Display calculi and other modal calculi: a comparison

Abstract

In this paper we introduce and compare four different syntactic methods for generating sequent calculi for the main systems of modal logic: the multiple sequents method, the higher-arity sequents method, the tree-hypersequents method and the display method. More precisely we show how the first three methods can all be translated in the fourth one. This result sheds new light on these generalisations of the sequent calculus and raises issues that will be examined in the last section.

Logicality, Double-Line Rules, and Modalities

“…In order to make the paper self-contained, we are now going to present display calculi for modal logics as they are presented in [22]. For the sake of conciseness, we omit the proofs since these can be found in [22] or in [32].…”

Display to Labeled Proofs and Back Again for Tense Logics

“…There are trade-offs in employing one formalism as opposed to another, motivating a study of the interrelationships between the current patchwork (see, e.g. [32]) of proof systems.…”

Hypersequents and Systems of Rules: Embeddings and Applications

“…Such embeddings can also provide useful reformulations of known calculi and allow the transfer of certain proof-theoretic results, thus alleviating the need for independent proofs in each system and avoiding duplicating work. Various embeddings between formalisms have appeared in the literature, see, e.g., [24,23,13,11,20,21,26] (and the bibliography thereof).…”