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

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