Two 4-valued implicative expansions of first-degree entailment logic: The relevant logic BN4VSP and the (relevant) entailment logic BN4AP

A logic L has the ‘variable-sharing property’ (VSP) if in all L-theorems of the form $A\rightarrow B$ , $A$ and $B$ share at least a propositional variable. A logic L has the ‘Ackermann property’ (AP) if in all L-theorems of the form $A\rightarrow (B\rightarrow C)$ , $A$ contains at least a conditional connective ( $\rightarrow $ ). Anderson and Belnap consider the VSP a necessary property of any relevant logic, and both the VSP and the AP necessary properties of any (relevant) entailment logic. Now, among relevant logicians, Brady's logic BN4 is widely viewed as the adequate 4-valued implicative logic. But BN4 lacks the VSP and the AP. The aim of this paper is to define the logics BN4 $^{\text {VSP}}$ and BN4 $^{\text {AP}}$ . The former one has the VSP, whereas the latter one has the VSP and the AP. Moreover, both logics have some properties that do not support their consideration as mere artificial constructs.