How did George Boole’s approach to logic differ from Gottlob Frege’s approach?

The approaches of George Boole and Gottlob Frege to logic are fundamentally distinguished by their differing conceptions of the role of logic, often categorized as "logic as calculus" versus "logic as language".1

George Boole: Logic as Calculus
Boole, a central figure in the algebraic tradition, viewed logic primarily as a calculus ratiocinator—a tool designed for drawing inferences. His approach, known as Boolean algebra, sought to ally logic with mathematics rather than philosophy. By separating symbols of quantity from those of operation, Boole created a system that could process propositions to derive conclusions contained within the premises. This algebraic method treats logic as a tool that can be theorized about, modified, and applied to various universes of discourse.12

Gottlob Frege: Logic as Language
In contrast, Frege viewed logic as a lingua universalis, or a universal medium of communication. His Begriffsschrift was intended to provide a rigorous, conceptual notation for all of mathematics and science. Because he saw his system as a universal language, he maintained that its semantics could not be discussed from within the logic itself, leading to a focus on the intensional logic of concepts rather than the extensional logic of classes favored by the British tradition. Frege was notably hostile toward Boolean methods, criticizing them for artificially mimicking notation suited for numerical analysis rather than developing a pure system for logical analysis.134