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

by Brady, Geraldine

• ISBN13: 9780444503343
• ISBN10: 044450334X
• Format: Hardcover
• Copyright: 2000-07-01
• Publisher: Elsevier Science Serials

Summary

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.

Table of Contents

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