Studies and Exercises in Formal Logic

by John Neville Keynes

S – P ;

and the following may be given as its special rules,—
(1) If the major is affirmative, the minor must be universal ;
(2) If either premiss is negative, the major must be universal ; 313
(3) If the minor is affirmative, the conclusion must be particular.[333]

[³³³] The special rules of the fourth figure are variously stated. They are given in the above form in the Port Royal Logic, pp. 202, 203. See, also, section [255].

The result of the application of these rules is again six valid moods, namely, AAI, AEE, AEO, EAO, EIO, IAI.

Our final conclusion then is that there are 24 valid moods, namely, six in each figure.

In Figure 1, AAA, AAI, EAE, EAO, AII, EIO.
In Figure 2, EAE, EAO, AEE, AEO, EIO, AOO.
In Figure 3, AAI, IAI, AII, EAO, OAO, EIO.
In Figure 4, AAI, AEE, AEO, EAO, IAI, EIO.

245. Weakened Conclusions and Subaltern Moods.—When from premisses that would have justified a universal conclusion we content ourselves with inferring a particular (as, for example, in the syllogism All M is P, All S is M, therefore, Some S is P), we are said to have a weakened conclusion, and the syllogism is said to be a weakened syllogism or to be in a subaltern mood (because the conclusion might be obtained by subaltern inference[334] from the conclusion of the corresponding unweakened mood).

[³³⁴] In treating the syllogism on the traditional lines it is assumed that S, M, P all represent existing classes. Subaltern inference is, therefore, a valid process.

In the [preceding] section it has been shewn that in each figure there are six moods which do not offend against any of the syllogistic rules: so that in all there are 24 distinct valid moods. Five of these, however, have weakened conclusions; and, since we are not likely to be satisfied with a particular conclusion when the corresponding universal can be obtained from the same premisses, these moods are of no practical importance. Accordingly when the moods of the various figures are enumerated (as in the mnemonic verses) they are usually omitted. Still, their recognition gives a completeness to the theory of the syllogism, which it cannot otherwise possess. There is also a symmetry in the result of 314 their recognition as yielding exactly six legitimate moods in each figure.[335]

[³³⁵] It has been remarked that 19 being a prime number at once suggests incompleteness or artificiality in the common enumeration.