Universal Proof Theory: Topology, Algebra and Categories in Logic Lecture Notes of the Coimbra TACL Summer School

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 flavors 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.