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converse part of the same rule, that a negative conclusion can only be proved by a negative premise; while EEA, EEE &c., break the 5th rule, which prohibits our reasoning at all from two negative premises. Examples of any of these moods can easily be invented, and their falsity would be very apparent; thus for AEA we might take
All Austrians are Europeans,
No Australians are Europeans; Therefore, all Australians are Austrians. Many of the 64 conceivable moods are excluded by the 7th and 8th rules of the syllogism. Thus AIA and EIE break the rule, that if one premise be particular the conclusion must be so also, while IIA, 100, OIO and many others, break the rule against two particular premises. Some combinations of propositions may break more than one rule; thus 000 has both negative premises and particular premises, and 00A also violates as well the 6th rule. It is an admirable exercise in the use of the syllogistic rules to write out all the 64 combinations and then strike out such as break any rule; the task if pursued systematically will not be so long or tedious as might seem likely. It will be found that there are only twelve moods which escape exclusion, and may so far be considered good forms of reasoning, and these are AAA EAE
A00 Of these however IỆО will have shortly to be rejected, because it will be found really to break the 4th rule, and involves Illicit process of the major term. There are,
then, only eleven moods of the syllogism which are really valid; and we may thus account for the whole of the sixty-four moods.
Number Excluded by
of moods. Negative premises, Rule 5
8 Negative conclusion 6..........
4 Illicit major ......... 4. Total excluded
53 Valid moods ....
Total We have by no means exhausted as yet all the possible varieties of the syllogism, for we have only determined the character, affirmative or negative, general or particular of the propositions, but have not decided the ways in which the terms may be disposed in them. The major term must be the predicate of the conclusion, but it may either be subject or predicate of the major premise, and similarly the minor term or subject of the conclusion, may be either the subject or predicate of the minor premise. There thus arise four different ways, or as they are called Figures, in which the terms can be disposed. These four figures of the syllogism are shewn in the following scheme, taking
X to denote the major term
........ middle 2........ ..minor
ist Fig. 2nd Fig. 3rd Fig. 4th Fig. Major Premise YX XY YX XY Minor
ZY ZY Y Z YZ Conclusion 2 X ZX ZX ZX
These figures must be carefully committed to memory, which will best be done by noting the position of the middle term. This term stands first as subject of the major premise in the ist Figure, second as predicate in both premises of the 2nd Figure, first again as subject of both premises in the 3rd Figure, and in an intermediate position in the 4th Figure. In the conclusion, of course, the major and minor terms have one fixed position, and when the middle term is once correctly placed in any figure we easily complete the syllogism.
The reader will hardly be pleased to hear that each of the eleven valid moods will have to be examined in each of the four figures separately, so that there are 44 cases still possible, from which the valid syllogisms have to be selected. Thus the mood AEE in the first figure would be as follows:
All Y's are X's,
No Z's are Y's;
No Z's are X's. This would break the 4th rule and be an Illicit Major, because X is distributed in the conclusion, which is a negative proposition, and not in the major premise. In the second figure it would be valid:
All X's are Y's,
No Z's are Y's;
No Z's are X's.
All Y's are X's,
No Z's are X's, and again breaks the 4th rule, as regards the major term. Lastly in the 4th figure it is valid, as the reader may easily satisfy himself.
When all the valid moods are selected out of the 44 possible ones, there are found to be altogether 24, which are as follows:
Valid Moods of the Syllogism.
First Second Third Fourth
ОАО [AAI] [EAO) EIO (EAO] [AEO]
[AEO] Five of the above moods are set apart and enclosed in brackets, because though valid they are of little or no use. They are said to have a weakened conclusion, because the conclusion is particular when a general one might have been drawn. Thus AAI, in the first figure is represented by the example:
All material substances gravitate,
All metals are material substances;
Therefore some metals gravitate. It is apparent that the conclusion only states a part of the truth, and that in reality all metals gravitate. It is not actually an erroneous conclusion, because it must be carefully remembered (p. 77) that the affirming of a subaltern or particular proposition does not deny the corresponding general proposition. It is quite true that some metals gravitate, and it must be true because all of them do so. But when we can as readily prove that all do gravitate it is desirable to adopt this conclusion.
If we agree with most logicians to overlook the existence of the five syllogisms with weakened conclusions,
there will remain nineteen which are at once valid and useful. In the next lesson certain ancient mnemonic lines will be furnished by which alone it would be possible for most persons to carry in the memory these 19 combinations; but the reader will in the mean time be able to gather from the statement of the moods in p. 140 the truth of the following remarks concerning the peculiar character of each figure of the syllogism.
The first figure is the only one which proves the pro-* position A, or has A for its conclusion. It is the only figure, too, which can prove any one of the four propositions A, E, I, O. As regards the premises, it is especially important to note that the major premise is always universal (A or E), and the minor premise affirmative (A or I): this peculiarity will be further considered in the next lesson.
The second figure only proves negative conclusions (E or 0), and the reason is easily apparent. As the middle term in this figure is the predicate of both premises it would necessarily be undistributed in both premises if these were affirmatives, and we should commit the fallacy exemplified in p. 137. It follows that one premise must be negative and of course one only, so that of the major and minor terms one must be included or excluded wholly from the middle, and the other at the same time excluded or included at least partially. To illustrate this we may take X, Y and Z to represent, as before, the major, middle and minor terms of a syllogism, and the four moods of this figure are then
no X's are Y's,
all X's are Y's, no Z's are Y's; .. no Z's are X's.