Proceedings of the Annual Meeting of the Cognitive Science Society

Researchers currently seek to explain the observed tractability of diagrammatic reasoning (DR) via the notions of "limited abstraction" and inexpressivity (Stenning and Oberlander, 1995; Stenning and Inder, 1995). We point out that these explanations are inadequate, in that they assume that each structure to be represented (i.e. each model) has a corresponding diagram. We show that inefficacy (in the sense of incorrectness) arises in DR because some (logically possible) models fail to have corresponding diagrams, due to non-trivial spatial constraints. Further, there are good explanations of why certain restricted languages are tractable, and we look to complexity theory to establish such results. The idea is that graphical representation systems may be fruitfully analysed as certain restricted quantifier fragments of first-order logic, similar to modal logics and vivid knowledge bases (Levesque, 1986; Levesque, 1988). This focus raises some problems for the expressive power of graphical systems, related to their topological and geometrical properties. A simple case study is carried out, which pinpoints the inexpressiveness of Euler's Circles and its variants. We conclude that there is little mileage in spatial (i.e. diagrammatic) approaches to abstract reasoning, except perhaps in relation to studies of human performance. Moreover, these results have ramifications for certain claims about mental representations, and the recent trend in cognitive semantics, where "meanings" and "concepts" are to be explicated spatially. We show that there should be combinations of "concepts" or "meanings" which are prohibited by the structure of the spaces they supposedly inhabit. The formal results thus suggest an empirical programme.