Diagrammatic notations, such as Venn diagrams, Petri-Nets and finite state automata, are in common use in mathematics and computer science. While the semantic domain of such systems is usually well formalized, the visual notation itself seldom is, so that they cannot be used as valid devices of formal reasoning. A complete formalization of such notations requires the construction of diagram systems with rigorously defined syntax and semantics. We discuss how diagram specification can be interpreted as multiset rewriting and, based on this, how it can be formalized in linear logic. We discuss the power of our approach through an illustration of its possible extension with reflective capabilities to manage negative conditions, and through the identification of a class of diagrammatic transformations which can be directly expressed in our framework.