Cut Elimination Theorem for Non-Commutative Hypersequent Calculus

Hypersequent calculi (HC) can formalize various non-classical logics. In [9] we presented a non-commutative variant of HC for the weakest temporal logic of linear frames Kt4.3 and some its extensions for dense and serial flow of time. The system was proved to be cut-free HC formalization of respective temporal logics by means of Schütte/Hintikka-style semantical argument using models built from saturated hypersequents. In this paper we present a variant of this calculus for Kt4.3 with a constructive syntactical proof of cut elimination.Zadanie „ Wdrożenie platformy Open Journal System dla czasopisma „ Bulletin of the Section of Logic” finansowane w ramach umowy 948/P-DUN/2016 ze środków Ministra Nauki i Szkolnictwa Wyższego przeznaczonych na działalność upowszechniającą naukę

Hypersequent Calculi for S5: The Methods of Cut Elimination

S5 is one of the most important modal logic with nice syntactic, semantic and algebraic properties. In spite of that, a successful (i.e. cut-free) formalization of S5 on the ground of standard sequent calculus (SC) was problematic and led to the invention of numerous nonstandard, generalized forms of SC. One of the most interesting framework which was very often used for this aim is that of hypersequent calculi (HC). The paper is a survey of HC for S5 proposed by Pottinger, Avron, Restall, Poggiolesi, Lahav and Kurokawa. We are particularly interested in examining different methods which were used for proving the eliminability/admissibility of cut in these systems and present our own variant of a system which admits relatively simple proof of cut elimination

S5-Style Non-Standard Modalities in a Hypersequent Framework

The aim of the paper is to present some non-standard modalities (such as non-contingency, contingency, essence and accident) based on S5-models in a framework of cut-free hypersequent calculi. We also study negated modalities, i.e. negated necessity and negated possibility, which produce paraconsistent and paracomplete negations respectively. As a basis for our calculi, we use Restall's cut-free hypersequent calculus for S5. We modify its rules for the above-mentioned modalities and prove strong soundness and completeness theorems by a Hintikka-style argument. As a consequence, we obtain a cut admissibility theorem. Finally, we present a constructive syntactic proof of cut elimination theorem

A purely syntactic and cut-free sequent calculus for the modal logic of provability

International audienceIn this paper we present a sequent calculus for the modal propositional logic GL (the logic of provability) obtained by means of the tree-hypersequent method, a method in which the metalinguistic strength of hypersequents is improved, so that we can simulate trees shapes. We prove that this sequent calculus is sound and complete with respect to the Hilbert-style system GL, that it is contraction free and cut free and that its logical and modal rules are invertible. No explicit semantic element is used in the sequent calculus and all the results are proved in a purely syntactic way

On a multilattice analogue of a hypersequent S5 calculus

In this paper, we present a logic MMLS5n which is a combination of multilattice logic and modal logic S5. MMLS5n is an extension of Kamide and Shramko’s modal multilattice logic which is a multilattice analogue of S4. We present a cut-free hypersequent calculus for MMLS5n in the spirit of Restall’s one for S5 and develop a Kripke semantics for MMLS5n, following Kamide and Shramko’s approach. Moreover, we prove theorems for embedding MMLS5n into S5 and vice versa. As a result, we obtain completeness, cut elimination, decidability, and interpolation theorems for MMLS5n. Besides, we show the duality principle for MMLS5n. Additionally, we introduce a modification of Kamide and Shramko’s sequent calculus for their multilattice version of S4 which (in contrast to Kamide and Shramko’s original one) proves the interdefinability of necessity and possibility operators. Last, but not least, we present Hilbert-style calculi for all the logics in question as well as for a larger class of modal multilattice logics

A Hypersequent Calculus with Clusters for Linear Frames

International audienceThe logic Kt4.3 is the basic modal logic of linear frames. Along with its extensions, it is found at the core of linear-time temporal logics and logics on words. In this paper, we consider the problem of designing proof systems for these logics, in such a way that proof search yields decision procedures for validity with an optimal complexity—coNP in this case. In earlier work, Indrzejczak has proposed an ordered hypersequent calculus that is sound and complete for Kt4.3 but does not yield any decision procedure. We refine his approach, using a hypersequent structure that corresponds to weak rather than strict total orders, and using annotations that reflect the model-theoretic insights given by small models for Kt4.3. We obtain a sound and complete calculus with an associated coNP proof search algorithm. These results extend naturally to the cases of unbounded and dense frames, and to the complexity of the two-variable fragment of first-order logic over total orders

The Varieties of Ought-implies-Can and Deontic STIT Logic

STIT logic is a prominent framework for the analysis of multi-agent choice-making. In the available deontic extensions of STIT, the principle of Ought-implies-Can (OiC) fulfills a central role. However, in the philosophical literature a variety of alternative OiC interpretations have been proposed and discussed. This paper provides a modular framework for deontic STIT that accounts for a multitude of OiC readings. In particular, we discuss, compare, and formalize ten such readings. We provide sound and complete sequent-style calculi for all of the various STIT logics accommodating these OiC principles. We formally analyze the resulting logics and discuss how the different OiC principles are logically related. In particular, we propose an endorsement principle describing which OiC readings logically commit one to other OiC readings

Rule-Generation Theorem and its Applications

In several applications of sequent calculi going beyond pure logic, an introduction of suitably defined rules seems to be more profitable than addition of extra axiomatic sequents. A program of formalization of mathematical theories via rules of special sort was developed successfully by Negri and von Plato. In this paper a general theorem on possible ways of transforming axiomatic sequents into rules in sequent calculi is proved. We discuss its possible applications and provide some case studies for illustration

Topics in the Proof Theory of Non-classical Logics. Philosophy and Applications

Chapter 1 constitutes an introduction to Gentzen calculi from two perspectives, logical and philosophical. It introduces the notion of generalisations of Gentzen sequent calculus and the discussion on properties that characterize good inferential systems. Among the variety of Gentzen-style sequent calculi, I divide them in two groups: syntactic and semantic generalisations. In the context of such a discussion, the inferentialist philosophy of the meaning of logical constants is introduced, and some potential objections – mainly concerning the choice of working with semantic generalizations – are addressed. Finally, I’ll introduce the case studies that I’ll be dealing with in part II. Chapter 2 is concerned with the origins and development of Jaśkowski’s discussive logic. The main idea of this chapter is to systematize the various stages of the development of discussive logic related researches from two different angles, i.e., its connections to modal logics and its proof theory, by highlighting virtues and vices. Chapter 3 focuses on the Gentzen-style proof theory of discussive logic, by providing a characterization of it in terms of labelled sequent calculi. Chapter 4 deals with the Gentzen-style proof theory of relevant logics, again by employing the methodology of labelled sequent calculi. This time, instead of working with a single logic, I’ll deal with a whole family of them. More precisely, I’ll study in terms of proof systems those relevant logics that can be characterised, at the semantic level, by reduced Routley-Meyer models, i.e., relational structures with a ternary relation between states and a unique base element. Chapter 5 investigates the proof theory of a modal expansion of intuitionistic propositional logic obtained by adding an ‘actuality’ operator to the connectives. This logic was introduced also using Gentzen sequents. Unfortunately, the original proof system is not cut-free. This chapter shows how to solve this problem by moving to hypersequents. Chapter 6 concludes the investigations and discusses the future of the research presented throughout the dissertation

Beyond semantic pollution: Towards a practice-based philosophical analysis of labelled calculi

This paper challenges the negative attitudes towards labelled proof systems, usually referred to as semantic pollution, by arguing that such critiques overlook the full potential of labelled calculi. The overarching objective is to develop a practice-based philosophical analysis of labelled calculi to provide insightful considerations regarding their proof-theoretic and philosophical value. To achieve this, successful applications of labelled calculi and related results will be showcased, and comparisons with other relevant works will be discussed. The paper ends by advocating for a more practice-based approach towards the philosophical understanding of proof systems and their role in structural proof theory