The Modal Cube Revisited: Semantics without Worlds (Technical Report)
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.
Cite
@article{arxiv.2505.12824,
title = {The Modal Cube Revisited: Semantics without Worlds (Technical Report)},
author = {Renato Leme and Carlos Olarte and Elaine Pimentel and Marcelo E. Coniglio},
journal= {arXiv preprint arXiv:2505.12824},
year = {2025}
}