268 Dr. A. Marquand on Logical 



The sixteen compartments here represent the formal division 

 of a universe into the classes ABCD, A~BGd, A 6 C D, &c. 



Thus by continued dichotomy we may reach a diagram for 

 any number of terms. A diagram for n terms, if n be any 



n 



even number greater than 2, requires 2 + 2 2 + 2 3 + . . . + 2 2 

 dividing lines; diagrams for n — 1 terms require 2 + 2 2 + 2 3 + . . . 



n n 



22 — 22" 1 such lines. 



As the number of terms increases, the labour of writing out 

 a quantity of letters may be considerably lessened by the use 

 of brackets. This will appear in the solution of the following 

 problem. 



The are eight arguments, A, B, C, D, E, F, G, H, thus 

 related to each other: — When E is true, F is true ; and when 

 F is true, either E is true or B and C are both false. When 

 either Gr is true or E and F are both false, D is true. If B is 

 false when either F or G (but not both) are true, then H is true 

 and either C is false or D true. It is true only when an even 

 number of the remaining arguments are true ; it is false only 

 when an odd number of the remaining arguments are false. 



Supposing any combination not inconsistent with the pre- 

 mises to exist, (1) What follows from A being true either when 

 B is true and D false or C false and F true ? and (2) From 

 what combination of arguments may we conclude that A and 

 H are both true when E and Gr are both false ? 



Representing truth by a capital and falsity by a small letter, 

 all possible combinations of the truth or falsity of the eight 

 arguments are indicated by the small squares in the following 

 diagram. 



The shaded squares indicate the A combinations which are 

 inconsistent with one or more of the premises ; the non-A 

 combinations, not being required in the conclusion, may be 

 neglected. The first part of the conclusion calls for the eight 

 combinations numbered 1 to 8 on the diagram. These are : — 



AB 



cDEFGH 



AB 



cDEFGA 



AB 



cd E ¥ g H 



AB 



cd E¥ g h 



Ab 



cDE F g H 



Ab 



cDe FGH 



Ab. 



cBe F Gth 



Ab 



cd e F g H 



and one or other of these is true. It will be observed that in 

 all the combinations C is false and F true. Many subordinate 



