[313] This argument holds good in the special case under consideration even if we interpret particulars, but not universals, as implying the existence of their subjects. For the validity of the above proof that two universal negatives yield no conclusion remains unaffected even if we allow to universals the maximum of existential import.
(ii) The rules that from two negative premisses nothing can be inferred and that if one premiss is negative the conclusion must be negative are mutually deducible from one another.
The following proof that the second of these rules is deducible from the first is suggested by De Morgan’s deduction of 293 the second corollary as given in section [200]. If two propositions P and Q together prove a third R, it is plain that P and the denial of R prove the denial of Q. For P and Q cannot be true together without R. Now, if possible, let P (a negative) and Q (an affirmative) prove R (an affirmative). Then P (a negative) and the denial of R (a negative) prove the denial of Q. But by hypothesis two negatives prove nothing.
It may be shewn similarly that if we start by assuming the second of the rules then the first is deducible from it.
(iii) Any syllogism involving directly an illicit process of major or minor involves indirectly a fallacy of undistributed middle, and vice versâ.[314]
[314] For this theorem and its proof I am indebted to Mr Johnson.
Let P and Q be the premisses and R the conclusion of a syllogism involving illicit major or minor, a term X which is undistributed in P being distributed in R. Then the contradictory of R combined with P must prove the contradictory of Q. But any term distributed in a proposition is undistributed in its contradictory. X is therefore undistributed in the contradictory of R, and by hypothesis it is undistributed in P. But X is the middle term of the new syllogism, which is therefore guilty of the fallacy of undistributed middle. It is thus shewn that any syllogism involving directly a fallacy of illicit major or minor involves indirectly a fallacy of undistributed middle.
Adopting a similar line of argument, we might also proceed in the opposite direction, and exhibit the rule relating to the distribution of the middle term as a corollary from the rule relating to the distribution of the major and minor terms.
203. Statement of the independent Rules of the Syllogism.—The theorems established in the preceding section shew that the first part of rule 4 (as given in section [201]) is a corollary from rule 3, and that rule 3 is in its turn a corollary from rule 1; also that rules 1 and 2 mutually involve one another, so that either one of them may be regarded as a corollary from the other. We are, therefore, left with either rule 1 or rule 2 and also with the second part of rule 4; and the independent rules of the syllogism may accordingly be stated as follows: 294
(α) Rule of Distribution:—The middle term must be distributed once at least in the premisses [or, as alternative with this, No term may be distributed in the conclusion which was not distributed in one of the premisses];
(β) Rule of Quality:—To prove a negative conclusion one of the premisses must be negative.[315]
[315] On examination it will be found that the only syllogism rejected by this rule and not also rejected directly or indirectly by the preceding rule is the following:—All P is M, All M is S, therefore, Some S is not P. In the technical language explained in the following chapter, this is AAO in figure 4. So far, therefore, as the first three figures are concerned, we are left with a single rule, namely, a rule of distribution, which may be stated in either of the alternative forms given above.