On the Insufficiency of Linear Diagrams for Syllogisms
In Volume 33:1 of the Notre Dame Journal of Formal Logic, a system for diagramming syllogistic inferences using straight line segments is presented by Englebretsen. In light of recent research on the representational power of diagrammatic representation systems by the authors, we point out some problems with the proposal, and indeed, with any proposal for representing logically possible situations diagrammatically. We shall first outline the proposed linear diagrammatic system of Englebretsen, and then show by means of counterexamples that it is inadequate as a representation scheme for general logical inferences (the task for which the system is intended). We also show that modifications to the system fail to remedy the problems. The considerations we present are not limited to the particular proposal of Englebretsen; we thus draw a more general moral about the use of spatial relations in representation systems. 1 Diagrammatic representation systems Diagrammatic representation systems are of increasing interest for at least two reasons. Philosophically, diagram systems interest those concerned with the nature of representation itself—in particular, those who argue that too much attention has been given to sequential symbol systems. These writers claim that diagrams represent by analogy or surrogacy—in virtue of sharing structure with the domains that they represent (see Barwise and Shimojima [2], Cummins [4], and Swoyer [21]). Practically, diagrammatic representations are frequently used in visual interfaces to databases, programming languages, and in logic teaching. Each of these domains demands careful consideration of the formal properties of the diagrammatic systems in question. For both of these reasons we propose to investigate the expressive power of one proposed diagram system and to determine its utility in reasoning tasks. 2 The diagrammatic system LD In [5], Englebretsen presents us with a system for diagramming syllogistic inferences. In this system, individuals are represented Received September 24, 1997; revised October 31, 1999 574 OLIVER LEMON and IAN PRATT