PERFORMANCE OF LOGICAL INFERENCE. 
503 
The combinations in which A may manifest itself are, according to the Laws of Thought, 
ABC, (a) 
A Be, ((3) 
AiC, (y) 
A be (i ) 
Bnt of these (y) and (l) are contradicted by (1) and (/ 3 ) by (2). Hence 
A is identical with ABC, 
and this term, ABC, contains the full description of A or iron under the conditions 
(1) and (2). 
Similarly, we may obtain the description of the term or class of things not-element , 
denoted by c. For by the Law of Duality c may be developed into its alternatives or 
possible combinations. 
ABc, . . . 
• • • 03) 
A be,. . . 
•••(*) 
a B c, . . . 
• • ■ K) 
a b c. . . . 
• • • (0 
Of these (/3) and (£) are contradicted by (2) and (&) by (1) ; so that, excluding these con- 
tradictory terms, a b c alone remains as the description or equivalent of the class c. 
Hence what is not-element , is always not-metallic and is also not-iron. 
19. In practising this process of indirect inference upon problems of even moderate 
complexity, it is found to be tedious in consequence of the number of alternatives which 
have to be written and considered time after time. Modes of abbreviation can, how- 
ever, be readily devised. In the problem already considered it is evident that the same 
combination sometimes occurs over again, as in the cases of (j3) and (ci ) ; and if we 
were desirous of deducing all the conclusions which could be drawn from the premises 
we should find the combination («) occurring in all the separate classes A, A B, B, 
B C, AC. Similarly, the combination a b C occurs in the classes a, b, C, a b, a C, b C, 
and it would be an absurd loss of labour to examine again and again whether the same 
combination is or is not contradicted by the premises. It is certain that all the com- 
binations of the terms A, B, C, a , b , c, which are possible under the universal conditions 
of thought and existence are but eight in number, as follows : — ■ 
(«) 
A 
B 
C 
(£) 
A 
B 
c 
(y) 
A 
b 
C 
(*) 
A 
b 
c 
(0 
a 
B 
C 
(?) 
a 
B 
G 
OO 
a 
b 
C 
00 
a 
b 
c 
