arXiv subjects

Elaine Pimentel

Publications and source records attributed to Elaine Pimentel.

At least 19 recordsLinked to original sources

arXiv2026

Efficient Decision Procedures for RNmatrix Semantics

Logical matrices provide a semantic framework in which connectives are interpreted by deterministic truth-functions. While elegant, this approach is often too restrictive to capture the semantics of many non-classical logics. Non-deterministic matrices (Nmatrices) generalise ordinary matrices by allowing connectives to return a set of possible truth values rather than a unique one, thereby increasing expressive power. Restricted non-deterministic matrices (RNmatrices) further refine this framework by imposing constraints on the rows of Nmatrices, filtering out "unsound" rows and retaining only "valid" ones. This yields a more expressive semantic framework that has been successfully used to provide sound and complete semantics for a wide range of logics, including paraconsistent, intuitionistic, and modal logics. Despite these advances, no efficient decision procedures based on RNmatrix semantics have been proposed. In this paper, we develop automated theorem provers based on RNmatrices by encoding their semantics as Satisfiability Modulo Theories (SMT) problems. The resulting provers decide validity and construct countermodels, achieving state-of-the-art performance for paraconsistent logics and competitive results for intuitionistic and modal logics.

cs.LO↗

arXiv2026

Bilateralism with incompatible proofs and refutations

Logical bilateralism challenges traditional concepts of logic by treating assertion and denial as independent yet opposed acts. While initially devised to justify classical logic, its constructive variants show that both acts admit intuitionistic interpretations. This paper presents a bilateral system where a formula cannot be both provable and refutable without contradiction, offering a framework for modelling epistemic entities, such as mathematical proofs and refutations, that exclude inconsistency. The logic is formalised through a bilateral natural deduction system with desirable proof-theoretic properties, including normalisation. We also introduce a base-extension semantics requiring explicit constructions of proofs and refutations while preventing them from being established for the same formula. The semantics is proven sound and complete with respect to the calculus. Finally, we show that our notion of refutation corresponds to David Nelson's constructive falsity, extending rather than revising intuitionistic logic and reinforcing the system's suitability for representing constructive epistemic reasoning.

cs.LO↗

arXiv2026

Glivenko's theorems from an ecumenical perspective

In this paper, we revisit Glivenko's theorems, foundational results relating classical and intuitionistic logic, from an ecumenical perspective. We begin by discussing the historical context and significance of Glivenko's original contributions, and then examine their extensions and reinterpretations within ecumenical logical frameworks. Our analysis focuses on three ecumenical systems: Prawitz's natural deduction system NE; the system NEK, closely related to one introduced by Krauss in an unpublished manuscript; and the ECI system proposed by Barroso-Nascimento.

cs.LO↗

arXiv2026

A Proof-Theoretic Approach to the Semantics of Classical Linear Logic

Linear logic (LL) is a resource-aware, abstract logic programming language that refines both classical and intuitionistic logic. Linear logic semantics is typically presented in one of two ways: by associating each formula with the set of all contexts that can be used to prove it (e.g. phase semantics) or by assigning meaning directly to proofs (e.g. coherence spaces). This work proposes a different perspective on assigning meaning to proofs by adopting a proof-theoretic perspective. More specifically, we employ base-extension semantics (BeS) to characterise proofs through the notion of base support. Recent developments have shown that BeS is powerful enough to capture proof-theoretic notions in structurally rich logics such as intuitionistic linear logic. In this paper, we extend this framework to the classical case, presenting a proof-theoretic approach to the semantics of the multiplicative-additive fragment of linear logic (MALL).

cs.LO↗

arXiv2025

The Modal Cube Revisited: Semantics without Worlds (Technical Report)

We present a non-deterministic semantic framework for all modal logics in the modal cube, extending prior works by Kearns and others. Our approach introduces modular and uniform multi-valued non-deterministic matrices (Nmatrices) for each logic, where necessitation is captured by the systematic use of level valuations. The semantics is grounded in an eight-valued system and provides a sound and complete decision procedure for each modal logic, extending and refining earlier semantics as particular cases. Additionally, we propose a novel model-theoretic perspective that links our framework to relational (Kripke-style) semantics, addressing longstanding conjectures regarding the correspondence between modal axioms and semantic conditions within non-deterministic settings. The result is a philosophically robust and technically modular alternative to standard possible-world semantics.

cs.LO↗

arXiv2025

A Sequent Calculus Perspective on Base-Extension Semantics (Technical Report)

We define base-extension semantics (Bes) using atomic systems based on sequent calculus rather than natural deduction. While traditional Bes aligns naturally with intuitionistic logic due to its constructive foundations, we show that sequent calculi with multiple conclusions yield a Bes framework more suited to classical semantics. The harmony in classical sequents leads to straightforward semantic clauses derived solely from right introduction rules. This framework enables a Sandqvist-style completeness proof that extracts a sequent calculus proof from any valid semantic consequence. Moreover, we show that the inclusion or omission of atomic cut rules meaningfully affects the semantics, yet completeness holds in both cases.

cs.LO↗

arXiv2025

An ecumenical view of proof-theoretic semantics

Debates concerning philosophical grounds for the validity of classical and intuitionistic logics often have the very nature of logical proofs as one of the main points of controversy. The intuitionist advocates for a strict notion of constructive proof, while the classical logician advocates for a notion which allows non-construtive proofs through reductio ad absurdum. A great deal of controversy still subsists to this day on the matter, as there is no agreement between disputants on the precise standing of non-constructive methods. Two very distinct approaches to logic are currently providing interesting contributions to this debate. The first, oftentimes called logical ecumenism, aims to provide a unified framework in which two "rival" logics may peacefully coexist, thus providing some sort of neutral ground for the contestants. The second, proof-theoretic semantics, aims not only to elucidate the meaning of a logical proof, but also to provide means for its use as a basic concept of semantic analysis. Logical ecumenism thus provides a medium in which meaningful interactions may occur between classical and intuitionistic logic, whilst proof-theoretic semantics provides a way of clarifying what is at stake when one accepts or denies reductio ad absurdum as a meaningful proof method. In this paper we show how to coherently combine both approaches by providing not only a medium in which classical and intuitionistic logics may coexist, but also one in which classical and intuitionistic notions of proof may coexist.

cs.LO↗

arXiv2024

Reasoning About Group Polarization: From Semantic Games to Sequent Systems

Group polarization, the phenomenon where individuals become more extreme after interacting, has been gaining attention, especially with the rise of social media shaping people's opinions. Recent interest has emerged in formal reasoning about group polarization using logical systems. In this work we consider the modal logic PNL that captures the notion of agents agreeing or disagreeing on a given topic. Our contribution involves enhancing PNL with advanced formal reasoning techniques, instead of relying on axiomatic systems for analyzing group polarization. To achieve this, we introduce a semantic game tailored for (hybrid) extensions of PNL. This game fosters dynamic reasoning about concrete network models, aligning with our goal of strengthening PNL's effectiveness in studying group polarization. We show how this semantic game leads to a provability game by systemically exploring the truth in all models. This leads to the first cut-free sequent systems for some variants of PNL. Using polarization of formulas, the proposed calculi can be modularly adapted to consider different frame properties of the underlying model.

cs.LO↗

arXiv2024

Multi-modalities and non-commutativity/associativity in functorial linear logic: a case study

Similar to modal connectives, the exponential ! in intuitionistic linear logic (ILL) is not canonical, in the sense that if $i\not= j$ then $!^i F\not\equiv !^j F$. Intuitively, this means that we can mark the exponential with labels taken from a set I organized in a pre-order $\preceq$, obtaining (possibly infinitely-many) exponentials ($!^i$ for $i\in I$). There are, however, two main differences between multi-modalities in normal modal logics and subexponentials in linear logic. i. substructural behaviour. Subexponentials carry the possibility of having different structural behaviors; ii. nature of modalities. Normal modal logics start from the weakest version, assuming only axiom K, then extensions are considered, by adding other axioms. Exponentials in linear logic "take for granted" the behaviors expressed by axioms T and 4. Regarding (i), originally subexponentials could assume only weakening and contraction axioms, but later non-commutative/non-associative systems allowing commutative/ associative subexponentials were presented. Concerning (ii), Guerrini et al unified the modal and LL approaches, with the exponentials assuming only the linear version of K, with the possibility of adding modal extensions to it. This discussion was brought to multi-modal case, where subexponentials consider not only the structural axioms for contraction and weakening, but also the subexponential version of axioms {K,4,D,T}. In this work, we intend to join these two studies. This means that $!^{i}$ can behave classically or not, model associative and commutative systems or not, but also with exponential behaviors different from those in LL. Hence, by assigning different modal axioms one obtains, in a modular way, a class of different substructural modal logics.

cs.LO↗

arXiv2023

Explorations in Subexponential Non-associative Non-commutative Linear Logic

In a previous work we introduced a non-associative non-commutative logic extended by multimodalities, called subexponentials, licensing local application of structural rules. Here, we further explore this system, exhibiting a classical one-sided multi-succedent classical analogue of our intuitionistic system, following the exponential-free calculi of Buszkowski, and de Groote, Lamarche. A large fragment of the intuitionistic calculus is shown to embed faithfully into the classical fragment.

cs.LO↗

arXiv2023

Explorations in Subexponential non-associative non-commutative Linear Logic (extended version)

In a previous work we introduced a non-associative non-commutative logic extended by multimodalities, called subexponentials, licensing local application of structural rules. Here, we further explore this system, considering a classical one-sided multi-succedent classical version of the system, following the exponential-free calculi of Buszkowski's and de Groote and Lamarche's works, where the intuitionistic calculus is shown to embed faithfully into the classical fragment.

cs.LO↗

arXiv2022

On an ecumenical natural deduction with stoup -- Part I: The propositional case

Natural deduction systems, as proposed by Gentzen and further studied by Prawitz, is one of the most well known proof-theoretical frameworks. Part of its success is based on the fact that natural deduction rules present a simple characterization of logical constants, especially in the case of intuitionistic logic. However, there has been a lot of criticism on extensions of the intuitionistic set of rules in order to deal with classical logic. Indeed, most of such extensions add, to the usual introduction and elimination rules, extra rules governing negation. As a consequence, several meta-logical properties, the most prominent one being harmony, are lost. Dag Prawitz proposed a natural deduction ecumenical system, where classical logic and intuitionistic logic are codified in the same system. In this system, the classical logician and the intuitionistic logician would share the universal quantifier, conjunction, negation and the constant for the absurd, but they would each have their own existential quantifier, disjunction and implication, with different meanings. Prawitz' main idea is that these different meanings are given by a semantical framework that can be accepted by both parties. In this paper, we propose a different approach adapting, to the natural deduction framework, Girard's mechanism of stoup. This will allow the definition of a pure harmonic natural deduction system for the propositional fragment of Prawitz' ecumenical logic.

cs.LO↗

arXiv2022

Separability and harmony in ecumenical systems

The quest of smoothly combining logics so that connectives from classical and intuitionistic logics can co-exist in peace has been a fascinating topic of research for decades now. In 2015, Dag Prawitz proposed a natural deduction system for an ecumenical first-order logic. We start this work by proposing a {\em pure} sequent calculus version for it, in the sense that connectives are introduced without the use of other connectives. For doing this, we extend sequents with an extra context, the stoup, and define the ecumenical notion of polarities. Finally, we smoothly extend these ideas for handling modalities, presenting pure labeled and nested systems for ecumenical modal logics.

cs.LO↗

arXiv2021

Proceedings of the Sixteenth Workshop on Logical Frameworks and Meta-Languages: Theory and Practice

Logical frameworks and meta-languages form a common substrate for representing, implementing and reasoning about a wide variety of deductive systems of interest in logic and computer science. Their design, implementation and their use in reasoning tasks, ranging from the correctness of software to the properties of formal systems, have been the focus of considerable research over the last two decades. This workshop brings together designers, implementors and practitioners to discuss various aspects impinging on the structure and utility of logical frameworks, including the treatment of variable binding, inductive and co-inductive reasoning techniques and the expressiveness and lucidity of the reasoning process.

cs.LO↗

arXiv2021

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-elimination, 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, and use rewrite- and narrowing-based reasoning -- 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.

cs.LO↗

arXiv2020

Hypersequent calculi for non-normal modal and deontic logics: Countermodels and optimal complexity

We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms $T$, $P$, $D$, and, for every $n\geq 1$, rule $RD^+_n$. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatisation by a syntactic proof of cut elimination. Then we define a terminating root-first proof search strategy based on the hypersequent calculi and show that it is optimal for coNP-complete logics. Moreover, we obtain that from every saturated leaf of a failed proof it is possible to define a countermodel of the root hypersequent in the bi-neighbourhood semantics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube.

cs.LO↗

arXiv2020

Ecumenical modal logic

The discussion about how to put together Gentzen's systems for classical and intuitionistic logic in a single unified system is back in fashion. Indeed, recently Prawitz and others have been discussing the so called Ecumenical Systems, where connectives from these logics can co-exist in peace. In Prawitz' system, the classical logician and the intuitionistic logician would share the universal quantifier, conjunction, negation, and the constant for the absurd, but they would each have their own existential quantifier, disjunction, and implication, with different meanings. Prawitz' main idea is that these different meanings are given by a semantical framework that can be accepted by both parties. In a recent work, Ecumenical sequent calculi and a nested system were presented, and some very interesting proof theoretical properties of the systems were established. In this work we extend Prawitz' Ecumenical idea to alethic K-modalities.

cs.LO↗

arXiv2019

A Game Model for Proofs with Costs

We look at substructural calculi from a game semantic point of view, guided by certain intuitions about resource conscious and, more specifically, cost conscious reasoning. To this aim, we start with a game, where player I defends a claim corresponding to a (single-conclusion) sequent, while player II tries to refute that claim. Branching rules for additive connectives are modeled by choices of II, while branching for multiplicative connectives leads to splitting the game into parallel subgames, all of which have to be won by player I to succeed. The game comes into full swing by adding cost labels to assumptions, and a corresponding budget. Different proofs of the same end-sequent are interpreted as more or less expensive strategies for I to defend the corresponding claim. This leads to a new kind of labelled calculus, which can be seen as a fragment of SELL (subexponential linear logic). Finally, we generalize the concept of costs in proofs by using a semiring structure, illustrate our interpretation by examples and investigate some proof-theoretical properties.

cs.LO↗