stratified modal logic formalization

This study addresses the distinction between logical contradiction and metaphysical impossibility, aiming to enable non-trivial reasoning about counterfactuals with impossible antecedents. To this end, it proposes a stratified possible worlds semantics that partitions worlds into logically normal and anti-logical ones, and introduces a metaphysical accessibility relation within the normal worlds. Coupled with a non-vacuous choice function mechanism, this framework avoids the vacuity problem commonly arising in counterfactual conditionals. Building on this semantics, the formal systems SCP and SCP1 are developed, for which soundness, completeness, and decidability are established. The resulting framework not only provides a fine-grained analysis of distinct kinds of impossibility but also demonstrates promising potential for extension to deontic and epistemic logics.

This paper addresses the foundational challenge of constructing a categorical framework for first-order modal logic, specifically how modal operators directly act on subobjects and interact with background factorization systems to model relational semantics. Method: It introduces, for the first time, a modal (quasi-)elementary topos structure; establishes a Joyal-style representation theorem formalizing “counterpart” semantics; and enhances categorical completeness via quotients and coproducts. Contributions/Results: It provides a systematic syntactic-to-categorical construction of first-order modal theories; delivers categorical characterizations of saturation conditions and definability problems; and furnishes modal logic with a unified, higher-order, and semantically rich categorical foundation—bridging syntax, semantics, and category theory in a principled way.

When extending lattice-based non-classical logics to modal languages, the semantic interpretation of the necessity operator □ lacks uniqueness. Method: We propose and formalize a natural algebraic interpretation: □φ is true at a world iff φ holds with value equal to the lattice meet (greatest lower bound) over all accessible worlds. Integrating algebraic logic (lattice theory) with possible-worlds semantics, we systematically compare logical properties across distinct semantic frameworks. Contributions: First, we establish necessary and sufficient conditions for the resulting modal system to validate axiom K under this meet-based interpretation. Second, we characterize how lattice-theoretic properties—such as completeness, distributivity, and compactness—affect the validity of modal principles (e.g., T, S4). Third, we uncover deep correspondences between lattice algebraic structure and modal operator semantics, thereby providing a unified semantic framework and foundational metatheoretic analysis for lattice-based modal logic.

This work addresses the limitations of traditional stratified definitions in logic, where the prohibition of negatively occurring defined predicates restricts expressive power for advanced applications such as relational reasoning. By relaxing the stratification condition, the paper integrates generic (nabla) quantification and general induction into an extended logical framework called G, achieving—for the first time—compatibility between weakly stratified definitions and these two mechanisms. Relying on monotonic fixed-point semantics and structural induction, the authors establish that this extension preserves logical consistency while substantially enhancing the capacity for formal reasoning about properties of programming languages. This result provides a theoretical foundation for extending the Abella proof assistant to support a broader class of inductive definitions.

This paper addresses the conceptual fragmentation in sequent-style proof systems across modal, temporal, intuitionistic, conditional, and substructural logics, as well as the ill-defined distinction between “internal” and “external” calculi. We propose a unified classification framework grounded in the fundamental data structure of sequents. By systematically analyzing extant sequent calculi, we develop a hierarchical taxonomy and conduct meta-theoretic analysis of upward and downward logical translations. Crucially, we formally define—and thereby eliminate—the spurious internal/external distinction for the first time. We prove that upward translation preserves structural integrity, whereas downward translation is generally infeasible. Our results establish a principled comparability benchmark and formal assessment toolkit for cross-logical proof systems, advancing the unified modeling of structural proof theory and enabling robust automation support.

This study addresses the problem of systematically constructing logical systems corresponding to program abstractions to support formal reasoning. By associating logical systems with finite abstract domains, the work proposes a general method: for a given abstract lattice, it constructs a logic whose Lindenbaum–Tarski algebra is isomorphic to the abstraction, and derives corresponding axioms and inference rules. This approach establishes, for the first time, a systematic connection between abstract interpretation and proof theory as well as algebraic logic, enabling logical modeling of non-Cartesian abstractions such as octagons. The resulting logical connectives and inference systems preserve the concretization map and, under suitable conditions, satisfy soundness and completeness. The framework naturally extends to Cartesian products, multi-variable settings, and non-Cartesian abstract domains.

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 work investigates the syntactic embedding of the classical modal logic S4 into Boolean Bunched Implication logic (BBI), preserving validity under hypothetical reasoning and axiomatic extensions. Inspired by Gödel’s embedding of intuitionistic logic into S4, the authors present, for the first time, a complete Hilbert-style deductive theorem proof for BBI. Building on this result, they construct robust syntactic embeddings of S4 into BBI and several of its extensions—including hybrid BBI and classical BI. The proposed method uniformly applies to arbitrary axiomatic extensions of both S4 and BBI, thereby uncovering deep semantic connections between modal logics and resource-sensitive logics.

This study addresses the unification of implication structures in intuitionistic logic and orthomodular logic while preserving both constructivity and quantum logical features. To this end, it introduces Sasaki implication as the central connective and constructs a novel logical system, termed iEx-logic, which is axiomatized for the first time. Employing methods from algebraic logic—integrating lattice theory, orthomodular algebras, and intermediate logics—the paper demonstrates that the lattice of extensions of iEx-logic decomposes as the direct product of the lattice of intermediate logics and the lattice of orthomodular logics. This work not only achieves a refined integration of these two logical frameworks but also reveals the robustness and decomposability of their underlying algebraic structure.

This study addresses the structural relationship between relevant logic and normal modal logic by systematically developing, for the first time, a formal translation mechanism from relevant logic into normal modal logic. By integrating semantic analyses from both logical frameworks, the work rigorously establishes a precise correspondence between their structures. The resulting framework not only uncovers deep connections between the two systems and yields several significant corollaries, but also opens new avenues for comparative research across logical systems. Furthermore, it identifies promising directions for future investigation that warrant deeper exploration.