# Syntactic completeness of proper display calculi

A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e. those calculi that support general and modular proof-strategies for cut elimination), and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analiticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e. those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (aka unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms, which can be equivalently and algorithmically transformed into analytic structural rules so that the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination and subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically, i.e. by showing that a (cut free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far this proof strategy for syntactic completeness has been implemented on a case-by-case base, and not in general. In this paper, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut free derivation can be effectively generated which has a specific shape, referred to as pre-normal form.

## Authors

• 1 publication
• 4 publications
• 6 publications
• 6 publications
• ### Syntactic cut-elimination and backward proof-search for tense logic via linear nested sequents (Extended version)

We give a linear nested sequent calculus for the basic normal tense logi...
07/02/2019 ∙ by Rajeev Goré, et al. ∙ 0

• ### Appendix for: Cut-free Calculi and Relational Semantics for Temporal STIT logics

This paper is an appendix to the paper "Cut-free Calculi and Relational ...
02/18/2019 ∙ by Kees van Berkel, et al. ∙ 0

• ### Modular Labelled Sequent Calculi for Abstract Separation Logics

Abstract separation logics are a family of extensions of Hoare logic for...
10/30/2017 ∙ by Zhé Hóu, et al. ∙ 0

• ### Display to Labelled Proofs and Back Again for Tense Logics

We introduce translations between display calculus proofs and labelled c...
11/06/2019 ∙ by Agata Ciabattoni, et al. ∙ 0

• ### Uniform labelled calculi for preferential conditional logics based on neighbourhood semantics

The preferential conditional logic PCL, introduced by Burgess, and its e...
02/14/2020 ∙ by Marianna Girlando, et al. ∙ 0

• ### Predicative proof theory of PDL and basic applications

Propositional dynamic logic (PDL) is presented in Schütte-style mode as ...
04/03/2019 ∙ by Lev Gordeev, et al. ∙ 0