 This approach has posed a fundamental challenge to some of the assumptions held about representation systems. Since the development of modern logic, important concepts, e.g., syntax, semantics, inference, logical consequence, validity, and completeness, have been applied to sentential representation systems only. However, none of these turned out to be intrinsic to these traditional symbolic logics only. For any representation system, whether it is sentential or diagrammatic, we can discuss two levels, a syntactic and a semantic level. What inference rules tell us is how to manipulate a given unit, whether symbolic or diagrammatic, to another. The definition of logical consequence is also free from any specific form of a representation system. The same argument goes for the soundness and the completeness proofs. When a system is proven to be sound, we should be able to adopt it in proofs. In fact, much current research explores the use of diagrams in automated theorem proving (see Barker-Plummer &amp; Bailin  1997;  and Jamnik et al.  1999).