A System of Paraconsistent Logic Equipped with Classical Negation

In a paper by Waragai and Shidori, a system of paraconsistent logic called PCL1 was proposed and a result showing the relation between PCL1 and the system of Modal Logic S5 was given. Though the classical negation plays an important role in the result, it was treated in an abstract way, so the present paper focuses on the concrete treatment of classical negation in PCL1 and makes the condition to enrich PCL1 with classical negation explicit. For this purpose, strong negation, which is defined by using a bottom particle in PCL1, is introduced and some of its results will be given. Also the notion of "behaving classically" in the extended system will be discussed.