In Anderson and Belnap (1976), Meyer stated that logics which hold a weak version of the variable sharing property –referred to here as quasi-relevant property, QRP– are good enough when some degree of relevance is needed. In this article, research regarding the relation between the QRP and 4-valued matrices is conducted. The aim is threefold: (1) to define the class of 4-valued quasi-relevant matrices and to determine which of those hold a natural conditional; (2) to specify which matrices from the set established in 1 verify Routley and Meyer basic logic B; (3) to strengthen these logics with truth-functional modal operators following a Łukasiewiczean approach to modal notions.
