# From Peirce to Skolem

**A Neglected Chapter in the History of Logic**

**By**

- Geraldine Brady, Department of Computer Science, University of Chicago, 1100 E 58th Street, Chicago, IL 60637 USA

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.

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### Book information

- Published: July 2000
- Imprint: NORTH-HOLLAND
- ISBN: 978-0-444-50334-3

### Reviews

The book is well written, and written for a ;arge audience. Many very detailed explanations of terminology, notation and proof techniques in the quotations of historicl texts are given.....

M. Guillaume, Mathematical Reviews, 2002

### Table of Contents

**Introduction****The Early Work of Charles S. Peirce**. Overview of the Mathematical Systems of Charles S. Peirce. Peirce's Influence on the Development of Logic. Peirce's Early Approaches to Logic. **Peirce's Calculus of Relatives: 1870**. Peirce's Algebra of Relations. Inclusion and Equality. Addition. Multiplication. Peirce's First Quantifiers. Involution. Involution and Mixed-quantifier Forms. Elementary Relatives. Quantification in the calculus of relatives in 1870. Summary. **Peirce on the Algebra of Logic: 1880**. Overview of Peirce's "On the algebra of logic". Discussion. The Origins of Logic. Syllogism and Illation. Forms of Propositions. The Algebra of the Copula. The Logic of Nonrelative Terms. Conclusion. **Mitchell on a New Algebra of Logic: 1883**. Mitchell's Rule of Inference. Single-Variable Monadic Logic. Single-Variable Monadic Propositions. Disjunctive Normal Form. Rules of Inference for Single-Variable Logic. Two-Variable Monadic Logic. Mitchell's Dimension Theory. Contrast to Peirce. Three-Variable Monadic Logic. Peirce on Mitchell. **Peirce on the algebra of relatives: 1883**. Background in Linear Associative Algebras. The Algebra of Relatives. Types of Relatives. Operations on Relatives. Syllogistic in the Relative Calculus. Prenex Predicate Calculus. Summary of Peirce's Accomplishments in 1883. Syntax and Semantics. Quantifiers. Peirce's Appraisal of His Algebra of Binary Relatives. **Peirce's Logic of Quantifiers: 1885**. On the Derivation of Logic from Algebra. Nonrelative Logic. Embedding Boolean algebra in Ordinary Algebra. Five Peirce Icons. Truth-functional Interpretations of Propositions. First-Order Logic. Infinite Sums and Products. Mitchell. Formulas and Rules. Second-Order Logic. **Schröder's Calculus of Relatives**. *Die algebra der Logik*: Volume 1. *Die Algebra der Logik*: Volume 2. *Die Algebra der Logik*: Volume 3. Peirce's Attack on the General solutions of Schröder. Lectures VI-X and Dedekind Chain Theory. Lectures XI-XII and Higher Order Logic. Norbert Wiener's Ph.D. Thesis. **Löwenheim's contribution**. Overview of Löwenheim's 1915 paper. Löwenheim's Theorem. Conclusions. Impact of Löwenheim's Theorem. Conclusions. Impact of Löwenheim's Paper. **Skolem's recasting**. **Appendices**. Schröder's Lecture I. Schröder's Lecture II. Schröder's Lecture III. Schröder's Lecture V. Schröder's Lecture IX. Schröder's Lecture XI. Schröder's Lecture XII. Norbert Wiener's Thesis. Bibliography. Index.