Charles Sanders Peirce, the son of the Harvard mathematics professor and discoverer of linear algebra Benjamin Peirce, was the first significant American figure in logic. Peirce had read the work of Aristotle, Whately, Kant, and Boole as well as medieval works and was influenced by his father’s sophisticated conceptions of algebra and mathematics. Peirce’s first published contribution to logic was his improvement in 1867 of Boole’s system. Although Peirce never published a book on logic (he did edit a collection of papers by himself and his students, the Studies in Logic of 1883), he was the author of an important article in 1870, whose abbreviated title was “On the Notation of Relatives,” and of a series of articles in the 1880s on logic and mathematics; these were all published in American mathematics journals.

It is relatively easy to describe Peirce’s main approach to logic, at least in his earlier work: it was a refinement of Boole’s algebra of logic and, especially, the development of techniques for handling relations within that algebra. In a phrase, Peirce sought a blend of Boole (on the algebra of logic) and De Morgan (on quantified relational inferences). Described in this way, however, it is easy to underestimate the originality and creativity (even idiosyncrasy) of Peirce. Although committed to the broadly “algebraic” tradition of Boole and his father, Peirce quickly moved away from the equational style of Boole and from efforts to mimic numerical algebra. In particular, he argued that a transitive and asymmetric logical relation of inclusion, for which he used the symbol “⤙,” was more useful than equations; the importance of such a basic, transitive relation was first stressed by De Morgan, and much of Peirce’s work can be seen as an exploration of the formal, abstract properties of this distinctively logical relation. He used it to express class inclusion, the “if . . . then” connective of propositional logic, and even the relation between the premises and conclusion of an argument (“illation”). Furthermore, Peirce slowly abandoned the strictly substitutional character of algebraic terms and increasingly used notation that resembled modern quantifiers. Quantifiers were briefly introduced in 1870 and were used extensively in the papers of the 1880s. They were borrowed by Schröder for his extremely influential treatise on the algebra of logic and were later adopted by Peano from Schröder; thus in all probability they are the source of the notation for quantifiers now widely used. In his earlier works, Peirce might have written “A ⤙ B” to express the universal statement “All A’s are B’s” (see illustration); however, he often wrote this as “Π_î A_i ⤙ Π_î” B_î (the class of all the i’s that are A is included in the class of all the i’s that are B) or, still later and interpreted in the modern way, as “For all i’s, if i is A, then i is B.” Peirce and Schröder were never clear about whether they thought these quantifiers and variables were necessary for the expression of certain statements (as opposed to using strictly algebraic formulas), and Frege did not address this vital issue either; the Boolean algebra without quantifiers, even with extensions for relations that Peirce introduced, was demonstrated to be inadequate only in the mid-20th century by Alfred Tarski and others.

Peirce developed this symbolism extensively for relations. His earlier work was based on versions of multiplication and addition for relations—called relative multiplication and addition—so that Boolean laws still held. Both Peirce’s conception of the purposes of logic and the details of his symbolism and logical rules were enormously complicated by highly developed and unusual philosophical views, by elaborate theories of mind and thought, and by his theory of mental and visual signs (semiotics). He argued that all reasoning was “diagrammatic” but that some diagrams were better (more iconic) than others if they more accurately represented the structure of our thoughts. His earlier works seems to be more in the tradition of developing a calculus of reason that would make reasoning quicker and better and permit one to validate others’ reasoning more accurately and efficiently. His later views, however, seem to be more in the direction of developing a “characteristic” language. In the late 1880s and 1890s Peirce developed a far more extensively iconic system of logical representation, his existential graphs. This work was, however, not published in his lifetime and was little recognized until the 1960s.

Peirce did not play a major role in the important debates at the end of the 19th century on the relationship of logic and mathematics and on set theory. In fact, in responding to an obviously quick reading of Russell’s restatements of Frege’s position that mathematics could be derived from logic, Peirce countered that logic was properly seen as a branch of mathematics, not vice versa. He had no influential students: the brilliant O.H. Mitchell died at an early age, and Christine Ladd Franklin never adapted to the newer symbolic tradition of Peano, Frege, and Russell. On the other hand, Peano and especially Schröder had read Peirce’s work carefully and adopted much of his notation and his doctrine of the importance of relations (although they were less fervent than De Morgan and Peirce). Peano and Schröder, using much of Peirce’s notation, had an enormous influence into the 20th century.

In Germany, the older formal and symbolic logical tradition was barely kept alive by figures such as Salomon Maimon, Semler, August Detlev Twesten, and Moritz Wilhelm Drobisch. The German mathematician and philologist Hermann Günther Grassmann published in 1844 his Ausdehnungslehre (“The Theory of Extension”), in which he used a novel and difficult notation to explore quantities (“extensions”) of all sorts—logical extension and intension, numerical, spatial, temporal, and so on. Grassmann’s notion of extension is very similar to the use of the broad term “quantity” (and the phrase “logic of quantity”) that is seen in the works of George Bentham and Sir William Hamilton from the same period in the United Kingdom; it is from this English-language tradition that the terms, still in use, of logical “quantification” and “quantifiers” derive. Grassmann’s work influenced Robert Grassmann’s Die Begriffslehre oder Logik (1872; “The Theory of Concepts or Logic”), Schröder, and Peano. The stage for a rebirth of German formal logic was further set by Friedrich Adolf Trendelenburg’s works, published in the 1860s and ’70s, on Aristotle’s and Leibniz’ logic and on the relationship of mathematics and philosophy. Alois Riehl’s much-read article “Die englische Logik der Gegenwart” (1876; “Contemporary English Logic”) introduced German speakers to the works of Boole, De Morgan, and Jevons.