Modal Operators over Constructive Logic

The article deals with modal extensions of Nelson's constructive logic with strong negation N4. Different logics are constructed with respect to the notions of semantical and formal dualities introduced in Odintsov and Wansing (2004, First-Order Logic Revisited, 269–286) and to a new notion of negative semantical duality. Soundness and completeness are proved for each logic. It is also shown that all the logics are conservative extensions of N4, and they possess the disjunction property and the constructible falsity property.