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Rosalie Iemhoff

Publications and source records attributed to Rosalie Iemhoff.

15 recordsLinked to original sources
arXiv2026

The G4i analogue of a G3i calculus

This paper provides a method to obtain terminating analytic calculi for a large class of intuitionistic modal logics. For a given logic L with a cut-free calculus G that is an extension of G3ip the method produces a terminating analytic calculus that is an extension of G4ip and equivalent to G. G4ip has been introduced by Dyckhoff in 1992 as a terminating analogue of the calculus G3ip for intuitionistic propositional logic. Thus this paper can be viewed as an extension of Dyckhoff's work to intuitionistic modal logic. (This is a corrected version (21 August 2026) of an earlier arXiv version that contained a mistake in Theorem 3.4.)

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arXiv2026

Proof Theory for Lax Logic

(A mistake has been discovered in this paper, implying that the main theorem stating that PLL has uniform interpolation cannot be considered proven (see Corollary 1 for the details of the mistake). A proof of the opposite is not known either. I am still trying to prove that PLL has uniform interpolation, as I somehow think that is true. We'll see. To be continued.) Original abstract: 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.

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arXiv2025

Six Proofs of Interpolation for the Modal Logic K

In this chapter, we present six different proofs of Craig interpolation for the modal logic K, each using a different set of techniques (model-theoretic, proof-theoretic, syntactic, automata-theoretic, using quasi-models, and algebraic). We compare the pros and cons of each proof technique.

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arXiv2025

Universal Proof Theory, TACL 2022 Lecture Notes

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 flavours 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.

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arXiv2025

A Deep-Inference Sequent Calculus for Basic Propositional Team Logic (Without Delving Too Deep)

We introduce a sequent calculus for the propositional team logic with both the split disjunction and the inquisitive disjunction consisting of a Gentzen-style system (G3-like) for classical propositional logic together with two deep-inference rules for the inquisitive disjunction. We show that the system satisfies various desirable properties: it admits height-preserving weakening, contraction and inversion; it supports a procedure for constructing cutfree proofs and countermodels similar to that for G3cp; and cut elimination holds as a corollary of cut elimination for the G3-style subsystem together with a normal form theorem for cutfree derivations. We also prove a sequent interpolation theorem for the system that yields a novel Lyndon's interpolation theorem for the logic as a corollary.

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arXiv2025

Skolemization In Intermediate Logics

Skolemization, with Herbrand's theorem, underpins automated theorem proving and various transformations in computer science and mathematics. Skolemization removes strong quantifiers by introducing new function symbols, enabling efficient proof search algorithms. We characterize intermediate first-order logics that admit standard (and Andrews) Skolemization. These are the logics that allow classical quantifier shift principles. For some logics not in this category, innovative forms of Skolem functions are developed that allow Skolemization. Moreover, we analyze predicate intuitionistic logic with quantifier shift axioms and demonstrate its Kripke frame-incompleteness. These findings may foster resolution-based theorem provers for non-classical logics. This article is part of a larger project investigating Skolemization in non-classical logics.

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arXiv2023

A new calculus for intuitionistic Strong Löb 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 Löb 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.

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arXiv2023

Proof Theory for Intuitionistic Strong Löb Logic

This paper introduces two sequent calculi for intuitionistic strong Löb 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.

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arXiv2022

Logics and Admissible Rules of Constructive Set Theories

We survey the logical structure of constructive set theories and point towards directions for future research. Moreover, we analyse the consequences of being extensible for the logical structure of a given constructive set theory. We finally provide examples of a number of set theories that are extensible.

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arXiv2022

Uniform Lyndon Interpolation for Basic Non-normal Modal and Conditional Logics

In this paper, a proof-theoretic method to prove uniform Lyndon interpolation for non-normal modal and conditional logics is introduced and applied to show that the logics $\mathsf{E}$, $\mathsf{M}$, $\mathsf{EN}$, $\mathsf{MN}$, $\mathsf{MC}$, $\mathsf{K}$, and their conditional versions, $\mathsf{CE}$, $\mathsf{CM}$, $\mathsf{CEN}$, $\mathsf{CMN}$, $\mathsf{CMC}$, $\mathsf{CK}$, in addition to $\mathsf{CKID}$ have that property. In particular, it implies that these logics have uniform interpolation. Although for some of them the latter is known, the fact that they have uniform Lyndon interpolation is new. Also, the proof-theoretic proofs of these facts are new, as well as the constructive way to explicitly compute the interpolants that they provide. On the negative side, it is shown that the logics $\mathsf{CKCEM}$ and $\mathsf{CKCEMID}$ enjoy uniform interpolation but not uniform Lyndon interpolation. Moreover, it is proved that the non-normal modal logics $\mathsf{EC}$ and $\mathsf{ECN}$ and their conditional versions, $\mathsf{CEC}$ and $\mathsf{CECN}$, do not have Craig interpolation, and whence no uniform (Lyndon) interpolation.

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arXiv2022

Uniform Lyndon interpolation for intuitionistic monotone modal logic

In this paper we show that the intuitionistic monotone modal logic $\mathsf{iM}$ has the uniform Lyndon interpolation property (ULIP). The logic $\mathsf{iM}$ is a non-normal modal logic on an intuitionistic basis, and the property ULIP is a strengthening of interpolation in which the interpolant depends only on the premise or the conclusion of an implication, respecting the polarities of the propositional variables. Our method to prove ULIP yields explicit uniform interpolants and makes use of a terminating sequent calculus for $\mathsf{iM}$ that we have developed for this purpose. As far as we know, the results that $\mathsf{iM}$ has ULIP and a terminating sequent calculus are the first of their kind for an intuitionistic non-normal modal logic. However, rather than proving these particular results, our aim is to show the flexibility of the constructive proof-theoretic method that we use for proving ULIP. It has been developed over the last few years and has been applied to substructural, intermediate, classical (non-)normal modal and intuitionistic normal modal logics. In light of these results, intuitionistic non-normal modal logics seem a natural next class to try to apply the method to, and we take the first step in that direction in this paper.

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arXiv2021

Reasoning in circles

Circular proofs, introduced by Daniyar Shamkanov, are proofs in which assumptions are allowed that are not axioms but do appear at least twice along a branch. Shamkanov has shown that a formula belongs to the provability logic GL exactly if it has a circular proof in the modal logic K4. Shamkanov uses Tait style proof systems and infinitary proofs. In this paper we prove the same result but then for sequent calculi and without the detour via infinitary systems. We also obtain a mild generalisation of the result, implying that its intuitionistic analogue holds as well.

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arXiv2017

Questions and dependency in intuitionistic logic

In recent years, the logic of questions and dependencies has been investigated in the closely related frameworks of inquisitive logic and dependence logic. These investigations have assumed classical logic as the background logic of statements, and added formulas expressing questions and dependencies to this classical core. In this paper, we broaden the scope of these investigations by studying questions and dependency in the context of intuitionistic logic. We propose an intuitionistic team semantics, where teams are embedded within intuitionistic Kripke models. The associated logic is a conservative extension of intuitionistic logic with questions and dependence formulas. We establish a number of results about this logic, including a normal form result, a completeness result, and translations to classical inquisitive logic and modal dependence logic.

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arXiv2015

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.

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