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premises, which may thus be symbolically expressed :
(1) A = AB,
C = CD,
The combination Ab C D is contradicted by (1) in substituting for A its value; ABCd by (2), a BcD by (3), and so on. There will be found to remain only four possible combinations :—
Now, if we wish to ascertain the nature of the term A, we learn at once that it can only exist in the presence of B and the absence of both C and D.
We ascertain also that D can only appear in the absence of both A and B, but that C may or may not be present with D. Where D is absent, C must also be absent, and so on.
47. Objections might be raised against this process of indirect inference, that it is a long and tedious one; and so it is, when thus performed. Tedium indeed is no argument against truth; and if, as I confidently assert, this method gives
us the means of solving an infinite number of problems, and arriving at an infinite number of conclusions, which are often demonstrable in no simpler way, and in fact in no other way whatever, no such objections would be of any weight. The fact however is, that almost all the tediousness and liability to mistake may be removed from the process by the use of mechanical aids, which are of several kinds and degrees. While practising myself in the use of the process, I was at once led to the use of the logical slate, which consists of a common writing slate, with several series. of the combinations of letters engraved upon it, thus:
When fully written out, these series consist respectively of 4, 8, 16, 32, and 64 combinations, and that series is chosen for any problem which just affords enough distinct terms. Each com
bination is then examined in connexion with each
of the premises, and the contradictory ones are struck through with the pencil.
48. It soon became apparent, however, that if these combinations, instead of being written in fixed order on a slate, were printed upon light moveable slips of wood, it would become easy by suitable mechanical arrangements to pick out the combinations in convenient classes, so as immensely to abbreviate the labour of comparison with the premises. This idea was carried out in the logical abacus, which I constructed several years ago, and have found useful and successful in the lecture-room for exhibiting the complete solution of logical arguments.
49. This logical abacus has been exhibited before the members of the Manchester Literary and Philosophical Society, and the following description of it is extracted from the Proceedings of the Society for 3d April, 1866, p. 161.
"The abacus consists of
"I. An inclined black board, furnished with four ledges, 3 ft. long, placed 9 in. apart.
"2. Series of flat slips of wood, the smallest set four in number, and other sets, 8, 16, and 32 in number, marked with combinations of letters, as follows:
A = Socrates,
Man is mortal,
Therefore Socrates is mortal.
"The third and fourth sets exhibit the corresponding combinations of the letters A, B, C, D, a, b, c, d, and A, B, C, D, E, a, b, c,d,e.
"The slips are furnished with little pins, so that, when placed upon the ledges of the board, those marked by any given letter may be readily picked out by means of a straight-edged ruler, and removed to another ledge.
50. "The use of the abacus will be best shown by an example. Take the syllogism in Bar
"The corresponding small italic letters then indicate the negatives,
a = not-Socrates,
b = not-man,
c = not-mortal,
and the premises may be stated as
A is B,
B is C.
"Now take the second set of slips containing all the possible combinations of A, B, C, a, b, c, and ascertain which of the combinations are possible under the conditions of the premises.
"Select all the slips marked A; and as all these ought to be B's, select again those which are notB or b, and reject them. Unite the remainder, and, selecting the B's, reject those which are not-C or c. There will now remain only four slips or combinations:
"If we require the description of Socrates, or A, we take the only combination containing A, and observe that it is joined with C: hence the Aristotelian conclusion, Socrates is mortal. We may also get any other possible conclusion. For instance, the class