4 Mr. Venn on the Diagrammatic and Mechanical 



this immensely simplifies the problem. "We can think of each 

 pair of circles without troubling ourselves about the other 

 pair ; we have nothing resembling implicit equations. But 

 suppose that, on the other hand, wo had a statement of the 

 relation of X to Y and Z combined with others giving that of 

 Y to Z and W, and, say, X to W, wo should hardly know 

 where to beo-in. Each statement beino- interlinked with the 

 others, no one of them could be disentangled and represented 

 separately. Xo doubt when the problem had been solved some- 

 how, and a full determination secured of the mutual relations 

 of the various classes, we could then set about undertaking to 

 draw our circles. But this is a very different thing from 

 working by help of the diagrams and employing them to aid 

 our conceptions in the actual task of solution. The simple 

 fact is that on this scheme, as already remarked, we have 

 no means of exhibiting imperfect knowledge. What is exhi- 

 bited is the final outcome of the relation, the actual exclusion 

 or inclusion of the classes ; and consequently we cannot repre- 

 sent our partial knowledge or the steps by which we attain to 

 complete information. This defect comes out even in such a 

 simple case as the ordinary disjunctive proposition " Every X 

 is either Y or Z." Such a statement gives us no information 

 as to the mutual relations of Y to Z ; and therefore, since we 

 have no means of marking by aid of our circles any thing but 

 the actual relations of these classes, we should have to draw 

 out a complete scheme of all the possibilities. This would 

 demand, to begin with, five different figures displaying the 

 five possible relations of Y to Z. We should then have to 

 proceed to draw our X circle in each case, applying it as well 

 as we could to each of these different figures. It will not need 

 a moment's consideration to realize how tedious and compli- 

 cated such a process would soon become when several class 

 terms have thus to be combined. 



We must therefore cast about for some new scheme of dia- 

 grammatic representation which shall be competent to indi- 

 cate imperfect knowledge on our part ; for this will at once 

 enable us to appeal to it step by step in the process of work- 

 ing out our conclusions. I have never seen any hint at 

 such a scheme, though the want seems • so evident that one 

 would suppose that something of the kind must have been 

 proposed before. The one here offered may be said to underlie 

 Boole's method *, and to be the appropriate diagrammatic 



* I tried at first, as others have done, to represent the complicated 

 propositions, there introduced, "by the old plan ; but the representations 

 failed altogether to answer the desired purpose ; and after some conside- 

 ration I hit upon the plan here described. 



