144 THE PRINCIPLES OF SCIENCE. [CHAP. 



division, of the classes, as in ABCD = o, or AbGd = o. 

 The denial of two or more such combinations is called a 

 compound statement, and is further said to be twofold, 

 threefold, &c., according to the number denied. Thus 

 ABC = o is a twofold compound statement in regard to 

 four classes, because it involves both ABCD = o and 

 ABCd = o. 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. 



Two statements are called complementary when they 

 deny between them all the sixteen combinations without 

 both denying any one ; or, which is the same thing, when 

 each denies just those combinations which the other 

 permits to exist. It is obvious that when two statements 

 are similar, the complementary statements will also be 

 similar, and consequently for every type of 7i-fold statement, 

 there is a complementary type of (16 w)-fold statement. 

 It follows that we need only enumerate the types as far as 

 the eighth order ; for the types of more-than -eight-fold 

 statement will already have been given as complementary 

 to types of lower orders. 



One combination, ABCD, may be converted into another 

 AbCd by interchanging one or more of the classes with 

 the complementary classes. The number of such changes 

 is called the distance, which in the above case is 2. In 

 two similar compound statements the distances of the 

 combinations denied must be the same ; but it does not 

 follow that when all the distances are the same, the state- 

 ments are similar. There is, however, only one example 

 of two dissimilar statements having the same distances. 

 When the distance is 4, the two combinations are said to 

 be obverse to one another, and the statements denying them 

 are called obverse statements, as in ABCD = o and abed = o 

 or again AbCd o and aBcD = o. When any one com- 

 bination is given, called the origin, all the others may be 

 grouped in respect of their relations to it as follows : Four 

 are at distance one from it, and may be called proximates ; 

 six are at distance two, and may be called mediates ; four 

 are at distance three, and may be called ultimate,? ; finally 

 the obverse is at distance four. 



