kripke semantics

This paper addresses the problem of providing a unified semantic framework for modal logics within the modal cube under non-deterministic environments, without recourse to possible worlds. Methodologically, it introduces a novel semantics based on multi-valued non-deterministic matrices (Nmatrices), specifically employing an eight-valued Nmatrix coupled with a hierarchical valuation scheme to modularly and worldlessly characterize necessity; it further establishes a rigorous correspondence between this semantics and standard Kripke semantics via combined algebraic and model-theoretic techniques. Key contributions include: (i) the first sound and complete worldless semantics for the entire modal cube; (ii) a definitive resolution of long-standing conjectures concerning the correspondence between modal axioms and semantic conditions; (iii) decidable proof procedures for all logics covered; and (iv) subsumption of prior work—e.g., Kearns’ semantics—as a special case, thereby establishing a new semantic foundation that is both philosophically robust and technically scalable.

This study addresses the challenge of effectively transferring soundness and completeness from highly expressive modal logics to weaker linguistic systems. To this end, it proposes a novel strategy that leverages semantic insensitivity to carry over soundness and employs faithful translations to embed the canonical model of the target logic into the framework of normal modal logic, thereby inheriting completeness. The approach unifies various notions of operator definability and allows the standard relational semantics to be inherited without explicitly specifying an accessibility relation. Consequently, it offers a general and streamlined method for constructing semantics for weak modal languages, substantially reducing the complexity of their metatheoretic analysis.

This work addresses the lack of a unified theoretical framework for bundled modalities in propositional modal logic, which has hindered systematic analyses of their expressivity and axiomatization. The paper proposes a general framework that defines bisimulation relations for arbitrary bundled modalities and introduces the class of convex bundled modalities together with their convex neighborhood semantics, thereby unifying expressivity analysis and axiomatization. It innovatively establishes a method for verifying the Hennessy–Milner property and successfully provides complete axiomatizations for three representative bundled modalities—“someone knows,” “group disagreement,” and “belief without knowledge”—corresponding respectively to S5, KD45, and S4.2 models. This constitutes the first systematic theoretical foundation for this area.

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.

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.

This study addresses the complexity in point-based constructions arising from varying morphism definitions in modal logic by establishing a Stone-type duality between an algebraic category equipped with paired modal operators and a category of topological spaces endowed with binary relations. By introducing a semi-continuity condition on relations, the work reveals a direct correspondence between modal axioms and relational properties of the underlying spaces, thereby significantly simplifying point-set manipulations in traditional dualities. This approach not only unifies several existing dualities between modal frameworks and topological semantics but also provides a precise bridge between algebraic and relational semantics, offering new categorical tools for the systematic study of modal logics.

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 work addresses the lack of the finite model property in traditional Gödel modal logics under standard Kripke semantics, which stems from their reliance on limit behaviors that yield non-constructive interpretations. To overcome this limitation, the paper introduces GW logic, equipped with a novel witnessed Kripke semantics that requires the truth of every modal formula to be explicitly witnessed by some accessible world, thereby eliminating non-constructive limit cases. Building on this semantics, the authors establish the first Gödel modal logic framework enjoying the finite model property and develop a corresponding refutation calculus together with a terminating backward proof-search algorithm. The calculus is proven sound and complete, enabling automated reasoning and countermodel generation, and substantially enhancing the constructivity and computability of the logical system.

This study investigates the interplay between expressive power and computational complexity in restricted operator fragments of propositional and modal logics. By integrating Post’s lattice theory with modal logic frameworks, the work introduces the notion of “simple modal fragments,” extending Boolean clone theory to the modal setting and establishing a unified parameterized analysis methodology. The paper systematically characterizes the boundaries of these fragments—parameterized by admissible logical operators—with respect to decidability, computational complexity (including dichotomy results), and learnability (encompassing teachability and exact learnability). This approach unifies two long-standing, independent research directions, offering a cohesive theoretical framework for understanding the structure and learnability of logical fragments.

This work establishes a duality between relations among computational systems—such as bisimulation—and relations among logical predicates, thereby enabling cross-system logical reasoning. By extending Tarski duality and Thomason duality to the relational level for the first time, and integrating tools from category theory, Kripke semantics, and infinitary modal logic, the authors construct a dual framework that systematically links system relations with predicate relations. Building on this foundation, they develop a novel proof system capable of formally relating formulas across distinct systems. The resulting framework provides a robust theoretical basis for program logics and verification of concurrent systems, while significantly broadening the scope of classical duality theory within relational semantics.