The diamond of contraries. (February 2015) - Gottfried, Björn
Abstract: A number of approaches for logical reasoning with diagrams have been proposed. This paper considers the question, how the expressiveness of such systems can be raised without losing the visual power of less expressive diagrams. The antagonism between expressiveness and diagrammatic simplicity is coped with by a set of jointly exhaustive and contrary relations, modelling definite knowledge within a new diagrammatic representation. The restriction on actual knowledge reduces the expressiveness of these diagrams, but strengthens their visual power by avoiding ambiguities and by providing a close correspondence between diagrammatic syntax and set-theoretic semantics. The extension towards compound diagrams enables the representation of uncertain knowledge, but have a negative impact on the clarity of these diagrams. It is shown how formulae of monadic first-order logic are treated within this approach. Abstract : Highlights: Linear diagrams are formalised for reasoning about formulae of first order monadic logic. The diamond of contraries is introduced as a foundation for representing complete knowledge. Compound diagrams are introduced that allow for disjunctive information, and thus, for treating incomplete knowledge. The diagrams support the simplification of formulae, their comparison, and they enable to test their satisfiability. It is shown how syllogisms can be comprehended by these diagrams.