[427] This result may be regarded as affording an additional argument in favour of the adoption of supposition (4).
343. Connexion between the truth and falsity of premisses and conclusion in a valid syllogism.—By saying that a syllogism is valid we mean that the truth of its conclusion follows from the truth of its premisses; and it is an immediate inference from this that if the conclusion is false one or both of the premisses must be false. The converse does not, however, hold good in either case. The truth of the premisses does not follow from the truth of the conclusion; nor does the falsity of the conclusion follow from the falsity of either or both of the premisses.
The above statements would probably be accepted as self-evident; still it is more satisfactory to give a formal proof of them, and such a proof is afforded by means of the three following theorems.[428]
[428] It is assumed throughout this section that our schedule of propositions does not include U. The theorems hold good, however, for the sixfold schedule, including Y and η, as well as for the ordinary fourfold schedule.
(1) Given a valid syllogism, then in no case will the combination of either premiss with the conclusion establish the other premiss.
We have to shew that if one premiss and the conclusion of a valid syllogism are taken as a new pair of premisses they do not in any case suffice to establish the other premiss.
Were it possible for them to do so, then the premiss given true would have to be affirmative, for if it were negative, the original conclusion would be negative, and combining these we should have two negative premisses which could yield no conclusion.
Also, the middle term would have to be distributed in the premiss given true. This is clear if it is not distributed in the other premiss; and since the other premiss is the conclusion of the new syllogism, if it is distributed there, it must also be distributed in the premiss given true or we should have an illicit process in the new syllogism. 395
Therefore, the premiss given true, being affirmative and distributing the middle term, cannot distribute the other term which it contains.[429] Neither therefore can this term be distributed in the original conclusion. But this is the term which will be the middle term of the new syllogism, and we shall consequently have undistributed middle.
Hence the truth of one premiss and the conclusion of a valid syllogism does not establish the truth of the other premiss; and à fortiori the truth of the conclusion cannot by itself establish the truth of both the premisses.[430]
[429] This statement, though not holding good for U, holds good for Y as well as A.
[430] Other methods of solution more or less distinct from the above might be given. A somewhat similar problem is discussed by Solly, Syllabus of Logic, pp. 123 to 126, 132 to 136. We have shewn that one premiss and the conclusion of a valid syllogism will never suffice to prove the other premiss, but it of course does not follow that they will never yield any conclusion at all; for a consideration of this question, see the following [section].
(2) The contradictories of the premisses of a valid syllogism will not in any case suffice to establish the contradictory of the original conclusion.
The premisses of the original syllogism must be either (α) both affirmative, or (β) one affirmative and one negative.
In case (α), the contradictories of the original premisses will both be negative; and from two negatives nothing follows.
In case (β), the contradictories of the original premisses will be one negative and one affirmative; and if this combination yields any conclusion, it will be negative. But the original conclusion must also be negative, and therefore its contradictory will be affirmative.
In neither case then can we establish the contradictory of the original conclusion.[431]