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From Peirce to Skolem. A neglected chapter in the history of logic. (English) Zbl 0958.01009

Studies in the History and Philosophy of Mathematics. 4. Amsterdam: North-Holland. xi, 468 p. (2000).
Standard historical treatments of the 19th century origins of modern logic very often neglect a line of development which goes from G. Boole’s algebraic treatment of logic, via Ch. S. Peirce’s theory of relatives and E. Schröder’s extension of it, up to L. Löwenheim and Th. Skolem.
The author of this nice book investigates just this development. She starts with an extended explanation of the work of Ch. S. Peirce from his “Description of a notation for the logic of relatives” [Mem. Am. Acad. Arts Sci. 9, 317-378 (1870)] to his “On the algebra of logic” [Am. J. Math. 7, 180-202 (1885; JFM 17.0044.02)]. This includes also a discussion of the contribution of Peirce’s student O. H. Mitchell, and it shows that Peirce essentially reached a preliminary version of first-order and second-order logic.
This discussion is completed by a description of the main content of E. Schröder’s 3-volume “Vorlesungen über die Algebra der Logik” (1890-1895; JFM 22.0073.02, JFM 23.0051.02, JFM 26.0074.01), as well as by a detailed discussion of L. Löwenheim’s famous “{Über Möglichkeiten im Relativkalkül}” [Math. Ann. 76, 447-470 (1915; JFM 45.0108.01)]. The considerations upon Schröder’s work are accompanied by a reference to the Ph.D. Thesis of N. Wiener. And finally the continuation of Löwenheim’s approach in the work of Th. Skolem is discussed.
The second half of the book is a very extended Appendix which offers to the reader parts of the work of E. Schröder in English translation: from the third volume of his “Vorlesungen …” Lecture IX (on Dedekind’s chain theory) is given completely, and the Lectures I, II, III, IV, and XII of that volume are translated partly.

MSC:

01A55 History of mathematics in the 19th century
01-02 Research exposition (monographs, survey articles) pertaining to history and biography
03-03 History of mathematical logic and foundations
01A60 History of mathematics in the 20th century
03G99 Algebraic logic
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