Negation in the context of gaggle theory

Studia Logica 80 (2-3):235 - 264 (2005)
Abstract
We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn's kite of different negations can be dealt with in the extensions of this basic logic Ki. Next we treat negation as “unnecessity” and use a characteristic semantics for different negations in a kite which is dual to Dunn's original one. Ku is the minimal logic that has a characteristic semantics. We also show that Shramko's falsification logic FL can be incorporated into some extension of this basic logic Ku. Finally, we unite the two basic logics Ki and Ku together to get a negative modal logic K-, which is dual to the positive modal logic K+ in [7]. Shramko has suggested an extension of Dunn's kite and also a dual version in [12]. He also suggested combining them into a “united” kite. We give a united semantics for this united kite of negations.
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