things are dry) is to be regarded as made up of these two 
ABCD = 0, ABCcl=0 (no hard, wet, black, nice things exist 
and no liard, wet, black, nasty things exist) and so is called 
a compound (in this case a two-fold) statement. The 
notion of types is defined in Art 1. 
1. Four classes, or terms, A,B,C,D, give rise to sixteen 
cross-divisions or marks, such as AhQd. A denial of the 
existence of one of these cross-divisions, or of anything 
having its mark (such as A5Cc? = 0), is called a simple state- 
ment. A denial of two or more cross-divisions is called a 
compound statement, and moreover two-fold, three-fold, 
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,h,c,d, they 
are called similar ; and all similar statements are said to 
belong to the same type. The problem before us is to enu- 
merate all the types of compound statement that can be 
made with four terms. 
2. Two statements are called comp)lementary when they 
deny between them all the sixteen marks without both 
denying 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 complementary statements will also be similar; and, 
consequently, for every type of 71-fold statement there is a 
complementary type of 16 — 7i-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. Every eight-fold statement is complementary to an 
eight-fold statement ; but these are not necessarily of the 
same type. 
8. One mark ABCD may be converted into another AhQd 
by interchanging one or more of the classes A,B,C,D with its 
