Abstract In the age of artificial intelligence (AI), the endeavour to translate legal concepts into machine language and leverage technology within legal systems heralds a fundamental transformation. However, the inherent challenges within this domain, particularly when confronted with the non-computable aspects of law, persist as significant obstacles. This article, by examining the structural and philosophical limitations of standard AI, introduces the concept of ‘non-computable law’—a concept positing that certain dimensions of law, such as human judgment, ethics, volition, and consciousness, transcend computational frameworks. Drawing upon novel approaches like quantum consciousness theories and biological technologies, this article explores potential foundations for creating a new generation of AI that is not only capable of processing legal data but also intrinsically capable of engaging with non-computable concepts. This article argues that designing intelligent systems based on artificial consciousness and cognition is a crucial step towards addressing the hard problems of computational law and realizing justice within complex legal systems. This research not only clarifies existing limitations but also offers strategies for the development of conscious AI, which could revolutionize the future trajectory of legal science and technology.