Representation of Propositions and Reasonings. 9 



A number of deductions will occur to the logical reader 

 which it may be left to him to work out. Some of them may 

 be just indicated. For instance, any two compartments be- 

 tween which we can communicate by crossing only one line, 

 can differ by the affirmation and denial of one term only, 

 ex. gr. XYZW and X Y Z W. Accordingly, when two such 

 are compounded, or, as we may say, "added " together, they 

 may be simplified by the omission of such term ; for the two 

 together make up all X Y W. Any compartments between 

 which we can only communicate by crossing two boundaries, 

 ex. gr. XYZW and X Y Z W, must differ in two respects ; 

 it would need four such compartments to admit of simplifi- 

 cation, the simplification then resulting in the opportunity 

 of dropping the reference to two terms ; ex. gr. X Y Z W, 

 XYZW, XYZW, XYZW, taken together lead simply 

 to X W. Many similar suggestions will present themselves. 



So far, then, this diagrammatic scheme has only been de- 

 scribed as representing terms or classes ; we have now to see 

 how it can be applied so as to represent propositions. Before 

 doing this it will be necessary to indicate a certain view as to 

 the Import of Propositions, because it is one which is not 

 familiar or generally accepted, though it is very relevant and 

 important for our present purpose. That view is briefly this 

 — that every universal proposition, whether or not it be ori- 

 ginally stated in a negative form, may be adequately repre- 

 sented by one or more negations. To give a complete justi- 

 fication of this view would involve a discussion which would 

 be quite unsuitable to a general article like this ; but a very 

 few remarks will serve to explain, and to a considerable ex- 

 tent to justify it. 



For instance, the common proposition "No X is Y," will 

 be read as just denying the existence of the combination X Y, 

 and therefore needs but little alteration. The proposition "All 

 X is Y" will be read as denying the combination " X that 

 is not Y" or X Y ; and the destruction of that combination 

 will here be regarded as its full import. " X is either Y or 

 Z " will be considered fully accounted for when we have said 

 that it denies " X that is neither Y nor Z " or X Y Z. " Every 

 X that is not Y must be both Z and W " destroys the two 

 combinations XYZ and XYW, and so on. In a full ex- 

 position of the method here indicated, rules might conve- 

 niently be given for thus breaking up complex propositions 

 into all the elementary denials which they implicitly contain ; 

 but the exercise of ordinary ingenuity will quite suffice thus 



