Sequent Systems for Negative Modalities
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Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used for handling reasoning under uncertainty caused by inconsistency or undeterminedness. Using such straightforward semantics, we study the classes of frames characterized by seriality, reflexivity, functionality, symmetry, transitivity, and some combinations thereof, and discuss what they reveal about sub-classical properties of negation. To the logics thereby characterized we apply a general mechanism that allows one to endow them with analytic ordinary sequent systems, most of which are even cut-free. We also investigate the exact circumstances that allow for classical negation to be explicitly defined inside our logics.
KeywordsNegative modalities sequent systems cut-admissibility Analyticity
Mathematics Subject Classification03B53 03B60 03B45 03F05
Open access funding provided by Max Planck Society. The authors acknowledge partial support by CNPq, by The Israel Science Foundation (Grant No. 817-15), by the Marie Curie project GeTFun (PIRSES-GA-2012-318986) funded by EU-FP7, and by the Humboldt Foundation. They also take the chance to thank Elaine Pimentel, Heinrich Wansing, Alessandra Palmigiano, Dorota Leszczyńska-Jasion, and two anonymous referees for their comments on an earlier incarnation of this manuscript.
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