 Pictures and diagrams seem at first to hover in the middle ground between theories and models. In practice model theorists often draw themselves pictures of structures, and use the pictures to think about the structures. On the other hand pictures don’t generally carry the labelling that is an essential feature of model-theoretic structures. There is a fast growing body of work on reasoning with diagrams, and the overwhelming tendency of this work is to see pictures and diagrams as a form of language rather than as a form of structure. For example Eric Hammer and Norman Danner (1996) describe a ‘model theory of Venn diagrams’; the Venn diagrams themselves are the syntax, and the model theory is a set-theoretical explanation of their meaning.  (A curious counterexample is the horizontal line diagrams of the 12th century Baghdad Jewish scholar Abū l-Barakāt they represent structures and not propositions, and Abū l-Barakāt uses them to express model-theoretic consequence in syllogisms.  Further details are in Hodges 2018 on model-theoretic consequence.)