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arXiv · 2505.12824

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

Renato Leme·Carlos Olarte·Elaine Pimentel·Marcelo E. Coniglio
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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.

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Renato Leme, Carlos Olarte, Elaine Pimentel, Marcelo E. Coniglio. 2025-05-19. The Modal Cube Revisited: Semantics without Worlds (Technical Report). https://arxiv.org/abs/2505.12824

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