Proof-theoretical investigation of Venn diagrams: A logic translation and free rides

Abstract

In the literature on diagrammatic reasoning, Venn diagrams are abstractly formalized in terms of minimal regions. In view of the cognitive process to recognize Venn diagrams, we modify slightly the formalization by distinguishing conjunctive, negative, and disjunctive regions among possible regions in Venn diagrams. Then we study a logic translation of the Venn diagrammatic system with the aim of investigating how our inference rules are rendered to resolution calculus. We further investigate the free ride property of the Venn diagrammatic system. Free ride is one of the most basic properties of diagrammatic systems and it is mainly discussed in cognitive science literature as an account of the inferential efficacy of diagrams. The soundness of our translation shows that a free ride occurs between the Venn diagrammatic system and resolution calculus. Furthermore, our translation provides a more in-depth analysis of the free ride. In particular, we calculate how many pieces of information are obtained in the manipulation of Venn diagrams. © 2012 Springer-Verlag.