# Diagrammatic Reasoning

Gerard Allwein

Diagrammatic reasoning, or better, reasoning with diagrams, has been with humans at least since the first written forms of communication. It probably began with the first person to draw a map to explain to another person how to get from here to there. The central feature is one of abstraction. The symbol in the diagram represents an abstraction of something. A diagram denotes diagrammatically when the internal structure of the diagram reflects either geometric or logical structure of the thing it denotes. How is this different from how a formal sentence denotes? It depends on whether the sentence is denoting a proposition or if it is denoting a truth value. If the former, then sentence is denoting diagrammatically the logical structure of the proposition. If the latter, then the sentence is not denoting diagrammatically, but instead denotes truth functionally. Put differently, there is no internal structure to a truth value which connects with the form of the sentence. In this instance, the form of the sentence might better be said as denoting the route to the truth value, and is what Frege called sense. In the sequel, I will use sentential representations in the sense of denoting truth values. In their guise as representing logical form, I will treat them as diagrammatic representations.