The Modal Cube Revisited: Semantics without Worlds (Technical Report)

TL;DR

This paper introduces a modular, non-deterministic semantic framework for all modal logics in the modal cube, providing a unified, sound, and complete decision procedure that links to relational semantics without relying on possible worlds.

Contribution

It develops a novel multi-valued Nmatrix approach for modal logics, extending prior semantics, and establishes a new connection to relational models, offering a robust alternative to possible-world semantics.

Findings

Provides a sound and complete decision procedure for each modal logic.

Introduces an eight-valued non-deterministic semantic system.

Links non-deterministic semantics to relational models, addressing longstanding conjectures.

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.