92 THE PRINCIPLES OF SCIENCE. [CHAP. 



we must ascertain which of these combinations will be 

 rendered self-contradictory by substitution; the second 

 and third will have to be struck out, and there will remain 

 only AB 



ba. 

 Hence we draw the following inferences 



A = AB, B = AB, a = ab, b = ab. 

 Exactly the same method must be followed when a 

 question involves a greater number of terms. Thus by the 

 Law of Duality the three terms A, B, C, give rise to eight 

 conceivable combinations, namely 



ABC (a) aBC (e) 



ABc (ft) aBc (ft 



AbC (7) abC (if) 



Abe (8) abc. (ff) 



The development of the term A is formed by the first four 

 of these; for B we must select (a), (ft), (e), () ; C 

 consists of (a), (7), (e), (ij) ; b of (7), (S),(^), (), and so on. 

 Now if we want to investigate completely the meaning 

 of the premises A = AB (i) 



B = BG (2) 



we examine each of the eight combinations as regards each 

 premise; (7) and (8) are contradicted by (i), and (ft) and 

 () by (2), so that there remain only 



ABC (a) 



aBC (e) 



abQ (if) 



abc. (0) 



To describe any term under the conditions of the premises 

 (i) and (2), we have simply to draw out the proper com- 

 binations from this list ; thus, A is represented only by 

 ABC, that is to say 



A = ABC, 



similarly c = abc. 



For B we have two alternatives thus stated, 



B = ABC -I- BC ; 

 and for b we have 



b = abC { abc. 



When we have a problem involving four distinct terms 

 we need to double the number of combinations, and as 

 we add each new term the combinations become twic,e 

 as numerous. Thus 



