PERFORMANCE OE LOGICAL INFERENCE. 
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of throwing any combinations which are in the third back into the first. It thus joins 
together all the combinations included in one or more alternatives of the subject, and 
prepares them for the due action of the predicate keys. 
50. It must be carefully observed that any doubly universal proposition of the form 
all A’s are all B’s, 
or, in another form of expression, 
A=B, 
can only be impressed upon the logical machine in the form of two ordinary propositions ; 
thus 
all A’s are B’s, 
and 
all B’s are A’s. 
The first of these excludes such A’s as may be not-B’s ; the second excludes such B’s 
as may be not- A’s. 
If we impress upon the keys of the machine the six propositions expressing the com- 
plete identity of A, B, C, and B, it is obvious that there would remain only the two 
combinations 
ABCD, 
abed , 
the identity of the positive terms involving the identity also of their negatives. 
The premise 
A or B~C or B 
would require to be read 
A or B is C or B, 
C or B is A or B. 
51. To give some notion of the degree of facility with which logical problems may 
be solved with the machine, I will adduce the logical problem employed by Boole to 
illustrate the powers of his system at p. 118 of the ‘ Laws of Thought.’ 
“ Suppose that an analysis of the properties of a particular class of substances has led 
to the following general conclusions, viz. : — 
“ 1st. That wherever the properties A and B are combined, either the property C, or 
the property B, is present also; but they are not jointly present. 
“ 2nd. That wherever the properties B and C are combined, the properties A and B 
are either both present with them, or both absent. 
“ 3rd. That wherever the properties A and B are both absent, the properties C and B 
are both absent also ; and vice versa, where the properties C and B are both absent, A 
and B are both absent also.” 
This somewhat complex problem is solved in Boole’s work by a very difficult and 
lengthy series of eliminations, developments, and algebraic multiplications. Two or 
three pages are required to indicate the successive stages of the solution, and the details 
of the algebraic work would probably occupy many more pages. Upon the machine 
the problem is worked by the successive pressure of the following keys : — 
4 a 2 
