How does logic as a calculus differ from logic as a universal language?

The distinction between logic as a calculus and logic as a universal language, a framework popularized by Jean van Heijenoort, highlights two fundamental approaches to logical systems.1

Logic as a Universal Language

In this conception, a logical system serves as a lingua universalis—a comprehensive medium of communication that encompasses all thought. Because this language is intended to be universal, its semantics cannot be discussed from within the system itself without resulting in triviality or nonsense. Consequently, this view leads to the thesis of the inexpressibility of logical semantics, limiting discourse to the system's formal or syntactic features. Thinkers such as Gottlob Frege, Bertrand Russell, and W.V.O. Quine are associated with this tradition.1

Logic as a Calculus

Conversely, the conception of logic as a calculus treats it primarily as a calculus ratiocinator, or a tool for drawing inferences. Unlike the universal language view, a calculus is an instrument that can be analyzed, theorized about, and modified as needed. This tradition, rooted in the algebraic work of George Boole, typically deals with all possible universes of discourse rather than assuming a single, all-encompassing framework.1