Representation of Propositions and Reasonings, 3 



number of possible resultant alternatives would be very con- 

 siderable. 



For instance, the proposition " All X is Y" needs both the 



diagrams, (X,YJ if x j Y ) ; for we cannot tell, from the mere 



verbal statement, whether there are any Y's which are not X. 

 Similarly the proposition " Some X is not Z" needs three 

 other diagrams, 



(These five relations, it may be remarked, comprise all the pos- 

 sible ways in which two terms may stand to one another.) 

 Hence the combination of the two given premises could not be 

 adequately represented by less than six figures. If more pre- 

 mises, and more complicated ones (such as we shall presently 

 proceed to illustrate), are introduced, the consequent confusion 

 would bo very serious. The fact is, as I have explained at 

 length in the article above referred to, that the five distinct 

 relations of classes to one another (viz. the inclusion of X in 

 Y, their coextension, the inclusion of Y in X, their intersec- 

 tion, and their mutual exclusion), which are thus pictured 

 by these circular diagrams, rest upon a totally distinct view 

 as to the import of a proposition from that which underlies 

 the statements of common life and common logic. The latter 

 statements naturally fall into four forms — the universal and 

 particular, affirmative and negative ; and it is quite impossible 

 to make the five divisions of the one scheme fit in harmoni- 

 ously with the four of the other. 



The second objection to which this scheme is obnoxious is 

 of a more practical character ; and viewed in that light it is, if 

 any thing, of a still more serious character. It consists in the 

 fact that we cannot readily break up a complicated problem 

 into successive steps which can be taken independently. We 

 have, in fact, to solve the problem first, by determining what 

 are the actual mutual relations of the classes involved, and 

 then to draw the circles to represent this final result; we cannot 

 work step by step towards the conclusion by aid of our figures. 



The extremely simple examples afforded by the syllogism 

 do not bring out this difficulty; and it is consequently verv apt 

 to be overlooked. Take, for instance, the pair of propositions, 

 " No Y is Z," " All X is Y." Here we have the relation of X 

 to Y, and of Y to Z, given independently of one another ; and 



B2 



