We introduce a systematic procedure to transform large classes of (Hilbert)

No abstract

What can (and cannot) be expressed by structural display rules? Given a display calculus, we present a systematic procedure for transforming axioms into structural rules. The conditions for the procedure are given in terms of (purely syntactic) abstract properties of the base calculus; thus, the method applies to large classes of calculi and logics. If the calculus satisfies certain additional properties, we prove the converse direction, thus characterising the class of axioms that can be captured by structural display rules. Determining if an axiom belongs to this class or not is shown to be decidable. Applied to the display calculus for tense logic, we obtain a new proof of Kracht’s Display Theorem I.

Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzen-style characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinite-valued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to first-order as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered.

We introduce necessary and sufficient conditions for a (single-conclusion) sequent calculus to admit (reductive) cut-elimination. Our conditions are formulated both syntactically and semantically.