An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis.

ALF [AGNvS94] is a structure editor for one of Martin-Löf's monomorphic type theories. It has been designed as an interactive environment for theorem proving and program development. Formulation of Natural Deduction via the Curry-Howard Isomorphism is a simple matter, and meta-theoretic proofs about Natural Deduction (e.g. normalisation proofs) are also straightforward. Adapting ALF's nomenclature to handle Sequent Calculus systems with proof terms is, however, a non-trivial exercise. If not actually impossible, it requires far more working around ALF's intended methods than working within them.

Abstract. After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: conservative translations, transfers and contextual translations. Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another.