PERFORMANCE OF LOGICAL INFERENCE. 
505 
saves the labour of writing out the combinations*. The same purpose may be effected 
by having series of combinations printed ready upon separate sheets of paper, a series 
of proper length being selected for the solution of any problem, and the inconsistent 
combinations being struck out with the pen as they are discovered on examination with 
the premises. 
22. A second step towards a mechanical logic was soon seen to be easy and desirable. 
The fixed order of the combinations in the written abecedarium renders it necessary to 
consider them separately, and to pick out by repeated acts of mental attention those 
which fall into any particular class. Considerable labour and risk of mistake thus arise. 
The Logical Abacus was devised to avoid these objections, and was constructed by placing 
the combinations of the abecedarium upon separate moveable slips of wood, which can 
then be easily classified, selected and arranged according to the conditions of the pro- 
blem. The construction and use of this Abacus have, however, been sufficiently described 
both in the ‘ Proceedings of the Manchester Literary and Philosophical Society’ for 3rd 
April, 1866, and more fully in my recently published work, called 4 The Substitution 
of Similars,’ which contains a figure of the Abacus. I will only remark, therefore, that 
while the logical slate or printed abecedarium is convenient for the private study of 
logical problems, the abacus is peculiarly adapted for the logical class-room. By its 
use the operations of classification and selection, on which Boole’s logic, and in fact any 
logic must be founded, can be represented, and the clearest possible solution of any 
question can be shown to a class of students, each step in the solution being made 
distinctly apparent. 
23. In proceeding to explain how the process of logical deduction by the use of the 
abecedarium can be reduced to a purely mechanical form, I must first point out that 
certain simple acts of classification are alone required for the purpose. If we take the 
eight conceivable combinations of the terms A, B, C, and compare them with a propo- 
sition of the form 
A is B, . , (1) 
we find that the combinations fall apart into three distinct groups, which may be thus 
indicated : — 
Excluded combinations B 
lc 
Included combinations consistent 
with premise (1) 
A 
lc 
A 
B 
c 
Included combinations inconsistent with 
premise (1) 
A 
b 
C 
A 
b 
c 
a a 
B b 
c C 
a 
b 
D 
The highest group contains those combinations which are all a s, and on account of 
* See Pure Logic, p. 68. 
3 z 
MDCCCLXX. 
