 Frege’s logical contributions grew out of a different soil, and were made (as far as can be determined) entirely independently of the Anglo-American algebraic tradition of Boole, De Morgan, and Peirce. Instead they have their root in the work on the foundations of real analysis by such German mathematicians as Dirichlet, Riemann, Weierstrass, and Heine. From this tradition Frege took, first, the idea of providing a rigorous foundation for mathematics (a project that, in his hands, became the project of showing that arithmetic can be grounded in the laws of logic); and, secondly, the central mathematical concepts of function and variable, which he employed in place of the Aristotelian concepts of predicate and subject. This latter step led him naturally to a logic of relations (since the functions considered in mathematics were multivariate); and his analysis of mathematical inference also led him to introduce a notation for quantificational logic. (Mathematicians such as Weierstrass, in his analysis of the limit concept, was already sensitive to the “nesting” of quantifiers, and the importance of their ordering: to the difference, for example, between saying “for every \(\varepsilon\) there exists a \(\delta\)”, and “there exists a \(\delta\) such that for every \(\varepsilon\)”. What was required now, and what Frege supplied, was a formal language to express and make explicit the quantificational inferences already present in the work of the German analysts.) So, at a single stroke, in the Begriffsschrift of 1879, Frege took the two major steps beyond traditional logic—relations and quantifiers—that the algebraic tradition had taken separately and decades apart. 