Structural completeness in propositional logics of dependence

In this paper we prove that three of the main propositional logics of dependence (including propositional dependence logic and inquisitive logic), none of which is structural, are structurally complete with respect to a class of substitutions under which the logics are closed. We obtain an analogues result with respect to stable substitutions, for the negative variants of some well-known intermediate logics, which are intermediate theories that are closely related to inquisitive logic

Proof Theory for Lax Logic

In this paper some proof theory for propositional Lax Logic is developed. A cut free terminating sequent calculus is introduced for the logic, and based on that calculus it is shown that the logic has uniform interpolation. Furthermore, a separate, simple proof of interpolation is provided that also uses the sequent calculus. From the literature it is known that Lax Logic has interpolation, but all known proofs use models rather than proof systems

Proof Theory for Intuitionistic Strong L\"ob Logic

This paper introduces two sequent calculi for intuitionistic strong L\"ob logic ${\sf iSL}_\Box$: a terminating sequent calculus ${\sf G4iSL}_\Box$ based on the terminating sequent calculus ${\sf G4ip}$ for intuitionistic propositional logic ${\sf IPC}$ and an extension ${\sf G3iSL}_\Box$ of the standard cut-free sequent calculus ${\sf G3ip}$ without structural rules for ${\sf IPC}$. One of the main results is a syntactic proof of the cut-elimination theorem for ${\sf G3iSL}_\Box$. In addition, equivalences between the sequent calculi and Hilbert systems for ${\sf iSL}_\Box$ are established. It is known from the literature that ${\sf iSL}_\Box$ is complete with respect to the class of intuitionistic modal Kripke models in which the modal relation is transitive, conversely well-founded and a subset of the intuitionistic relation. Here a constructive proof of this fact is obtained by using a countermodel construction based on a variant of ${\sf G4iSL}_\Box$. The paper thus contains two proofs of cut-elimination, a semantic and a syntactic proof.Comment: 29 pages, 4 figures, submitted to the Special Volume of the Workshop Proofs! held in Paris in 201

The depth of intuitionistic cut free proofs

Abstract We prove a quadratic upper bound for the depth of cut free proofs in propositional intuitionistic logic formalized with Gentzen's sequent calculus. We discuss bounds on the necessary number of reuses of left implication rules. We exhibit an example showing that this quadratic bound is optimal. As a corollary, this gives a new proof that propositional validity for intuitionistic logic is in PSPACE

A new calculus for intuitionistic Strong L\"ob logic: strong termination and cut-elimination, formalised

We provide a new sequent calculus that enjoys syntactic cut-elimination and strongly terminating backward proof search for the intuitionistic Strong L\"ob logic $\sf{iSL}$, an intuitionistic modal logic with a provability interpretation. A novel measure on sequents is used to prove both the termination of the naive backward proof search strategy, and the admissibility of cut in a syntactic and direct way, leading to a straightforward cut-elimination procedure. All proofs have been formalised in the interactive theorem prover Coq.Comment: 21-page conference paper + 4-page appendix with proof

Properties of Intuitionistic Provability Logics

Abstract. We study the modal properties of intuitionistic modal logics that belong to the provability logic or the preservativity logic of Heyting Arithmetic. We describe thefragment of some preservativity logics and we present fixed point theorems for the logics iL and iP L, and show that they imply the Beth property. These results imply that the fixed point theorem and the Beth property hold for both the provability and preservativity logic of Heyting Arithmetic. We present a frame correspondence result for the preservativity principle W p that is related to an extension of Löb's principle