Representation of Propositions and Reasonings. 5 



representation for it. He makes no employment of diagrams 

 himself, nor any suggestion for them. 



One essential characteristic of Boole's method, as many 

 readers of this article will probably know, is the complete sub- 

 division of our field of inquiry into all the elementary classes 

 which can possibly be yielded by combination of all the terms 

 involved. Let there be two terms, X and Y ; then we have 

 to take account of the four subclasses, X that is Y, X that is 

 not Y, Y that is not X, and what is neither X nor Y. 

 Writing, for simplicity, X for not-X, the four classes are 

 XY, XY, XY, XY. Three class terms similarly yield eight 

 subclasses, which admit of equally ready symbolic representa- 

 tion, and so on. Generally, if there be n classes involved in 

 any given combination of logical premises there will be 2 n sub- 

 classes, every one of which must, somehow or other, be taken 

 account of in any complete investigation of the problem. 



This consideration seems to suggest a more hopeful scheme 

 of diagrammatic representation. Whereas the Eulerian plan 

 endeavoured at once and directly to represent propositions, or 

 relations of class terms to one another, we shall find it best to 

 begin by representing only classes, and then proceed to modify 

 these in some way so as to make them indicate what our pro- 

 positions have to say. How, then, shall we represent all the 

 subclasses which two or more class terms can produce? 

 Bear in mind that what we have to indicate is the successive 

 duplication of the number of subdivisions produced by the 

 introduction of every successive term, and we shall see our 

 way to a very important departure from the Eulerian concep- 

 tion. All that we have to do is to draw our figures, say 

 circles, so that each successive one which we introduce shall 

 intersect once, and once only, all the subdivisions already ex- 

 isting, and we then have what may be called a general frame- 

 work indicating every possible combination producible by the 

 given class terms. This successive duplication of the number 

 of subclasses was the essential characteristic when we were 

 dealing with such symbols as X and Y. For suppose theso 

 two terms only involved, and there resulted the four minor 

 classes indicated by XY, XY, XY, and XY. Now suppose 

 that a third term Z makes its appearance. This at once calls 

 for a subdivision of each of these four into its Z and Z parts 

 respectively. Thus XY is split up into XYZ and XYZ, and 

 so with the others, whence we get the eight subdivisions de- 

 manded. Provided our diagrams represent this characteristic 

 clearly and unambiguously , they will do all that we can require 

 of theim 



