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This paper gives a broad account of the various sequent-based proofformalisms in the proof-theoretic literature. We consider formalisms forvarious modal and tense logics, intuitionistic logic, conditional logics, andbunched logics. After providing an overview of the logics and proof formalismsunder consideration, we show how these sequent-based formalisms can be placedin a hierarchy in terms of the underlying data structure of the sequents. Wethen discuss how this hierarchy can be traversed using translations.Translating proofs up this hierarchy is found to be relatively easy whiletranslating proofs down the hierarchy is substantially more difficult. Finally,we inspect the prevalent distinction in structural proof theory between'internal calculi' and 'external calculi'. It is observed that these classesresist a rigorous separation, and we critically assess the properties that(calculi from) these classes are purported to possess.

Internal and External Calculi: Ordering the Jungle without Being Lost in Translations