Negative modalities in the light of paraconsistency

Hitoshi Omori

Toshiharu Waragai

pp. 539-555

in: Arnold Koslow, Arthur Buchsbaum (eds), The road to universal logic II, Basel, Birkhäuser, 2015

Abstract

Modality and non-classical negation have some interesting connections. One of the most famous connections is the relation between S4-modality and intuitionistic negation. In this chapter, we focus on the negative modalities in the perspective of paraconsistency. The basic idea here is to consider the negative modality defined as "not necessarily" or equivalently "possibly not" where "not" is classical negation, and "necessarily" and "possibly" are modalities in modal logics. This chapter offers a solution to the problem of axiomatizing systems of modal logic such as D and S4 in terms of negative modalities. One of the upshots of this solution is that we may consider the semantics of paraconsistency with the help of various considerations known in the literature of modal logics related to D and S4.