Gottlob Frege’s Grundgesetze der Arithmetik (Basic Laws of Arithmetic) was originally published in two volumes: the first in 1893, the second in 1903. It was to be the pinnacle of Frege’s life’s work. The aim was to demonstrate that arithmetic and analysis are reducible to logic—a position later called “Logicism”. Frege’s project began with the publication of Begriffsschrift in 1879, which contains the first version of his logical system, also named ‘Begriffsschrift’ (concept-script), the mature formulation of which Frege would present in Basic Laws. Begriffsschrift was groundbreaking in many ways: it contained the first occurrence of the quantifier in formal logic, including a treatment of second-order quantification; the first formal treatment of multiple and embedded generality; and it also offered the first formulation of a logical system containing relations. Frege’s work is widely acknowledged to constitute the greatest advance in logic since Aristotle. As W.V. Quine put it: “Logic is an old subject, and since 1879 it has been a great one”.
Begriffsschrift was followed by Die Grundlagen der Arithmetik (Foundations of Arithmetic) in 1884. Having previously completed a manuscript of a more formal treatment of Logicism around 1882—a lost ancestor of Grundgesetze—Frege developed a philosophical foundation for his position in Grundlagen. At the end of Grundlagen, Frege was, however, left with the monumental task properly to establish Logicism: he needed to identify a small number of basic laws of logic; offer a small number of indisputably sound rules of inference; and, finally, provide gapless proofs in his formal system of the basic laws of arithmetic, using only the identified basic laws and rules of logic together with suitable explicit definitions. This was the task of his magnum opus: Grundgesetze der Arithmetik.
In 2003 we started the Grundgesetze translation project. The aim was to produce the first complete translation of Frege’s magnum opus, while retaining the original pagination and the original formalism. Our translation is published by Oxford University Press and available here.
More information on Gottlob Frege can be found on Stanford Encyclopedia of Philosophy