Representation of Propositions and Reasonings. 11 



by saying that it just destroys the class XYZW or "X that 

 is Y and Z but not W," and does nothing else*. 



The method of employing the diagrams in order to express 

 propositions will readily be understood. It is merely this : — 

 Ascertain what each given proposition denies, and then put 

 some kind of mark upon the corresponding partition in the 

 figure. The most effective means of doing this is just to shade 

 it out. For instance, the proposition "All X is Y" is inter- 

 preted to mean that there is no such class of things in exist- 

 ence as " X that is not-Y " or XY. All, then, that we have to 

 do is to scratch out that subdivision in the two-circle figure, 



X Y 



thus, ^TT^ • ^ we want to represent "All X is all Y," 

 we take this as adding on another denial, viz. that ofXY, and 

 we proceed to scratch out that division also, thus, 



The main characteristic of this scheme, viz. the facility with 

 which it enables us to express each separate accretion of know- 

 ledge, and so to break up any complicated group of data, and 

 attack them in detail, will begin to show itself even in such a 

 simple instance as this. On the common plan we should have 

 to begin again with a new figure in each case respectively, 

 viz. for "All X is Y," and "All X is all Y; " whereas here we 

 use the same figure each time, merely modifying it in accord- 

 ance with the new information. Or take the disjunctive " All 

 X is either Y or Z." It is very seldom even attempted to 

 represent this diagrammatically (and then, so far as I have 

 seen, only if the alternatives are mutually exclusive); but it is 

 readily enough exhibited when we regard it as merely extin- 

 guishing any X that is neither Y nor Z — thus, 



If to this were added the statement that " none but the X's 

 are either Y or Z," we should then abolish the XY and the XZ, 



and have £7SA • Scratch out, again, the XYZ compart- 



* Though this interpretation, however, of the import of propositions 

 seems desirable for a really generalized system of logic, it is by no means 

 necessary to adopt it in order to explain and justify the use of the dia- 

 grammatic method here proposed. 



