apply 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 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 lack of higher-order semantic frameworks for intuitionistic and non-classical modal logics. We introduce *higher-order Kripke models*, defining standard Kripke models as 0th-order, and recursively constructing an *n*-th-order model whose domain comprises all (*n*−1)-th-order models; modal operators are interpreted via accessibility relations between lower-order models. This recursive hierarchy enables the first higher-order abstraction of “possible worlds”, endowing them with the intuitive interpretation of “alternative timelines”. We construct first-order models for intuitionistic modal logic *IK* and a novel logic *MK*, verifying the modularity and extensibility of semantic clauses. The framework uniformly supports modal extensions of diverse non-classical logics—including intuitionistic, paraconsistent, and many-valued systems—and yields several open conjectures concerning expressive power and strong completeness.

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 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.

为解决知识图谱与基础模型推理方式不匹配的问题，提出语义贝叶斯世界模型，通过概率信念更新和行动干预实现统一推理架构。

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.

This study addresses the absence of modal extension semantics for generalized Nelson logics by constructing a Kripke-style modal extension for the semi-relevant logic RM based on Dunn’s binary relational semantics, which is subsequently generalized to the entire family of generalized Nelson logics. Methodologically, this work proposes a unified modal axiomatization framework that systematically integrates Kripke semantics with axiomatic techniques. The principal contribution lies in establishing soundness and completeness proofs for the modal extensions of all generalized Nelson logics. By bridging the modal gap in relevant epistemic logics, this research provides a general paradigm for the modalization of non-classical logics.

This work addresses the absence of a unified semantic framework for belief change that is independent of logical syntax, which has hindered systematic integration of classical and non-prioritized models. The paper proposes an abstract possible-world semantics (AWS) grounded in set theory, taking possible worlds as primitive elements and drawing on Grove’s sphere systems to define purely semantic operators for contraction and revision. This framework provides, for the first time, a logic-language-free unification of AGM, KM, and multiple belief change models. It not only simplifies and generalizes existing theories but also offers an isomorphic account of various belief change operations within propositional logic, thereby achieving a systematic and generalized foundation for belief set dynamics.

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.