Modal Companions for the Special Extensions of Nelson’s Constructive Logic

The Belnapian version $$\mathsf{BS4}$$ of the normal modal logic $$\mathsf{S4}$$ is related to Nelson’s constructive logic $$\mathsf{N4}^{\bot}$$ in approximately the same way as the logic $$\mathsf{S4}$$ is related to the intuitionistic logic. For this reason, it is natural to define modal companions for logics extending $$\mathsf{N4}^{\bot}$$ as extensions of the Belnapian modal logic $$\mathsf{BS4}$$ . It is proved that, for every special extension $$L$$ of $$\mathsf{N4}^{\bot}$$ , the logic $$\tau^BL$$ , where $$\tau^B$$ is a natural modification of the mapping $$\tau$$ assigning the least modal companion to each superintuitionistic logic, is the least modal companion of $$L$$ in the lattice of extensions of $$\mathsf{BS4}$$ .