The paper discusses a proposal of a formal description of ARD+ a conceptual design and hierarchical refinement method for knowledge-based systems. Formalization of method syntax is given, as well as description of its semantics. ARD+ is a rework of the classic ARD presented elsewhere, that provides a conceptual design for rule-based systems. ARD+ is based on the concept of gradual design where consecutive design steps correspond to different knowledge abstraction levels in the design. Application examples, along with directions for future research are given.