How do the views of logic as calculus and logic as language lead to different approaches to semantics?

The distinction between logic as language and logic as calculus fundamentally shapes how logicians approach semantics.1

Logic as Language

When logic is viewed as a lingua universalis—a universal medium of communication—it is seen as the all-encompassing framework for expressing truths about the world. Under this conception, the semantics of the system cannot be discussed from within the logic itself, as doing so would lead to triviality or nonsense. Consequently, this view often leads to a thesis of the inexpressibility of logical semantics, restricting discourse to the purely formal or syntactic features of the system. Proponents of this tradition, such as Gottlob Frege and Bertrand Russell, treated logical truths as the most general truths about the world.1

Logic as Calculus

In contrast, the conception of logic as a calculus ratiocinator treats it primarily as a tool for drawing inferences. Because this logic is viewed as a formal instrument rather than a universal language, it can be theorized about, modified, and analyzed from an external perspective. This approach allows for the development of model theory, where different universes of discourse can be explored and interpreted.12

While syntax and semantics are distinct, the calculus approach allows semantic questions—such as consistency or completeness—to be transformed into sharper, manageable syntactic problems.3