The Modal Cube Revisited: Semantics without Worlds

2025-05-19

AI Summary

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.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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.

Problem

• Develops non-deterministic semantics for modal cube logics
• Introduces modular multi-valued matrices with level valuations
• Links framework to Kripke semantics addressing axiom correspondences

Innovation

• Modular multi-valued non-deterministic matrices for modal logics
• Eight-valued system with sound and complete decision procedure
• Novel model-theoretic perspective linking to Kripke-style semantics