Many do not doubt that Peirce's Existential Graphs are diagrammatic, as opposed to symbolic. However, when we are pressured to draw a distinction between the two different forms of representation, we find ourselves at a loss and our intuition quite vague. In this paper, I locate fundamental differences between two logically equivalent systems, Peirce's Alpha system and propositional languages. Suppose we have only two sentential connectives, ¬ and ^. In spite of its truth-functional completeness, we don't want to use this (...) language for the translation of English sentences or as a deductive calculus. We would adopt this language only when we intend to develop logical theories. That is, it is convenient to have fewer connectives for a meta-theory, but not for practical use. So, there seems to be a trade-off between a language with fewer connectives and a language with more connectives. This view has been accepted without question. In this paper, I will argue that this trade-off is limited to linear symbolic systems and that we could have a diagrammatic system with fewer operations but no need to suffer from problems like those of a sentential language with fewer connectives. How is that possible? A comparison between Peirce's Alpha system and a propositional language is presented to answer this question. The case study identifies the following unique property of a non-linear diagrammatic system as a main source of the discrepancy between two different types of representation: One and the same diagram can be read off in more than one way by carving it up in many ways, but without ambiguity. (shrink)