Representation of Propositions and Reasonings. 15 



In the same way the implied total abolition of any one class 

 is thus made extremely obvious. Take, for example, the fol- 

 lowing premises, and let us ask quite generally for any obvious 

 conclusion which follows from them: — 



[" Every Y is either X and not Z, or Z and not X ; 

 < Every WY is either both X and Z, or neither of the two; 

 [All XY is either W or Z, and all YZ is either X or W. 



It will be seen on reflection that these statements involve re- 

 spectively the abolition of the following classes, viz. : — (1) of 

 YXZ,_YXZ ; (2) of WYXZ and WYXZ; (3) of XYWZ, 

 YZXW. Shade out the corresponding compartments in the 

 diagram, and it presents the following appearance — 



It is then clear at a glance that the collective effect of the 

 given premises is just to deny that there can be any such 

 class of things as Y in existence, though they leave every one 

 of the remaining eight combinations perfectly admissible. 

 This, then, is the diagrammatic answer to the proposed ques- 

 tion. 



It will be easily seen that such methods as those here de- 

 scribed readily lend themselves to mechanical performance. I 

 have no high estimate myself of the interest or importance of 

 what are sometimes called logical machines, and this on two 

 grounds. In the first place, it is very seldom that intricate 

 logical calculations are practically forced upon us ; it is rather 

 we who look about for complicated examples in order to illus- 

 trate our rules and methods. In this respect logical calcula- 

 tions stand in marked contrast with those of mathematics, 

 where economical devices of any kind may subserve a really 

 valuable purpose by enabling us to avoid otherwise inevitable 

 labour. Moreover, in the second place, it does not seem tome 

 that any contrivances at present known or likely to be disco- 

 vered really deserve the name of logical machines. It is but 

 a very small part of the entire process which goes to form a 

 piece of reasoning which they are capable of performing. For, 

 if we begin from the beginning, that process would involve 

 four tolerably distinct steps. There is, first, the statement of 

 our data in accurate logical language. This step deserves to 

 be reckoned, since the variations of popular language are so 



