E ∴ No Americans are trees.
It would not be difficult to determine by actual experiment, just how many moods could be formed, and of these, how many would admit of valid conclusions. It may be seen that there are sixty-four permutations of the four logical propositions, taken three at a time. These are in part:
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) |
| A | A | A | A | A | A | A | A |
| A | A | A | A | E | E | E | E |
| A | E | I | O | A | E | I | O |
| (9) | (10) | (11) | (12) | (13) | (14) | (15) | (16) |
| A | A | A | A | A | A | A | A |
| I | I | I | I | O | O | O | O |
| A | E | I | O | A | E | I | O |
And so the permutations could be continued. Substituting E for the major premise of the above group would give another group of sixteen, while a like substitution of I and O would result in two more groups, sixteen in each.This gives sixty-four in all.[10]
3. TESTING THE VALIDITY OF THE MOODS.
In order to put the moods to good use, it is necessary to ascertain which ones yield a valid conclusion in any figure. If each were valid in all of the four figures, there would be 256. But it is obvious that such is not the case.
Referring to the sixteen permutations given above, we find that the “negative-conclusion” rule makes invalid 2, 4, 5, 7, 10, 12, 13 and 15; whereas the rule for particulars throws out 9 and 14. This leaves the following as the probable valid moods in one or more of the figures: 1, 3, 6, 8, 11, 16. But to be certain of this the investigation must be continued. The mood A
A
A has stood the test of the rules for negative and particular conclusions; now let us test this mood from the standpoint of the distribution of terms, using it in all four figures:
| First | Second | Third | Fourth | ||||
| A | M — G | G — M | M — G | G — M | |||
| A | S — M | S — M | M — S | M — S | |||
| A | S — G | S — G | S — G | S — G |
As an A proposition distributes its subject only, we underscore the subject of each proposition in all the figures. (This underscoring is a simple way to indicate distribution.)
We now find that the mood is valid in the first figure, because the middle term is distributed at least once;namely, in the major premise, and there is no term distributed in the conclusion which is not already distributed in the premise where it occurs. On the other hand, the mood A
A
A is invalid in the second, because of “undistributed middle,” and invalid in the third and fourth, because S is distributed in the conclusion but not distributed in the premise where it occurs (illicit minor).