In 1879 the young German mathematician Gottlob Frege—whose mathematical specialty, like Boole’s, had actually been calculus—published perhaps the finest single book on symbolic logic in the 19th century, Begriffsschrift (“Conceptual Notation”). The title was taken from Trendelenburg’s translation of Leibniz’ notion of a characteristic language. Frege’s small volume is a rigorous presentation of what would now be called the first-order predicate logic. It contains a careful use of quantifiers and predicates (although predicates are described as functions, suggestive of the technique of Lambert). It shows no trace of the influence of Boole and little trace of the older German tradition of symbolic logic. One might surmise that Frege was familiar with Trendelenburg’s discussion of Leibniz, had probably encountered works by Drobisch and Hermann Grassmann, and possibly had a passing familiarity with the works of Boole and Lambert, but was otherwise ignorant of the history of logic. He later characterized his system as inspired by Leibniz’ goal of a characteristic language but not of a calculus of reason. Frege’s notation was unique and problematically two-dimensional; this alone caused it to be little read (see illustration).

Frege was well aware of the importance of functions in mathematics, and these form the basis of his notation for predicates; he never showed an awareness of the work of De Morgan and Peirce on relations or of older medieval treatments. The work was reviewed (by Schröder, among others), but never very positively, and the reviews always chided him for his failure to acknowledge the Boolean and older German symbolic tradition; reviews written by philosophers chided him for various sins against reigning idealist dogmas. Frege stubbornly ignored the critiques of his notation and persisted in publishing all his later works using it, including his little-read magnum opus, Grundgesetze der Arithmetik (1893–1903; The Basic Laws of Arithmetic).

His first writings after the Begriffsschrift were bitter attacks on Boolean methods (showing no awareness of the improvements by Peirce, Jevons, Schröder, and others) and a defense of his own system. His main complaint against Boole was the artificiality of mimicking notation better suited for numerical analysis rather than developing a notation for logical analysis alone. This work was followed by the Die Grundlagen der Arithmetik (1884; The Foundations of Arithmetic) and then by a series of extremely important papers on precise mathematical and logical topics. After 1879 Frege carefully developed his position that all of mathematics could be derived from, or reduced to, basic “logical” laws—a position later to be known as logicism in the philosophy of mathematics. His view paralleled similar ideas about the reducibility of mathematics to set theory from roughly the same time—although Frege always stressed that his was an intensional logic of concepts, not of extensions and classes. His views are often marked by hostility to British extensional logic and to the general English-speaking tendencies toward nominalism and empiricism that he found in authors such as J.S. Mill. Frege’s work was much admired in the period 1900–10 by Bertrand Russell who promoted Frege’s logicist research program—first in the Introduction to Mathematical Logic (1903), and then with Alfred North Whitehead, in Principia Mathematica (1910–13)—but who used a Peirce-Schröder-Peano system of notation rather than Frege’s; Russell’s development of relations and functions was very similar to Schröder’s and Peirce’s. Nevertheless, Russell’s formulation of what is now called the “set-theoretic” paradoxes was taken by Frege himself, perhaps too readily, as a shattering blow to his goal of founding mathematics and science in an intensional, “conceptual” logic. Almost all progress in symbolic logic in the first half of the 20th century was accomplished using set theories and extensional logics and thus mainly relied upon work by Peirce, Schröder, Peano, and Georg Cantor. Frege’s care and rigour were, however, admired by many German logicians and mathematicians, including David Hilbert and Ludwig Wittgenstein. Although he did not formulate his theories in an axiomatic form, Frege’s derivations were so careful and painstaking that he is sometimes regarded as a founder of this axiomatic tradition in logic. Since the 1960s Frege’s works have been translated extensively into English and reprinted in German, and they have had an enormous impact on a new generation of mathematical and philosophical logicians.

Ernst Schröder

German symbolic logic (in a broad sense) was cultivated by two other major figures in the 19th century. The tradition of Hermann Grassmann was continued by the German mathematician and algebraist Ernst Schröder. His first work, Der Operations-kreis des Logikkalkuls (1877; “The Circle of Operations of the Logical Calculus”), was an equational algebraic logic influenced by Boole and Grassmann but presented in an especially clear, concise, and careful manner; it was, however, intensional in that letters stand for concepts, not classes or things. Although Jevons and Frege complained of what they saw as the “mysterious” relationship between numerical algebra and logic in Boole, Schröder announced with great clarity: “There is certainly a contrast of the objects of the two operations. They are totally different. In arithmetic, letters are numbers, but here, they are arbitrary concepts.” He also used the phrase “mathematical logic.” Schröder’s main work was his three-volume Vorlesungen über die Algebra der Logik (1890–1905; “Lectures on the Algebra of Logic”). This is an extensive and sometimes original presentation of all that was known about the algebra of logic circa 1890, together with derivations of thousands of theorems and an extensive bibliography of the history of logic. It is an extensional logic with a special sign for inclusion “” (paralleling Peirce’s “⤙”; see illustration), an inclusive notion of class union, and the usual Boolean operations and rules.

The first volume is devoted to the basic theory of an extensional theory of classes (which Schröder called Gebiete, logical “domains,” a term that is somewhat suggestive of Grassmann’s “extensions”). Schröder was especially interested in formal features of the resulting calculus, such as the property he called “dualism” (carried over from his 1877 work): any theorem remains valid if the addition and multiplication, as well as 0 and 1, are switched—for example, A Ā = 0, A + Ā = 1, and the pair of De Morgan laws. The second volume is a discussion of propositional logic, with propositions taken to refer to domains of times in the manner of Boole’s Laws of Thought but using the same calculus. Schröder, unlike Boole and Peirce, distinguished between the universes for the separate cases of the class and propositional logics, using respectively 1 and {dotted 1}. The third volume contains Schröder’s masterful but leisurely development of the logic of relations, borrowing heavily from Peirce’s work. In the first decades of the 20th century, Schröder’s volumes were the only major works in German on symbolic logic other than Frege’s, and they had an enormous influence on important figures writing in German, such as Thoralf Albert Skolem, Leopold Löwenheim, Julius König, Hilbert, and Tarski. (Frege’s influence was felt mainly through Russell and Whitehead’s Principia Mathematica, but this tradition had a rather minor impact on 20th-century German logic.) Although it was an extensional logic more in the English tradition, Schröder’s logic exhibited the German tendency of focusing exclusively upon deductive logic; it was a legacy of the English textbook tradition always to cover inductive logic in addition, and this trait survived in (and often cluttered) the works of Boole, De Morgan, Venn, and Peirce.