Atomic Entailment and Atomic Inconsistency and Classical Entailment
  In this paper we put forward a new solution of the well-known problem of relevant logics, i.e. we construct an atomic entailment. Hence, we construct a system of predicate calculus based on the atomic entailment. Next, we establish the definition of atomic inconsistency. The atomic inconsistency establishes an infinite class of inconsistent, but non-trivial systems. In this paper we construct the new definition of the classical entailment, into the bargain. 
T. J. Stepien, L. T. Stepien
math.LO