Conclusion

Discussing the question “Why is paraconsistency worthy?” J.-Y. Béziau [11] emphasized that paraconsistent logic is an important contribution to the theory of negation and to modern logic in general. The distinction between triviality and inconsistency made in paraconsistent logic is similar to the distinction between implication and inference relation and allows one to elucidate new features of traditional logical notions. Suppose that the investigations presented in this book also contribute to the general theory of logical systems. For two explosive logics with the same positive fragment and with essentially different kinds of negation, we investigated how the lattice of extensions of a logic changes when the explosion axiom is deleted, i.e., if we pass from a logic to its paraconsistent analog. It turns out that in both cases the lattices of extensions extend in a rather regular manner. In the class of extensions of a paraconsistent logic, one can distinguish the subclass of explosive logics, i.e., the class of extensions of the original explosive logic; the subclass of logics, which can be used to represent the structures of contradictions in all extensions of the considered paraconsistent logic (see Remark after Proposition 10.2.5). Finally, all other logics can be obtained via a combination of logics from the above two subclasses. The manner of combination can be explicated via a suitable representation theory for algebras modelling the logics under consideration.