Logica Universalis 7 (1):71-86 (2013)

Marcelo E. Coniglio
University of Campinas
In this paper we introduce non-normal modal extensions of the sub-classical logics CLoN, CluN and CLaN, in the same way that S0.5 0 extends classical logic. The first modal system is both paraconsistent and paracomplete, while the second one is paraconsistent and the third is paracomplete. Despite being non-normal, these systems are sound and complete for a suitable Kripke semantics. We also show that these systems are appropriate for interpreting □ as “is provable in classical logic”. This allows us to recover the theorems of propositional classical logic within three sub-classical modal systems
Keywords Non-normal modal logics  paraconsistent logics  paracomplete logics
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DOI 10.1007/s11787-012-0076-3
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An Essay in Classical Modal Logic.Krister Segerberg - 1971 - Uppsala, Filosofiska Föreningen Och Filosofiska Institutionen Vid Uppsala Universitet.
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.
The Logic of Provability.George S. Boolos - 1993 - Cambridge University Press.
Algebraic Semantics for Modal Logics I.E. J. Lemmon - 1966 - Journal of Symbolic Logic 31 (1):46-65.

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