Absurdity as unary operator

It was shown in the previous work of the author that one can avoid the paradox of minimal logic { ϕ , ¬ ϕ } ¬ ψ defining the negation operator via reduction not a constant of absurdity, but to a unary operator of absurdity. In the present article we study in details what does it mean that negation in a logical system can be represented via an absurdity or contradiction operator. We distinguish different sorts of such presentations. Finally, we consider the possibility to represent the negation via absurdity and contradiction operators in such well known systems of paraconsistent logic as D.Batens's logic CLuN and Sette's logic P 1.