The Algebra of Logic : Schröder

The monument to the work initiated by Boole, the algebraization of logic, is the three volumes Algebra der Logik by Schröder (1841–1902), which appeared in the years 1890–1910, filling over 2,000 pages. Although the spirit of the subject came from the work of Boole and De Morgan, Schröder's volumes are really a tribute to the work of C.S. Peirce, along with Schröder's contributions. In addition to the substantial job of organizing the literature, the lasting contributions of Schröder's volumes are 1) his emphasis on the Elimination Problem and 2) his fine presentation of the Calculus of Binary Relations. Volumes I and II are devoted to the Calculus of Classes, with the standard operations of union, intersection and complement, adhering to Boole's arithmetic notation for union (+) and intersection (·). Schröder was very much influenced by Peirce's work, and followed him in making the relation of subclass (⊆) the primitive notion, whose properties are given axiomatically (what we now call the axioms for a bounded lattice, presented as a partially ordered set), then defining the other operations and equality from it. One of the historically interesting items in Vol. I is Schröder's discovery that the distributive law does not follow from the assumptions Peirce put on ⊆. Schröder's proof is via a model, and indeed a rather complicated one (based on 990 quasigroup equations). Subsequently Dedekind published his first paper on dualgroups (= lattices) in 1897, giving a much shorter proof using a five element example (to show that a lattice need not be distributive). The main goal of Schröder's work is stated most clearly in Vol. III, p. 241, where he says that getting a handle on the consequences of any premisses, or at least the fastest methods for obtaining these consequences, seems to me to be the noblest, if not the ultimate goal of mathematics and logic. Schröder is very fond of examples and is only too aware that one can get into computational difficulties with the Calculus of Classes. The examples worked out at the end of Vol. I show how demanding the methods of Jevons and Venn become as the number of variables increases. Such difficulties 1