In the first part of this paper (Méndez and Robles 2008) a minimal and an intuitionistic negation is introduced in a wide spectrum of relevance logics extending Routley and Meyer's basic positive logic B+. It is proved that although all these logics have the characteristic paradoxes of consistency, they lack the K rule (and so, the K axiom). Negation is introduced with a propositional falsity constant. The aim of this paper is to build up logics definitionally equivalent to those in (...) the aforementioned paper, negation being now introduced with the unary connective. Relational ternary semantics are provided for the new logics and soundness and completeness results are proved. (shrink)