Linear diagrams for syllogisms (with relationals).
George Englebretsen · Notre Dame Journal of Formal Logic · 1991
A system for diagramming syllogisms is developed here.Unlike Venn, and other planar diagrams, these diagrams are linear.This allows one to diagram inferences which exceed the virtual four term limit on nonlinear systems.It also can be extended (by the use of vectors) to inferences involving all kinds of relational expressions.... be he a Triangle, Square, Pentagon, Hexagon, Circle, what you willa straight Line he looks and nothing else.E. A. Abbott, Flatland 1Euler and Venn diagrams are simple and effective devices for illustrating syllogistic validity.Their potential is limited, however, since they cannot apply to arguments with more than four terms.Attempts at extending the scope of plane figure diagrams (e.g., by Lewis Carroll) have been only marginally successful.(For a survey of such attempts, see Gardner [2].)Aristotle probably used some sort of diagram method in teaching the syllogistic.And many ancient commentaries made use of linear diagrams, though our understanding of just how they worked is sketchy (see Ross [13], pp.301-302 and Kneale [5], pp.71-72).If the ancient syllogists used linear diagrams rather than planar ones, and if they diagrammed not only simple syllogisms but sorites, polysyllogisms, and compound syllogisms, then it is likely that there is a satisfactory linear method for logical diagrams which can readily go beyond the virtual four term limit on plane diagrams.(In the eighteenth century, Lambert attempted linear diagrams for syllogistic [5] (pp.111-150).See the critiques of his systems by Venn [15] (pp.504-527), Peirce [7], and Keynes [4] (pp. 243-247).)Here, I will describe such a linear diagram method, illustrate some of its uses, and do what no syllogist has done before -extend the method to relations.I will conclude with a brief polemical remark in favor of syllogistic in general.