A proof-theoretical investigation on diagrammatic reasoning

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Recently, logical reasoning based on diagrammatic or graphical representations has been investigated by researchers from various areas. For the study of diagrammatic reasoning, it is important to combine methods and analyses of traditional symbolic logic and cognitive science. In such studies, researches so far have concentrated on semantic studies on the static nature of diagrams, i. e., the nature at the level of representation. And there are few proof-theoretical investigations on the dynamic nature of diagrams, i. e., the nature at the level of proofs or at the level of manipulation of diagrams.In this study, I introduced a proof-theoretical framework to investigate the dynamic nature of diagrammatic representations, in particular of Euler and Venn diagrams. Then, I formalized, by applying one of the basic proof-theoretical techniques of logic translation, the notion of "free ride" in my proof-theoretical framework. Free ride is one of the most basic properties of diagrams that is mainly discussed in the literature of cognitive science as an account of inferential efficacy of diagrams. Then, based on the formalization of free ride, I investigated a proof-theoretical characterization of the structure of Euler diagrammatic proofs and that of Venn diagrammatic proofs, compared with the usual natural deduction proofs of the traditional symbolic logic.