(ii) To find a pair of valid syllogisms having a common conclusion, such that a premiss in the one contradicts a premiss in the other.
Let Q and Qʹ (which we may assume to be respectively affirmative and negative) be the premisses in question, and Pʹ the conclusion; also let Q and Qʹ contain the terms Y and Z, while Pʹ contains the terms X and Z, so that Z is the middle term, and X and Y the extreme terms, of each syllogism.
It follows immediately that Pʹ is negative; also that Y 398 must be undistributed in Pʹ, since it is necessarily undistributed either in Q or in Qʹ.
Hence Pʹ = YoX.
Since X is distributed in Pʹ it must also be distributed in the premiss which is combined with Qʹ ; and as this premiss must be affirmative, it cannot also distribute Z, which must therefore be distributed in Qʹ (and 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 the same conclusion is obtainable from another pair of premisses, of which one contradicts one of the original premisses (namely, that in black type):—
In figure 1: EAO, EIO;
In figure 2: EAO, AEO, EIO, AOO;
In figure 3: EIO;
In figure 4: AEO, EIO.
(iii) To find a pair of valid syllogisms having a common premiss, such that the conclusion of one contradicts the conclusion of the other.[435]
[435] This problem was suggested by the following question of Mr Panton’s, which puts the same problem in another form: If the conclusion be substituted for a premiss in a valid mood, investigate the conditions which must be fulfilled in order that the new premisses should be legitimate.
Let P be the common premiss, Q and Qʹ (respectively affirmative and negative) the contradictory conclusions; also let P contain the terms X and Y, while Q and Qʹ contain the terms Y and Z, so that X is the middle term, and Y and Z the extreme terms, of each syllogism.
Since Q is affirmative, P must be affirmative; and since either Q or Qʹ will distribute Y, P must distribute Y.
Hence P = YaX.
The premiss which, combined with P, proves Q must be affirmative and must distribute X ; it cannot therefore distribute Z, and Z must accordingly be undistributed in Q (and distributed in Qʹ). 399
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 the contradictory of the conclusion is obtainable, although one of the premisses (that in black type) is retained:—
In figure 1: AAA, AAI, EAE, EAO;
In figure 2: EAE, EAO, AEE;
In figure 4: AAI, AEE.[436]
[436] It will be observed that each of the above problems yields nine cases. Between them they cover all the 24 valid moods; but there are three moods (namely, EAO in figures 1 and 2 and AAI in figure 3) which occur twice over. The 15 unstrengthened and unweakened moods are equally distributed, namely, the four yielding I conclusions (together with OAO) falling under (i); the six yielding O conclusions (except OAO) under (ii); the five yielding A or E conclusions under (iii). All the moods of figure 1 (except those with an I premiss) fall under (iii); all the moods of figure 2 (except those with an E conclusion) under (ii); all the moods of figure 3 (except the one not having an A premiss) under (i).
The three sets of moods worked out above are mutually derivable from one another. Thus,
| (i) | (ii) | (iii) | ||
| P and Q ∴ R | = | Q and Rʹ ∴ Pʹ | = | Rʹ and P ∴ Qʹ |
| P and Qʹ ∴ Tʹ | = | Qʹ and T ∴ Pʹ | = | T and P ∴ Q |
In this table (i) represents the possible cases in which, one premiss being retained, the other premiss may be replaced by its contradictory. We can then deduce (ii) the cases in which, the conclusion being retained, one premiss may be replaced by its contradictory; and (iii) the cases in which, one premiss being retained, the conclusion may be replaced by its contradictory. We might of course equally well start from (ii) or from (iii), and thence deduce the two others.
Comparing the first syllogism of (i) with the second syllogism of (iii) and vice versâ, we see further that (i) gives the cases in which, one premiss being retained, the conclusion may be replaced by the other premiss; and that (iii) gives the cases in which, one premiss being retained, the other premiss may be replaced by the conclusion.