82 PROP. W. K. CLIFFORD ON THE 



explanation of the nature and purpose of the problem than 

 is to be found in art. i. It may, however, be premised 

 that the letters A, B, C, D denote four classes or terms (for 

 example, hard, wet, black, nice) , and that, according to a 

 convenient notation of De Morgan's, the small letters a, b, 

 c, d denote the complementary classes or contrary terms 

 (not hard, not wet, not black, not nice) . A simple statement 

 is of the form ABCD = o (no hard, wet, black, nice things 

 exist, or, which is the same thing, all hard, wet, black things 

 are nasty). The statement ABC = o (no hard, wet, black 

 things exist, or all hard, black things are dry) is to be re- 

 garded as made of these two, ABCD = o, ABCc?=o (no 

 hard, wet, black, nice things exist, and no hard, wet, black, 

 nasty things exist) and so is called a compound (in this case 

 a twofold) statement. The notion oi types is defined in art. i. 



1 . Four classes or terms A, B, C, D, give rise to sixteen 

 cross divisions or marks, such as A6C</. A denial of the 

 existence of one of these cross divisions, or of any thing 

 having its mark (such as P>^bCd=o), is called a simple 

 statement. A denial of two or more cross divisions is called 

 a compound statement, and, moreover, twofold, threefold, 

 etc., according to the number denied. 



When two compound statements can be converted into 

 one another by interchange of the classes A, B, C, D with 

 each other or with their complementary classes a, b, c, d, 

 they are called similar ; and all similar statements are said 

 to belong to the same type. The problem before us is to 

 enumerate all the types of compound statement that can 

 be made with four terms. 



2. Two statements are called complementary when they 

 deny between them all the sixteen marks without both de- 

 nying any mark, or, which is the same thing, when each 

 denies just those marks which the other permits to exist. 

 It is obvious that when two statements are similar, the 



