Perp and Star in the Light of Modal Logic

This paper is an exploration in the light of modal logic of Dunn’s ideas about two treatments of negation in non-classical logics: perp and star. We take negation as an impossibility modal operator and choose the base positive logic to be distributive lattice logic (DLL). It turns out that, if we add one De Morgan law and contraposition to DLL (call this system K−), then we can prove a natural completeness and hence treat perp in this modal setting. Moreover, star can be dealt with in the extensions of K−. Based on these results, a complete table of star and perp semantics for Dunn’s kite of negations is given. In the last section, we discuss perp and star in relevance logic and their related logics. The Routley star is interpreted at the end of this paper.