

(248)

Appendices

1. Schroder's Lecture I

207

(16)

2. Schroder's Lecture II

223

(28)

3. Schroder's Lecture III

251

(6)

4. Schroder's Lecture V

257

(38)

5. Schroder's Lecture IX

295

(44)

6. Schroder's Lecture XI

339

(40)

7. Schroder's Lecture XII

379

(50)

8. Norbert Wiener's Thesis

429

(16)

Bibliography

445

(16)

Index

461

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From Peirce to Skolem : A Neglected Chapter in the History of Logic

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