🤖 AI Summary
This study addresses the fundamental question in general proof theory of whether logical systems admit proof systems possessing desirable meta-properties—namely, cut elimination, analyticity, and termination. For classical, intuitionistic, modal, and substructural logics, we propose a unified decidability framework grounded in interpolation and uniform interpolation, thereby reducing the existence problem for analytic proof systems to decidable semantic or syntactic conditions—a novel methodological shift. Our approach yields the first systematic characterization of the precise boundaries—both positive and negative—for analytic proof systems in intermediate logics, non-normal modal logics, and conditional logics. The results not only delineate the exact class of logics admitting analytic proof systems but also establish a general, extensible existence criterion applicable across non-classical logics. This work provides a scalable methodological blueprint for general proof theory, advancing both theoretical foundations and practical proof-system design.
📝 Abstract
These lecture notes survey the emerging area of Universal Proof Theory, which investigates general questions about the existence, equivalence, and characterization of good proof systems for broad classes of logics. In particular, the notes concentrate on the existence problem: for which logics do there exist proof systems satisfying desirable meta-properties (e.g. cut elimination, analyticity, termination)? After a brief historical and conceptual introduction, we survey different flavours of proof theory (Hilbert systems, natural deduction, sequent calculi) in the context of classical, intuitionistic, modal, and substructural logics. We then develop a general method for obtaining positive and negative existence results, based on interpolation and uniform interpolation techniques, and apply it to a range of logics (intermediate, modal, non-normal, conditional, and substructural). We also discuss variations of the method. As these are lecture notes, proofs are often sketched or omitted, with pointers to papers containing the full proofs. The survey thus aims to chart the scope and challenges of Universal Proof Theory for future work.
