The idea of translation between logic systems was first proposed exactly one century ago by Kolmogorov. In 1925, he showed that it is possible to translate, in a homogeneous way, the language of propositional classical logic into the language of propositional intuitionistic logic, while preserving theoremhood. Between 1929 and 1933, and independently of Kolmogorov, Glivenko, Lewis and Langford, Gödel, and Gentzen also introduced interpretations of classical logic inside intuitionistic logic. Gödel also introduced in 1933 the well-known translation of intuitionistic logic into the modal system S4.

The original translations (usually called 'interpretations', and mainly developed to show the relative consistency of classical logic with respect to intuitionistic logic) were afterwards generalized and studied from an abstract perspective by diverse authors. The use of interesting translations between logics is very useful for studying and analyzing the inter-relations between diverse logic systems. For instance, the notion of equivalence between logical systems based on translations, introduced in 2001 by Blok and Pigozzi, is a useful tool in their general theory of the algebraization of logics.

Translations between logics constitute an important theoretical component in the theory of combination of logics. The problem of combining logics has attracted significant attention from researchers in logic, computer science and philosophy. Besides leading to interesting and useful applications in situations in which it is necessary to work with different logics at the same time, combinations of logics are also of great interest from the theoretical and philosophical perspective. Several techniques for combining logics have been introduced in the literature. In modal logic, the most important methods are Products (Segerberg, 1973; Šehtman, 1978) and Fusion (Thomason, 1984). Another key method is Fibring (Gabbay, 1999), which is a general technique for combining logics, not limited to modal logics. This method was later generalized by A. Sernadas and his collaborators, resulting in a wast repertoire of formal frameworks for combining logics through fibring, based on concepts and tools from category theory.

In this talk, we will discuss the relevance of different notions of translation in the context of the combination of logics. This is particularly important when combining different fragments of a given logic, and it is closely related to the collapsing problem when combining classical and intuitionistic logic. Several examples will be discussed.