[431] It is possible, however, that some conclusion may be obtainable. See section [359].
(3) One premiss and the contradictory of the other premiss of a valid syllogism will not in any case suffice to establish the contradictory of the original conclusion.[432]
[432] It does not follow that one premiss and the contradictory of the other premiss of a valid syllogism will never yield any conclusion at all. See the following [section].
396 This follows at once from the first of the theorems established in this section. Let the premisses of a valid syllogism be P and Q, and the conclusion R, P and the contradictory of Q will not prove the contradictory of R ; for if they did, it would follow that P and R would prove Q ; but this has been shewn not to be the case.
We have now established by strictly formal reasoning Aristotle’s dictum that although it is not possible syllogistically to get a false conclusion from true premisses, it is quite possible to get a true conclusion from false premisses.[433] In other words, the falsity of one or both of the premisses does not establish the falsity of the conclusion of a syllogism. The second of the above theorems deals with the case in which both the premisses are false; the third with that in which one only of the premisses is false.
[433] Hamilton (Logic, I. p. 450) considers the doctrine “that if the conclusion of a syllogism be true, the premisses may be either true or false, but that if the conclusion be false, one or both of the premisses must be false” to be extralogical, if it is not absolutely erroneous. He is clearly wrong, since the doctrine in question admits of a purely formal proof.
344. Arguments from the truth of one premiss and the falsity of the other premiss in a valid syllogism, or from the falsity of one premiss to the truth of the conclusion, or from the truth of one premiss to the falsity of the conclusion.—In this section we shall consider three problems, mutually involved in one another, which are in a manner related to the theorems contained in the preceding section. It has, for example, been shewn that one premiss and the contradictory of the other premiss will not in any case suffice to establish the contradictory of the original conclusion; the object of the first of the following problems is to enquire in what cases they can establish any conclusion at all.
(i) To find a pair of valid syllogisms having a common premiss, such that the remaining premiss of the one contradicts the remaining premiss of the other.[434]
[434] This problem was suggested by the following question of Mr O’Sullivan’s, which puts the same problem in another form: Given that one premiss of a valid syllogism is false and the other true, determine generally in what cases a conclusion can be drawn from these data.
397 We have to find cases in which P and Q, P and Qʹ (the contradictory of Q) are the premisses of two valid syllogisms. In working out this problem and the problems that follow, it must be remembered that if two propositions are contradictories, they will differ in quality, and also in the distribution of their terms, so that any term distributed in either of them is undistributed in the other and vice versâ. We may, therefore, assume that Q is affirmative and Qʹ negative. Let P contain the terms X and Y, while Q and Qʹ contain the terms Y and Z, so that Y is the middle term, and X and Z the extreme terms, of each syllogism.
Since Qʹ is negative, P must be affirmative; and since Y must be undistributed either in Q or in Qʹ, it must be distributed in P.
Hence P = YaX.
Qʹ must distribute Z: for the conclusion (being negative) must distribute one term, and X is undistributed in P. It follows that Z is undistributed in Q.
Hence Q = YaZ or YiZ or ZiY ;
Qʹ = YoZ or YeZ or ZeY.
If the different possible combinations are worked out, it will be found that the following are the syllogisms satisfying the condition that if one premiss (that in black type) is retained, while the other is replaced by its contradictory, a conclusion is still obtainable:—
In figure 1: AII;
In figure 3: AAI, AAI, IAI, AII, EAO, OAO;
In figure 4: IAI, EAO.