constructive neighbourhood semantics

This paper addresses the fundamental semantic divergence between CK and IK—two prominent intuitionistic modal logics—regarding the treatment of the possibility operator (◇), resolving their inconsistency over the ◇-free fragment and clarifying their conservativity over pure necessity (□) axiom systems. Methodologically, it extends CK’s Kripke semantics into a unified semantic framework, enabling the first precise characterization of frame conditions for IK and several classical axioms (e.g., T, 4, B). Building on this, the paper establishes definitive (non-)conservativity results for over a dozen intuitionistic modal logics with respect to ◇-free intuitionistic modal logic. All results are supported by machine-checked formal proofs in Coq, thereby settling multiple long-standing open problems on conservativity. The work provides a rigorous semantic foundation and principled guidance for axiomatization in intuitionistic modal logic.

Intuitionistic monotonic modal logic lacks a systematic formalization and semantic foundation. Method: This paper introduces IK—the first intuitionistic monotonic modal logic based on intuitionistic neighborhood semantics—and establishes its sound and complete axiomatization. (1) It proposes a dynamic neighborhood structure to model semantic evolution under intuitionistic accessibility; (2) it formalizes IK via a faithful translation into intuitionistic first-order logic; (3) it develops a multi-modal extension of IK and proves its embeddability into standard multi-modal IK. Contribution/Results: This work provides the first unified semantic and syntactic framework for intuitionistic monotonic modal logic, precisely characterizes its expressive boundaries relative to classical monotonic modal logic and other intuitionistic monotonic systems, and introduces a novel modeling paradigm for non-classical modal logics.

This work addresses the long-standing lack of a systematic semantic characterization and effective proof system for intuitionistic monotonic modal logic (IM). It introduces, for the first time, a constructive neighborhood semantics that provides exact models for IM and its natural extensions. Building upon the structured sequent calculus for classical monotonic modal logic M, the paper develops a tailored structured sequent calculus for IM. Through proof-theoretic analysis, the calculus is shown to be sound, complete, and decidable. Furthermore, the study uncovers a deep analogy between IM and intuitionistic modal logic K, thereby establishing IM as the faithful intuitionistic counterpart of M.

This paper addresses the semantic divide between classical and intuitionistic modal logic by introducing a novel frame semantics based on partially ordered “possibilities” instead of traditional possible worlds. Methodologically, it constructs possibility frames—interpreting formulas via regular open sets in Alexandrov topologies—and develops a categorical duality theory between such frames and non-atomic CV-Boolean algebraic operators (CV-BAOs). It establishes, for the first time, a duality between full possibility frames and CV-BAOs using filters rather than ultrafilters, thereby avoiding the Axiom of Choice; it also introduces principal possibility frames to characterize V-BAOs. The main contributions are: (i) unifying classical and intuitionistic modal semantics within a single framework; (ii) systematically establishing dualities between possibility frames and various classes of BAOs; (iii) completing definability, correspondence, and strong completeness theories; and (iv) proving that every BAO is fully characterized by a filter-descriptive possibility frame.

Prior proof-theoretic semantics for intuitionistic linear logic (ILL) were restricted to the multiplicative fragment and lacked a inferentialist account of the exponential modality “!”. Method: This paper introduces a novel proof-theoretic semantics for full ILL—including “!”—based on base extensions, providing the first systematic inferentialist interpretation of “!” that respects resource-sensitive reasoning. Contribution/Results: Within this framework, we establish soundness and completeness for ILL, precisely characterizing the structural role of “!” in resource-aware deduction. Our semantics fully covers all logical connectives and the “!” modality, thereby closing a longstanding theoretical gap in the proof-theoretic semantics of ILL. Moreover, the base-extension methodology offers a scalable paradigm for developing inferentialist semantics in substructural logics, advancing foundational work on meaning-as-use in resource-conscious systems.

Traditional Kripke semantics relies on classical reasoning, making it difficult to formalize within constructive type theory. While Goldblatt’s covering semantics offers a constructive alternative, it is constrained by a “modal locality” condition that complicates model construction. This work proposes a conservative extension of relational covering semantics that eliminates this restriction, enabling simpler and more standard model constructions and accommodating various intuitionistic modal logics featuring independent □ and ◇ operators. Building on this framework, we provide a constructive completeness proof that avoids intricate order-theoretic arguments and fully formalize both the semantics and multiple logical systems in Agda. This significantly enhances the applicability and formalizability of modal semantics within constructive settings.

This study addresses the problem of system collapse in the non-normal modal logic CLoN when weak negation coexists with deontic principles. To resolve this, the paper proposes a dual neighborhood semantic framework, equipping each modal operator with two neighborhood functions and integrating rejection-set techniques to adequately capture modal semantics under weak negation. This approach constitutes the first application of dual neighborhood semantics to CLoN, successfully validating non-trivial modal axioms involving weak negation. The resulting logical system accommodates standard deontic principles while formally representing moral dilemmas without trivialization, thereby establishing a robust semantic and axiomatic foundation for deontic logics capable of handling moral conflicts.

This study addresses the axiomatization of multimodal logics induced by three natural neighborhood functions within the framework of full product neighborhoods, focusing on the frame classes validating the modal logics T and D. By combining three monomodal logics and introducing an interaction axiom (mix), the authors construct the trimodal logics Tx+T and Dx+D. The work innovatively extends known product logic results for S4 and D4 to the weaker systems T and D, revealing that the role of (mix) varies across logical contexts: in S4, (mix) is equivalent to (sub), thereby enabling an axiomatization of the full product logic over topological spaces. The main contributions establish that Tx+T equals the product logic T×T×T augmented with (mix), and similarly, Dx+D equals D×D×D plus (mix).

This study addresses intuitionistic justification logic with satisfaction operators by introducing, for the first time, two distinct semantic frameworks: basic modular models and modular models incorporating Kripke semantics. These constructions provide a rigorous semantic foundation that transcends the limitations of prior syntax-driven approaches. By integrating modular semantic structures with possible-worlds semantics, the work establishes a realization theorem linking this justification logic to its corresponding intuitionistic modal logic at the semantic level. The research not only proves soundness and completeness for both model classes but also achieves the first semantic correspondence between intuitionistic justification logic with satisfaction operators and intuitionistic modal logic, thereby filling a significant gap in the semantic understanding of this logical system.

This study investigates the expressive power of intuitionistic modal logic IK within intuitionistic first-order logic. By introducing a notion of IK-bisimulation and combining relational semantics with model-theoretic techniques—including an intuitionistic version of Łoś’s theorem, elementary embeddings, and countable saturation—the work provides the first intrinsic characterization of IK: it precisely captures the fragment of intuitionistic first-order logic invariant under IK-bisimulation. This result establishes the exact model-theoretic status of IK and yields a Hennessy–Milner theorem in the intuitionistic setting, thereby furnishing a crucial tool for the development of intuitionistic first-order model theory.