14 Mr, Venn on the Diagrammatic and Mechanical 



what more complicated set of data than those for which it was 

 designed. But these data are really of the same kind as when 

 we take the two propositions " All X is Y," " All Y is Z," and 

 draw the customary figure. When the problem, however, has 

 been otherwise solved, it is easy enough to draw a figure of the 

 Qld-fashioned ; or " inclusion-and-exclusion " kind, to represent 



the result, as follows, ^^^J Y ; DU ^ one may safely 



z 

 assert that not many persons would have seen their way to 

 drawing it at first hand for themselves*. 



One main source of aid which diagrams can afford is worth 

 noticing here. It is that sort of visual aid which is their 

 especial function. Take the following problem : — " Every X is 

 either Y or Z ; every Y is either Z or W ; every Z is either 

 W or X ; and every W is either X or Y: what further con- 

 dition, if any, is needed in order to ensure that every XY shall 

 be W ? " It is readily seen that the first statement abolishes 

 any X that is neither Y nor Z, and similarly with the others ; 

 so that the four abolished classes are XYZ, YZ W, ZWX, and 

 WXY. Shade them out in our diagram, and it stands thus : — 



It is then obvious that, of the surviving component parts of 

 XY, one only (viz. XYZW) is not W. If, then, this be de- 

 stroyed, all XY will be "W ; that is, the necessary and suffi- 

 cient condition is that "all XYZ is W." 



* Even then we Lave said more in this figure than we are entitled to 

 say. For instance, we have implied that there is some X which is W, 

 and so forth. The other scheme does not thus commit us ; for though the 

 extinction of a class is final, its being let alone merely spares it condition- 

 ally. It holds its life subject to the sentence, it may be, of more premises 

 to come. This must be noticed, as it is an important distinction between 

 the customary plan and the one here proposed. The latter makes the di- 

 stinction between rejection and non-rejection — such non-rejection being 

 provisional, and not necessarily indicating ultimate acceptance. The 

 former has to make the distinction between rejection and acceptance; for 

 the circles must either intersect or not, and their non-intersection indicates 

 the definite abandonment of the class common to both. Hence the prac- 

 tical impossibility of appealing to such diagrams for aid in representing 

 complicated groups of propositions. 



