N4┴-Lattices

In the previous chapter, we considered two variants of Nelson’s paraconsistent logic, N4 and N4┴, which were defined in different languages. N4┴ is a conservative extension of N4 in the language with the additional constant ┴ allowing us to define in this logic the intuitionistic negation. The addition of this constant results in an easy modification of the semantics. Models of N4 are isomorphic to twist-structures over implicative lattices and due to the fact that implicative lattices do not necessarily have the least element and the lattices modelling N4 do not have the greatest element either. In fact, no constant can be naturally defined in N4. Extending the language with ┴ we obtain the class of models isomorphic to twist-structures over Heyting algebras, i.e., implicative lattices with the least element 0. A twist-structure over bounded lattice is also bounded, it contains necessarily elements (0, 1) and (1, 0), which are the least and the greatest elements. Thus, the semantics for N4┴ is given by the class of bounded lattices. It turns out that the introduction of ┴ has essential consequences for the class of extensions εN4┴. As we will see in Chapter 10, adding the constant ┴ enriches the class of N4┴-extensions as compared to εN4, and, which is more important, provides it with a regular structure close to some extent to the structure of the class of extensions of minimal logic studied in the first part of the book. This is why we will work mainly with the logic N4┴ and with N4┴-lattices. However, all results in this chapter remains true if we replace "N4┴-lattice" with "N4-lattice" and “Heyting algebra” with “implicative lattice”.