Sequent Systems Research Articles

Sequent Systems for Consequence Relations of Cyclic Linear Logics

Linear Logic is a versatile framework with diverse applications in computer science and mathematics. One intriguing fragment of Linear Logic is Multiplicative-Additive Linear Logic (MALL), which forms the exponential-free component of the larger framework. Modifying MALL, researchers have explored weaker logics such as Noncommutative MALL (Bilinear Logic, BL) and Cyclic MALL (CyMALL) to investigate variations in commutativity. In this paper, we focus on Cyclic Nonassociative Bilinear Logic (CyNBL), a variant that combines noncommutativity and nonassociativity. We introduce a sequent system for CyNBL, which includes an auxiliary system for incorporating nonlogical axioms. Notably, we establish the cut elimination property for CyNBL. Moreover, we establish the strong conservativeness of CyNBL over Full Nonassociative Lambek Calculus (FNL) without additive constants. The paper highlights that all proofs are constructed using syntactic methods, ensuring their constructive nature. We provide insights into constructing cut-free proofs and establishing a logical relationship between CyNBL and FNL.

Analyticity with extra-logical information

Abstract In this paper, a new approach to the issue of extra-logical information within analytic (i.e. obeying the sub-formula property) sequent systems is introduced. We prove that incorporating extra-logical axioms into a purely logical system can preserve analyticity, provided these axioms belong to a suitable class of formulas that can be decomposed into a set of equivalent initial sequents and are permutable over the cut rule. Our approach is applicable not only to first-order classical and intuitionistic logics, but also to substructural logics. Furthermore, we establish a limit for the augmented systems under analysis: exceeding the boundaries of their respective classes of extra-logical axioms leads to either a loss of analyticity or a loss of structural properties.

A Hierarchy of Determinative Sequent Systems with Different Substitution Rules

A Hierarchy of Determinative Sequent Systems with Different Substitution Rules

Cyclic Hypersequent System for Transitive Closure Logic

We propose a cut-free cyclic system for transitive closure logic (TCL) based on a form of hypersequents, suitable for automated reasoning via proof search. We show that previously proposed sequent systems are cut-free incomplete for basic validities from Kleene Algebra (KA) and propositional dynamic logic (text {PDL}), over standard translations. On the other hand, our system faithfully simulates known cyclic systems for KA and text {PDL}, thereby inheriting their completeness results. A peculiarity of our system is its richer correctness criterion, exhibiting ‘alternating traces’ and necessitating a more intricate soundness argument than for traditional cyclic proofs.

Cut-Free Gentzen Sequent Calculi for Tense Logics

The cut-free single-succedent Gentzen sequent calculus GKt for the minimal tense logic Kt is introduced. This sequent calculus satisfies the displaying property. The proof proceeds in terms of a Kolmogorov translation and three intermediate sequent systems. Finally, we show that tense logics axiomatized by strictly positive implication have cut-free Gentzen sequent calculi uniformly.

Nested sequents for intermediate logics: the case of Gödel-Dummett logics

We present nested sequent systems for propositional Gödel-Dummett logic and its first-order extensions with non-constant and constant domains, built atop nested calculi for intuitionistic logics. To obtain nested systems for these Gödel-Dummett logics, we introduce a new structural rule, called the linearity rule, which (bottom-up) operates by linearising branching structure in a given nested sequent. In addition, an interesting feature of our calculi is the inclusion of reachability rules, which are special logical rules that operate by propagating data and/or checking if data exists along certain paths within a nested sequent. Such rules require us to generalise our nested sequents to include signatures (i.e. finite collections of variables) in the first-order cases, thus giving rise to a generalisation of the usual nested sequent formalism. Our calculi exhibit favourable properties, admitting the height-preserving invertibility of every logical rule and the (height-preserving) admissibility of a large collection of structural and reachability rules. We prove all of our systems sound and cut-free complete, and show that syntactic cut-elimination obtains for the intuitionistic systems. We conclude the paper by discussing possible extensions and modifications, putting forth an array of structural rules that could be used to provide a sizable class of intermediate logics with cut-free nested sequent systems.

Soundness and completeness results for LEA and probability semantics

Abstract In Artemov (2020, J. Logic Comput., 30, 61–76), a logical system called the logic of evidence aggregation (LEA) was introduced, along with an intended semantics for it called probability semantics. The goal was to describe probabilistic evidence aggregation in the setting of formal logic. However, as noted in that paper, LEA is not complete with respect to probability semantics. This leaves the tasks open to find sound and complete semantics for LEA and a proper axiomatization for probability semantics. In this paper, we do both. We define a class of basic models called deductive basic models. We show LEA is sound and complete with respect to the class of deductive basic models. We also define an axiomatic system LEA$_+$ extending LEA and show it is sound and complete with respect to probability semantics. Along the way, we develop an intermediate system LEA$_-$ which, it turns out, is equivalent to propositional lax logic with a classical logic base. Sequent systems and decidability results for LEA$_-$ and LEA$_+$ are also presented.

A rewriting logic approach to specification, proof-search, and meta-proofs in sequent systems

This paper develops an algorithmic-based approach for proving inductive properties of propositional sequent systems such as admissibility, invertibility, cut-admissibility, and identity expansion. Although undecidable in general, these structural properties are crucial in proof theory because they can reduce the proof-search effort and further be used as scaffolding for obtaining other meta-results such as consistency. The algorithms –which take advantage of the rewriting logic meta-logical framework– are explained in detail and illustrated with examples throughout the paper. They have been fully mechanized in the L-Framework, thus offering both a formal specification language and off-the-shelf mechanization of the proof-search algorithms coming together with semi-decision procedures for proving theorems and meta-theorems of the object system. As illustrated with case studies in the paper, the L-Framework achieves a great degree of automation when used on several propositional sequent systems, including single conclusion and multi-conclusion intuitionistic logic, classical logic, classical linear logic and its dyadic system, intuitionistic linear logic, and normal modal logics.

A linear logic framework for multimodal logics

AbstractOne of the most fundamental properties of a proof system is analyticity, expressing the fact that a proof of a given formula F only uses subformulas of F. In sequent calculus, this property is usually proved by showing that the $\mathsf{cut}$ rule is admissible, i.e., the introduction of the auxiliary lemma H in the reasoning “if H follows from G and F follows from H, then F follows from G” can be eliminated. The proof of cut admissibility is usually a tedious, error-prone process through several proof transformations, thus requiring the assistance of (semi-)automatic procedures. In a previous work by Miller and Pimentel, linear logic ( $\mathsf{LL}$ ) was used as a logical framework for establishing sufficient conditions for cut admissibility of object logical systems (OL). The OL’s inference rules are specified as an $\mathsf{LL}$ theory and an easy-to-verify criterion sufficed to establish the cut-admissibility theorem for the OL at hand. However, there are many logical systems that cannot be adequately encoded in $\mathsf{LL}$ , the most symptomatic cases being sequent systems for modal logics. In this paper, we use a linear-nested sequent ( $\mathsf{LNS}$ ) presentation of $\mathsf{MMLL}$ (a variant of LL with subexponentials), and show that it is possible to establish a cut-admissibility criterion for $\mathsf{LNS}$ systems for (classical or substructural) multimodal logics. We show that the same approach is suitable for handling the $\mathsf{LNS}$ system for intuitionistic logic.

A Classification of Improper Inference Rules

In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper as opposed to proper ones. Improper inference rules are more complicated than proper ones and more difficult to understand. In 2022, we provided a sequent system based solely on the application of proper rules. In the present paper, on the basis of our system from 2022, we classify improper inference rules.

On Sizes of Linear and Tree-Like Proofs for any Formulae Families in Some Systems of Propositional Calculus

The sizes of linear and tree-like proofs for any formulae families are investigated in some systems of propositional calculus: in different sequent systems (with quantifier rules, with the substitution rule, with the cut rule, without the cut rule, monotone) and in the generalization splitting system. The comparison of results obtained here with the bounds obtained formerly for the steps of proofs for the same formulas in the mentioned systems shows the importance of the size of proof among the other characteristics of proof complexities.

A SIMPLE SEQUENT SYSTEM FOR MINIMALLY INCONSISTENT LP

AbstractMinimally inconsistent LP (MiLP) is a nonmonotonic paraconsistent logic based on Graham Priest’s logic of paradox (LP). Unlike LP, MiLP purports to recover, in consistent situations, all of classical reasoning. The present paper conducts a proof-theoretic analysis of MiLP. I highlight certain properties of this logic, introduce a simple sequent system for it, and establish soundness and completeness results. In addition, I show how to use my proof system in response to a criticism of this logic put forward by J. C. Beall.

A SUBSTRUCTURAL GENTZEN CALCULUS FOR ORTHOMODULAR QUANTUM LOGIC

AbstractWe introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers a calculus for classical logic.

From axioms to synthetic inference rules via focusing

An important application of focused variants of Gentzen's sequent calculus proof rules is the construction of (possibly) large synthetic inference rules. This paper examines the synthetic inference rules that arise when using theories composed of bipolars, and we do this in both classical and intuitionistic logics. A key step in transforming a formula into synthetic inference rules involves attaching a polarity to atomic formulas and some logical connectives. Since there are different choices in how polarity is assigned, it is possible to produce different synthetic inference rules for the same formula. We show that this flexibility allows for the generalization of different approaches for transforming axioms into sequent rules present in the literature. We finish the paper showing how to apply these results to organize the proof theory of labeled sequent systems for several propositional modal logics.

Analytic Non-Labelled Proof-Systems for Hybrid Logic: Overview and a couple of striking facts

This paper is about non-labelled proof-systems for hybrid logic, that is, proofsystems where arbitrary formulas can occur, not just satisfaction statements. We give an overview of such proof-systems, focusing on analytic systems: Natural deduction systems, Gentzen sequent systems and tableau systems. We point out major results and we discuss a couple of striking facts, in particular that nonlabelled hybrid-logical natural deduction systems are analytic, but this is not proved in the usual way via step-by-step normalization of derivations.

Exponentially Handsome Proof Nets and Their Normalization

Handsome proof nets were introduced by Retor\'e as a syntax for multiplicative linear logic. These proof nets are defined by means of cographs (graphs representing formulas) equipped with a vertices partition satisfying simple topological conditions. In this paper we extend this syntax to multiplicative linear logic with units and exponentials. For this purpose we develop a new sound and complete sequent system for the logic, enforcing a stronger notion of proof equivalence with respect to the one usually considered in the literature. We then define combinatorial proofs, a graphical proof system able to capture syntactically the proof equivalence, for the cut-free fragment of the calculus. We conclude the paper by defining the exponentially handsome proof nets as combinatorial proofs with cuts and defining an internal normalization procedure for this syntax.

A note on cut-elimination for classical propositional logic

In Schwichtenberg (Studies in logic and the foundations of mathematics, vol 90, Elsevier, pp 867–895, 1977), Schwichtenberg fine-tuned Tait’s technique (Tait in The syntax and semantics of infinitary languages, Springer, pp 204–236, 1968) so as to provide a simplified version of Gentzen’s original cut-elimination procedure for first-order classical logic (Gallier in Logic for computer science: foundations of automatic theorem proving, Courier Dover Publications, London, 2015). In this note we show that, limited to the case of classical propositional logic, the Tait–Schwichtenberg algorithm allows for a further simplification. The procedure offered here is implemented on Kleene’s sequent system G4 (Kleene in Mathematical logic, Wiley, New York, 1967; Smullyan in First-order logic, Courier corporation, London, 1995). The specific formulation of the logical rules for G4 allows us to provide bounds on the height of cut-free proofs just in terms of the logical complexity of their end-sequent.

A Sequent Systems without Improper Derivations

In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper, and others proper. Improper inference rules are more complicated and are often harder to understand than the proper ones. In the present paper, we distinguish between proper and improper derivations by using sequent systems. Specifically, we introduce a sequent system \(\vdash_{\bf Sc}\) for classical propositional logic with only structural rules, and prove that \(\vdash_{\bf Sc}\) does not allow improper derivations in general. For instance, the sequent \(\Rightarrow p \to q\) cannot be derived from the sequent \(p \Rightarrow q\) in \(\vdash_{\bf Sc}\). In order to prove the failure of improper derivations, we modify the usual notion of truth valuation, and using the modified valuation, we prove the completeness of \(\vdash_{\bf Sc}\). We also consider whether an improper derivation can be described generally by using \(\vdash_{\bf Sc}\).

The (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework)

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined in terms of a \(p\)-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints.

A note on an alternative Gentzenization of RW+∘

AbstractIn our paper [3], a sequent system for the contraction‐less positive relevant logic with co‐tenability, , is considered. Its sequent system with the rule of cut, which does not involve the truth constant t, is presented, and the proof that it admits the elimination of cut is given. However, in the proof of the cut‐elimination theorem some forms of proofs, which may cause difficulties for the cut‐elimination argument, are not considered. The purpose of this paper is to present them and to re‐prove that admits the elimination of cut.