289 (4) No term may be distributed in the conclusion which was not distributed in one of the premisses.
The breach of this rule is called illicit process of the major, or illicit process of the minor, as the case may be; or, more briefly, illicit major or illicit minor.
(5) From two negative premisses nothing can be inferred.
This rule may, like rule 3, be very well illustrated by means of the Eulerian diagrams.
(6) If one premiss is negative, the conclusion must be negative; and to prove a negative conclusion, one of the premisses must be negative.[309]
[309] This rule and the second corollary given in the following section are sometimes combined into the one rule, Conclusio sequitur partem deteriorem ; i.e., the conclusion follows the worse or weaker premiss both in quality and in quantity, a negative being considered weaker than an affirmative and a particular than a universal.
200. Corollaries from the Rules of the Syllogism.—From the rules given in the preceding section, three corollaries may be deduced:—[310]
[310] The formulation of these corollaries may in some cases help towards the more immediate detection of unsound syllogisms.
(i) From two particular premisses nothing can be inferred.
Two particular premisses must be either
(α) both negative,
or (β) both affirmative,
or (γ) one negative and one affirmative.
But in case (α), no conclusion follows by rule 5.
In case (β), since no term can be distributed in two particular affirmative propositions, the middle term cannot be distributed, and therefore by rule 3 no conclusion follows.
In case (γ), if any valid conclusion is possible, it must be negative (rule 6). The major term, therefore, will be distributed in the conclusion; and hence we must have two terms distributed in the premisses, namely, the middle and the major (rules 3, 4). But a particular negative proposition and a particular affirmative proposition between them distribute only one term. Therefore, no conclusion can be obtained.
(ii) If one premiss is particular, the conclusion must be particular.
290 We must have either
(α) two negative premisses, but this case is rejected by rule 5;
or (β) two affirmative premisses;
or (γ) one affirmative and one negative.
In case (β) the premisses, being both affirmative and one of them particular, can distribute but one term between them. This must be the middle term by rule 3. The minor term is, therefore, undistributed in the premisses, and the conclusion must be particular by rule 4.
In case (γ) the premisses will between them distribute two and only two terms. These must be the middle by rule 3, and the major by rule 4 (since we have a negative premiss, necessitating by rule 6 a negative conclusion, and therefore the distribution of the major term in the conclusion). Again, therefore, the minor cannot be distributed in the premisses, and the conclusion must be particular by rule 4.