upper part of its space; when the letter is a small italic representing a negative term, the pin is fixed in the lower part of the space. Now, if one of the series of combinations be ranged upon a ledge of the black-board, the sharp edge of a flat rule can be inserted beneath the pins belonging to any one letter—say A, so that all the combinations marked A can be lifted out and placed upon a separate ledge. Thus we have represented the act of thought which separates the class A from what is not-A. The operation can be repeated; out of the A’s we can in like manner select those which are B’s, obtaining the AB’s; and in like manner we may select any other classes such as the aB’s, the ab’s, or the abc’s.
If now we take the series of eight combinations of the letters A, B, C, a, b, c, and wish to analyse the argument anciently called Barbara, having the premises
| A = AB | (1) |
|---|---|
| B = BC, | (2) |
we proceed as follows—We raise the combinations marked a, leaving the A’s behind; out of these A’s we move to a lower ledge such as are b’s, and to the remaining AB’s we join the a’s which have been raised. The result is that we have divided all the combinations into two classes, namely, the Ab’s which are incapable of existing consistently with premise (1), and the combinations which are consistent with the premise. Turning now to the second premise, we raise out of those which agree with (1) the b’s, then we lower the Bc’s; lastly we join the b’s to the BC’s. We now find our combinations arranged as below.
| A | a | a | a | ||||
|---|---|---|---|---|---|---|---|
| B | B | b | b | ||||
| C | C | C | c | ||||
| A | A | A | a | ||||
| B | b | b | B | ||||
| c | C | c | c |
The lower line contains all the combinations which are inconsistent with either premise; we have carried out in a mechanical manner that exclusion of self-contradictories which was formerly done upon the slate or upon paper. Accordingly, from the combinations remaining in the upper line we can draw any inference which the premises yield. If we raise the A’s we find only one, and that is C, so that A must be C. If we select the c’s we again find only one, which is a and also b; thus we prove that not-C is not-A and not-B.
When a disjunctive proposition occurs among the premises the requisite movements become rather more complicated. Take the disjunctive argument
The premises are represented accurately as follows:—
| A = AB ꖌ AC ꖌ AD | (1) |
|---|---|
| A = A c | (2) |
| A = A d . | (3) |
As