206. Other apparent exceptions to the Rules of the Syllogism.—It is curious that the logicians who have laid so much stress on the case considered in the preceding section do not appear to have observed that, as soon as we admit more than three terms, other apparent breaches of the syllogistic rules may occur in what are perfectly valid reasonings. Thus, the premisses All P is M and All S is M, in which M is not distributed, yield the conclusion Some not-S is not-P;[321] and 298 hence we might argue that undistributed middle does not invalidate an argument. Again, from the premisses All M is P, All not-M is S, we may infer Some S is not P,[322] although there is apparently an illicit process of the major. It is unnecessary after what has been said in the preceding section to give examples of valid reasonings in which we have a negative premiss with an affirmative conclusion, or two affirmative premisses with a negative conclusion, or a particular major with a negative minor. Any valid syllogism which is affirmative throughout will yield the first and, if it has a particular major, also the last of these by the obversion of the minor premiss, and the second by the obversion of the conclusion. The only syllogistic rules, indeed, which still hold good when more than three terms are admitted are the rule providing against illicit minor and the first two corollaries.

[321] By the contraposition of both premisses this reasoning is reduced to the valid syllogistic form, All not-M is not-P, All not-M is not-S, therefore, Some not-S is not-P.

[322] By the inversion of the first premiss, this reasoning is reduced to the valid syllogistic form, Some not-M is not P, All not-M is S, therefore, Some S is not P. Compare section [104].

But of course none of the above examples really invalidate the syllogistic rules; for these rules have been formulated solely with reference to reasonings of a certain form, namely, those which contain three and only three terms. In every case the reasoning inevitably conforms to the rule which it appears to violate, as soon as, by the aid of immediate inferences, the superfluous number of terms has been eliminated.

207. Syllogisms with two singular premisses.—Bain (Logic, Deduction, p. 159) argues that an apparent syllogism with two singular premisses cannot be regarded as a genuine syllogistic or deductive inference; and he illustrates his view by reference to the following syllogism:

Socrates fought at Delium,
Socrates was the master of Plato,
therefore, The master of Plato fought at Delium.

The argument is that “the proposition ‘Socrates was the master of Plato and fought at Delium,’ compounded out of the two premisses, is nothing more than a grammatical abbreviation,” whilst the step hence to the conclusion is a mere omission of something that had previously been said. “Now, we never 299 consider that we have made a real inference, a step in advance, when we repeat less than we are entitled to say, or drop from a complex statement some portion not desired at the moment. Such an operation keeps strictly within the domain of Equivalence or Immediate Inference. In no way, therefore, can a syllogism with two singular premisses be viewed as a genuine syllogistic or deductive inference.”

This argument leads up to some interesting considerations, but it proves too much. In the following syllogisms the premisses may be similarly compounded together:

All men are mortal,⎱All men are mortal and rational ;
All men are rational,⎰
therefore, Some rational beings are mortal.
All men are mortal,⎱All men including kings are mortal ;
All kings are men,⎰
therefore, All kings are mortal.[323]

[323] Compare with the above the following syllogism which has two singular premisses:—The Lord Chancellor receives a higher salary than the Prime Minister; Lord Herschell is the Lord Chancellor; therefore, Lord Herschell receives a higher salary than the Prime Minister. These premisses would presumably be compounded by Bain into the single proposition, “The Lord Chancellor, Lord Herschell, receives a higher salary than the Prime Minister.”