The intuitionistic description logic iALC was created to model and reason over the domain of Law based on Kelsenian Jurisprudence [1]. Over the past decade, this logic has been used in several ways to either model norms or formalise legal reasoning [2, 3, 4, 5, 6, 7, 8, 9, 10]. In this work we intend to complement previous research done with this logic by filling some gaps found while working with it. The first gap occurs in iALC needing an intuitive way to explain reasoning for non-logicians. It has a sound and complete (concerning intuitionistic conceptual models [3]) Sequent Calculus (SC) [6] that has seen less usage than expected due to its non-intuitive way of presenting a proof. We present a (quasi-)normalising, sound and complete (w.r.t. TBox validity for intuitionistic conceptual models) Natural Deduction (ND) System to cover this difficulty in explaining SC to non-logicians, especially those in the domain of Law, which are essential to us. We do not achieve full normalisation due to a kind of derivation which cannot be normalised - aside from this exception, the rest of the system can provide uniform derivations. The second gap is being unable to deal with non-monotonic reasoning (NMR). Usually, one considers monotonic reasoning, in which, if one can conclude something from a set of premises, there is no way to add another premise to avoid said conclusion. This is not the case in a court of law, for instance, in which different parties aim to convince a judge or jury of opposite consequences by adding different premises to the case itself. We provide an exploratory investigation of an extension of iALC to deal with NMR to represent legal reasoning in aspects of the Law, such as the judicial process, which is non-monotonic by nature. We present desirable properties and a possible application of such a system via a case study. We explain further the motivation for both the ND system and the NMR extension and the decisions taken for both.