Philosophical Monologues: On Various Translations Between Classical, Intuitionistic, and Linear Logic

Thursday, July 23, 2026

On Various Translations Between Classical, Intuitionistic, and Linear Logic

Ferreira, G., Oliva, P. & Protin, C.L. On Various Translations Between Classical, Intuitionistic, and Linear Logic. Stud Logica (2026). https://doi.org/10.1007/s11225-026-10251-y

Several different proof translations exist between classical and intuitionistic logic (negative translations), and intuitionistic and linear logic (Girard translations). Our aims in this paper are: (1) to consider extensions of intuitionistic linear logic corresponding to each of these systems, and (2) using this common logical basis, to develop a uniform approach to devising and simplifying proof translations. Through this process of “simplification” we recover most of the well-known translations in the literature.

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Logic, therefore, is not a tool; it is an open-ended, evolving science having an ontology of its own. Mathematics is but one of its parts, and its full scope is yet to be discovered. Given the definition of validity, every necessary thought whose analysis contains only purely logical objects is valid. So every time we discover a purely logical analysis of a concept previously thought to belong to a discipline outside logic, the acknowledged scope of logic must be expanded accordingly. The purely logical analysis of number is just one case in point, and I submit that there are many more (...) If there is anything in the analyses I will offer, then the conception of logic that emerges is very far indeed from that of Aristotle, for whom logic is primarily a tool; instead, it is more like that of Plato, for whom logic is akin to reason itself. George Bealer