Deductive Logic
by George William Joseph Stock, M.A.
Part 3: Chapter 12
Additional Information
 Year Published: 1888
 Language: English
 Country of Origin: England
 Source: Stock, G. W. J. (1888). Deductive Logic. Oxford, England; Pembroke College.

Readability:
 Flesch–Kincaid Level: 11.0
 Word Count: 557
 Genre: Informational
 Keywords: math, math history
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PART III.—OF INFERENCES
CHAPTER XII.
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.