Ordinarily in logical studies, logical consequence is abstractly represented in the form of a closure operator on the formula algebra or, more generally on any algebra of a specific type. This representation requires that the logic satisfies the axioms of inflationarity and monotonicity. However, in a number of applied contexts one needs to study logical systems that do not necessarily conform to these requirements, most notably non-monotonic logics. We provide the rudiments of an abstract experimental framework that could serve as a platform for formalizing and studying an abstract algebraic hierarchy of non-monotonic logics akin to the Leibniz hierarchy, which forms the cornerstone of the traditional theory of Abstract Algebraic Logic.