Conservativity for a hierarchy of Euler and Venn reasoning systems

This paper introduces a hierarchy of Euler and Venn diagrammatic reasoning systems in terms of their expressive powers in topological-relation-based formalization. At the bottom of the hierarchy is the Euler diagrammatic system introduced in Mineshima-Okada-Sato-Takemura [13, 12], which is expressive enough to characterize syllogistic reasoning in terms of unification and deletion rules. At the top of the hierarchy is a Venn diagrammatic system such as Swoboda-Allwein’s Euler/Venn diagrammatic system [23]. In order to understand the hierarchy uniformly, we introduce an algebraic structure, which also provides another description of our unification rule of Euler diagrams. We prove that each system S’ of the hierarchy is conservative over any lower system S with respect to validity—in the sense that S’ is an extension of S, and the semantic consequence relations of S and S’ are equivalent for diagrams of S. Furthermore, we prove that a region-based Venn diagrammatic system is conservative over our topological-relation-based Euler diagrammatic system with respect to provability.