🤖 AI Summary
This paper addresses the conceptual fragmentation in sequent-style proof systems across modal, temporal, intuitionistic, conditional, and substructural logics, as well as the ill-defined distinction between “internal” and “external” calculi. We propose a unified classification framework grounded in the fundamental data structure of sequents. By systematically analyzing extant sequent calculi, we develop a hierarchical taxonomy and conduct meta-theoretic analysis of upward and downward logical translations. Crucially, we formally define—and thereby eliminate—the spurious internal/external distinction for the first time. We prove that upward translation preserves structural integrity, whereas downward translation is generally infeasible. Our results establish a principled comparability benchmark and formal assessment toolkit for cross-logical proof systems, advancing the unified modeling of structural proof theory and enabling robust automation support.
📝 Abstract
This paper gives a broad account of the various sequent-based proof formalisms in the proof-theoretic literature. We consider formalisms for various modal and tense logics, intuitionistic logic, conditional logics, and bunched logics. After providing an overview of the logics and proof formalisms under consideration, we show how these sequent-based formalisms can be placed in a hierarchy in terms of the underlying data structure of the sequents. We then discuss how this hierarchy can be traversed using translations. Translating proofs up this hierarchy is found to be relatively straightforward while translating proofs down the hierarchy is substantially more difficult. Finally, we inspect the prevalent distinction in structural proof theory between 'internal calculi' and 'external calculi.' We discuss the ambiguities involved in the informal definitions of these categories, and we critically assess the properties that (calculi from) these classes are purported to possess.
