<p>Over the past few years, <i>CL</i> diagrams have gained popularity in diagrammatic reasoning, drawing inspiration from Lange’s <i>C</i>ubus <i>L</i>ogicus. The intuitive understanding of <i>CL</i> diagrams is based on simple structures that are straightforward both to draw and to comprehend. This structure supports embedding of information and inferencing. Furthermore, these diagrams are more than just heuristic tools; <i>CL</i> diagrams can actually be extended to full blown formal systems. The present paper shows that a formal system for <i>CL</i> diagrams can have an expressivity similar to standard Boolean algebra of propositions or propositional logic without losing their intuitive properties. It is also possible to combine this interpretation of <i>CL</i> with elements from the class calculus. The syntax and semantics are presented separately, and the system’s soundness and completeness are proven using a variant of bitstring semantics.</p>