_Of the Determination of the Legitimate Moods of Syllogism._
§ 599. It will be remembered that there were found to be 64 possible
moods, each of which might occur in any of the four figures, giving us
altogether 256 possible varieties of syllogism. The task now before us
is to determine how many of these combinations of mood and figure are
legitimate.
§ 600. By the application of the preceding rules we are enabled to
reduce the 64 possible moods to 11 valid ones. This may be done by a
longer or a shorter method. The longer method, which is perhaps easier
of comprehension, is to write down the 64 possible moods, and then
strike out such as violate any of the rules of syllogism.
AAA -AEA- -AIA- -AOA-
-AAE- AEE -AIE- -AOE-
AAI -AEI- AII -AOI-
-AAO- AEO -AIO- AOO
-EAA- -EEA- -EIA- -EOA-
EAE -EEE- -EIE- -EOE-
-EAI- -EEI- -EII- -EOI-
EAO -EEO- EIO -EOO-
[Illustration]
§ 601. The batches which are crossed are those in which the premisses
can yield no conclusion at all, owing to their violating Rule 6 or 9;
in the rest the premises are legitimate, but a wrong conclusion is
drawn from each of them as are translineated.
§ 602. IEO stands alone, as violating Rule 4. This may require a
little explanation.
Since the conclusion is negative, the major term, which is its
predicate, must be distributed. But the major premiss, being 1, does
not distribute either subject or predicate. Hence IEO must always
involve an illicit process of the major.
§ 603. The II moods which have been left valid, after being tested by
the syllogistic rules, are as follows--
AAA. AAI. AEE. AEO. AII. AOO.
EAE. EAO. EIO.
IAI.
OAO.
§ 604. We will now arrive at the same result by a shorter and more
scientific method. This method consists in first determining what
pairs of premisses are valid in accordance with Rules 6 and g, and
then examining what conclusions may be legitimately inferred from them
in accordance with the other rules of syllogism.
§ 605. The major premiss may be either A, E, I or O. If it is A, the
minor also may be either A, E, I or O. If it is E, the minor can only
be A or I. If it is I, the minor can only be A or E. If it is O, the
minor can only be A. Hence there result 9 valid pairs of premisses.
AA. AE. AI. AO.
EA. EI.
IA. IE.
OA.
Three of these pairs, namely AA, AE, EA, yield two conclusions apiece,
one universal and one particular, which do not violate any of the
rules of syllogism; one of them, IE, yields no conclusion at all; the
remaining five have their conclusion limited to a single proposition,
on the principle that the conclusion must follow the weaker part.
Hence we arrive at the same result as before, of II legitimate moods--
AAA. AAI. AEE. AEO. EAE. EAO.
AII. AOO. EIO. IAI. OAO.