On the algebra of possibly paraconsistent sets

Abstract

In this paper we define the notion of possibly paraconsistent sets. We introduce algebraic operations
 on them and we analyze their properties. Moreover, it is shown that our class can be considered as isomorphic to
 the classes of intuitionistic and weak rough sets (with the assumption that intersection, union and complement
 are understood in an appropriate manner). Hence, the framework of paraconsistent sets can be treated as a
 new semantics for three-valued logic. Two less typical operations on possibly paraconsistent sets are studied
 too. They do not give us de Morgan algebra but rather a bisemilattice with only one absorption law. Finally,
 we pay attention to the fact that possibly paraconsistent sets can be treated as neutrosophic crisp sets of type
 2. As for the exact isomorphism, it should be a matter of further research.