Modal logic with non-deterministic semantics: Part I—Propositional case

Logic Journal of the IGPL 28 (3):281-315 (2020)
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Dugundji proved in 1940 that most parts of standard modal systems cannot be characterized by a single finite deterministic matrix. In the eighties, Ivlev proposed a semantics of four-valued non-deterministic matrices (which he called quasi-matrices), in order to characterize a hierarchy of weak modal logics without the necessitation rule. In a previous paper, we extended some systems of Ivlev’s hierarchy, also proposing weaker six-valued systems in which the (T) axiom was replaced by the deontic (D) axiom. In this paper, we propose even weaker systems, by eliminating both axioms, which are characterized by eight-valued non-deterministic matrices. In addition, we prove completeness for those new systems. It is natural to ask if a characterization by finite ordinary (deterministic) logical matrices would be possible for all those Ivlev-like systems. We will show that finite deterministic matrices do not characterize any of them.



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Dugundji’s Theorem Revisited.Marcelo E. Coniglio & Newton M. Peron - 2014 - Logica Universalis 8 (3-4):407-422.


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Marcelo E. Coniglio
University of Campinas

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Semantical Analysis of Modal Logic I. Normal Propositional Calculi.Saul A. Kripke - 1963 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 9 (5‐6):67-96.
Algebraic semantics for modal logics I.E. J. Lemmon - 1966 - Journal of Symbolic Logic 31 (1):46-65.

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