Representation of Propositions and Reasonings. 13 



our plan than on the old one. The superiority, if any, in such 

 an example must rather be sought in the completeness of the 

 pictorial information in other respects — as, for instance, that, 

 of the four kinds of X which may have to be taken into con- 

 sideration, one only, viz. the XYZ, or the " X that is Y but 

 is not Z," is left surviving. Similarly with the possibilities of 

 Y and Z : the relative number of these, as compared with the 

 actualities permitted by the data, are detected at a glance. 

 As a more suitable example consider the following — 



{All X is either Y and Z, or not-Y, 

 If any XY is Z, then it is W, 

 Xo WX is YZ ; 



and suppose we are asked to exhibit the relation of X and Y 

 to one another as regards their inclusion and exclusion. The 

 problem is essentially of the same kind as the syllogistic one ; 

 but we certainly could not draw the figures in the same off- 

 hand way we did there. Since there are four terms, we sketch 

 the appropriate 4-ellipse figure, and then proceed to analyze 

 the premises in order to see what classes are destroyed by 

 them. The reader will readily see that thefirst premise anni- 

 hilates all " XY which is not Z," or XYZ ; the second de- 

 stroys "XYZ which is not W," or XYZW; and the third 

 " WX which is YZ," or WXYZ. Shade out these three 

 classes, and we see the resultant figure at once, viz. 



It is then evident that all XY has been thus made away with; 

 that is, X and Y must be mutually exclusive, or, as it would 

 commonly be thrown into propositional form, " No X is Y." 

 I will not say that it would be impossible to draw Eulerian 

 circles to represent all this, just as we draw them to represent 

 the various moods of the syllogism; but it would certainly be an 

 extremely intricate and perplexing task to do so. This is 

 mainly owing to the fact already alluded to, viz. that we can- 

 not break the process up conveniently into a series of easy 

 steps each of which shall be complete and accurate as for as it 

 goes. But it should be understood that the failure of the older 

 method is simply due to its attempted application to a some- 



